Mass Deposition Rates of Carbon Dioxide onto a Cryogenically Cooled Surface

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript investigates the mass deposition rates of carbon dioxide onto cryogenically cooled surfaces through a combined experimental and modeling approach. It thoroughly explores the effects of heat transfer, gas composition, and operating conditions on the desublimation process. The topic is highly relevant to cryogenic CO2 capture technologies, fits well with the journal's scope, and the study is comprehensive. Based on its merits, I am inclined to recommend acceptance of this manuscript for publication after minor revisions, addressing the following points:

1. Introduction: The introduction provides a good background but could be strengthened. Briefly mentioning other recent studies  on cryogenic CO2 capture or frost formation to better situate the current work within the most recent research landscape. More explicitly stating the specific knowledge gap that this study aims to address, beyond the general need for detailed data on desublimation rates.

2. About the figures. The inclusion of many figures with experimental and modeling results is excellent. However, the quality and formatting should be made more consistent. Ensure all axis labels, legends, and data points are large and clear enough to be easily read. Figure 4: The photographic sequence is very useful. Please consider adding a scale bar or indicating the initial diameter of the cold plate (20 mm) directly on one of the images to provide a clear sense of scale for the frost growth.

3. Page 2, Line 46: "Several more innovative..." - This paragraph lists technologies but seems to be cut off ("Cryogenic distillation is one of the most mature technologies, but is typically..."). Please check the intended text here.

4. Page 6, Section 2.3: The data processing method for handling the radial growth is well described. Consider adding a brief sentence summarizing the main assumption (e.g., that the density is uniform across the layer) justifying the use of the volume ratio to apportion mass.

5. Page 16, Line 384: "Although the results are somewhat scattered again..." - This phrasing is somewhat informal. Consider stating "Although significant scatter is observed in the data, a clear trend is visible...".

6. Page 18, Line 438: Equation 13 uses d_p for particle diameter, but the nomenclature list uses d for diameter and d_p is not explicitly defined there. Please ensure consistency between equations and nomenclature.

7. The discussion on the significance for the cryogenic packed bed concept (Section 4.3) is valuable. Briefly elaborating on why heat transfer through the solid layer is negligible in this case (small thickness due to moving front vs. continuous build-up in a heat exchanger) would make the contrast even clearer for the reader.

Author Response

Reviewer 1: 

Comments and Suggestions for Authors 

The manuscript investigates the mass deposition rates of carbon dioxide onto cryogenically cooled surfaces through a combined experimental and modeling approach. It thoroughly explores the effects of heat transfer, gas composition, and operating conditions on the desublimation process. The topic is highly relevant to cryogenic CO2 capture technologies, fits well with the journal's scope, and the study is comprehensive. Based on its merits, I am inclined to recommend acceptance of this manuscript for publication after minor revisions, addressing the following points: 

 

Comment 1: Introduction: The introduction provides a good background but could be strengthened. Briefly mentioning other recent studies on cryogenic CO2 capture or frost formation to better situate the current work within the most recent research landscape. More explicitly stating the specific knowledge gap that this study aims to address, beyond the general need for detailed data on desublimation rates. 

Response 1: We agree with the statement of Reviewer 1. We have expanded the introduction with other recent studies on frost formation. The following sentences have been added or changed to improve the literature study of the work: 

Page 2, Line 65: “Naletov et al. already demonstrated in an experimental study that the rate of CO2 deposition is strongly limited by the heat transfer limitation induced by the formed frost layer [11]. 

Page 2, Line 88: “A study by Du et al. highlighted again the importance of heat transfer on the desublimation rate, while also showing the influence of gas flow rate and cooling temperature on the heat exchange coefficient [19].” 

Page 2, Line 91: “A follow-up study by Cai et al. investigated the desublimation dynamics at elevated pressure [20]. Results showed that at increased pressure the morphological structure of the formed solid CO2 changes from crystalline to a laminar/layered structure.” 

Page 2, Line 93: “The experimental studies on CO2 desublimation demonstrated that lower cooling temperatures drive faster and denser frost growth, causing a rapidly increasing thermal-resistance layer that progressively degrades heat-transfer performance [21].” 

Page 2, Line 96: “Wang et al. experimentally tested and modelled the CO2 desublimation on the walls of a tubular counter-current heat exchanger, showing reduction due to both heat and mass transfer [22].” 

Page 3, Line 99: “This work presents a comprehensive study on CO2 desublimation rates in precisely controlled conditions to gain better understanding of the roles of kinetics, mass transfer, and heat transfer under various CO2 (partial) pressures, surface and gas phase temperatures, and flow conditions.” 

 

Comment 2: About the figures. The inclusion of many figures with experimental and modeling results is excellent. However, the quality and formatting should be made more consistent. Ensure all axis labels, legends, and data points are large and clear enough to be easily read. Figure 4: The photographic sequence is very useful. Please consider adding a scale bar or indicating the initial diameter of the cold plate (20 mm) directly on one of the images to provide a clear sense of scale for the frost growth. 

Response 2: We agree with the statement of Reviewer 1. We have increased font sizes in Figures 6-12, also different colors were added to data points related to different experiments and the marker size of the data points has been increased. Additionally, a scale bar has been added to Figure 4. 

 
 
Comment 3: Page 2, Line 46: "Several more innovative..." - This paragraph lists technologies but seems to be cut off ("Cryogenic distillation is one of the most mature technologies, but is typically..."). Please check the intended text here. 

Response 3: We have looked and looked again over the paragraph, but it seems in order; the paragraph starting with “Several cryogenic CO2 separation processes...” aims to inform the reader about the various processes. The “Several more innovative...” sentence intends to list a few other processes that rely on desublimation of CO2, so the technologies by Song et al, Clodic and Younes and Tuinier were all referred to. 

 

Comment 4: Page 6, Section 2.3: The data processing method for handling the radial growth is well described. Consider adding a brief sentence summarizing the main assumption (e.g., that the density is uniform across the layer) justifying the use of the volume ratio to apportion mass. 

Response 4: As suggested by Reviewer 1, we have added an extra phrase at page 6, line 189: “…and to assume a constant density throughout the entire layer.”. 

 

Comment 5: Page 20, Line 416: "Although the results are somewhat scattered again..." - This phrasing is somewhat informal. Consider stating "Although significant scatter is observed in the data, a clear trend is visible...". 

Response 5: The proposed change by Reviewer 1 has been implemented. 

 

Comment 6: Page 18, Line 438: Equation 13 uses d_p for particle diameter, but the nomenclature list uses d for diameter, and d_p is not explicitly defined there. Please ensure consistency between equations and nomenclature. 

Response 6: We thank Reviewer 1 for pointing this out. We have added d_p to the nomenclature list. 

 

Comment 7: The discussion on the significance of the cryogenic packed bed concept (Section 4.3) is valuable. Briefly elaborating on why heat transfer through the solid layer is negligible in this case (small thickness due to moving front vs. continuous build-up in a heat exchanger) would make the contrast even clearer for the reader. 

Response 7: We agree with adding the clarification suggested by Reviewer 1. We have added the following sentences to Page 22, line 498: “This is mainly due to the fact that there is a limited amount of cold energy stored in one particle, effectively limiting the potential solid CO2 formed on each particle. This is very different from what happens on a continuously cooled surface, like in the case of a more traditional heat-exchanger type process.” 

Reviewer 2 Report

Comments and Suggestions for Authors

The abstract lacks the mention to any statistical or numeric data.

Are both vales 4&5 for venting?

There is no mention of the rates or composition in the procedure section. Also distinguish if the mentioned pressure gauge or a pressure transducer.

Is the frost image analysis qualitative or quantitative?

What defines an ideal mixing?

Where any repeated runs where performed?

 where the concentrations hold fixed or where their variations encountered?

In the result section it mentioned that mass deposition rate is very high, How high? Decreased by how much?

There is great lack of the mention of quantitative measurements in the result section. If there is something increasing or decreasing, please state by how much or the type of influence. Also state the reason for the nature of the influence.

What is the range of pressure and temperature to achieve equilibrium assumptions?

Clarify the transition between eq4&5.

Justify the choice of uniform frost temperature assumption.

Similarly to the previous the simulation results lack quantitative assessment. If something increases or decrease state by how much %.

“cold plate temperature had a larger effect” discuss why it is expected.

Discuss the model limitations.

Where Peclet or Biot dimensionless number calculate?

What was the Reynold number value?

Author Response

Reviewer 2: 

Comments and Suggestions for Authors 

Comment 1: The abstract lacks the mention to any statistical or numeric data. 

Response 1: We have added some numeric data to the abstract, the following sentences were added/modified: 

Page 1, line 11: ”Experiments were carried out with both pure CO2 and CO2/N2 mixtures, growing frost layers up to 8 mm thick” 

Page 1, line 22: “…, calculations showed that a frost layer of 3.24∙10-5 m is formed, resulting in a Biot number well below 0.01, indicating that heat transfer in the frost layer is not limiting.” 

 

Comment 2: Are both vales 4&5 for venting? 

Response 2: Valves 4 and 5 are the outlets of the stirred cell, they are connected to the vent as indicated in Figure 1. We have added the following sentence to page 3, line 121: “The stirred cell is also connected to two outlets (of which one with a vacuum pump),…” 

 

Comment 3: There is no mention of the rates or composition in the procedure section. Also distinguish if the mentioned pressure gauge or a pressure transducer. 

Response 3: During the experiments, the CO2 flow rate is regulated such that the desired pressure is kept. This corresponds to the CO2 consumed by desublimation in the stirred cell. So the inlet flow rates are not parameters that define an experiment. We believe that this is clearly explained in the Procedure section (2.2) and Data processing section (2.3). The gas phase composition used for each experiment is defined by the partial pressure for CO2 an N2, which are shown in Table 1. To clarify this further we have added a second column to Table 1 with the CO2 mole fraction, calculated by dividing the PCO2 (column 3) by the sum of the PCO2 (column 3) and PN2 (column 4). 

 

Comment 4: Is the frost image analysis qualitative or quantitative? 

Response 4: The frost image analysis is quantitative. We have highlighted this more by adding the following sentence to page 6, line 182: “These pictures were used to quantitatively calculate the volume of solid CO2 formed on the cold plate.” 

 

Comment 5: What defines an ideal mixing? 

Response 5: Ideal mixing is defined as a state where there is perfect uniformity of the mixture throughout the vessel. We have added the following sentence between brackets at page 5, line 159: “… (i.e. perfect uniformity of composition and temperature throughout the vessel) …”. 

 

Comment 6: Where any repeated runs where performed? 

Response 6: Some experiments were executed in duplicate to ensure the reproducibility of the setup. The following sentence has been added at page 3, line 133: “We have conducted one benchmark case in duplo to verify the reproducibility of the setup.”  

 

Comment 7: where the concentrations hold fixed or where their variations encountered? 

Response 7: The concentrations were kept constant for each experiment. 

 

Comment 8: In the result section it mentioned that mass deposition rate is very high, How high? Decreased by how much? 

Response 8: We believe that sentences that describe a graph, as it is the case for the sentence highlighted by Reviewer 2, there is no need to state specific values, as the exact values can be observed in the graph. We have further clarified the reason for the observed behavior by adding the following sentence (page 8, line 235): “This is due to the initial surface temperature of the cold plate being very low and the absence of any solid CO2 on the cold plate hindering mass and heat transfer.” 

 

Comment 9: There is great lack of the mention of quantitative measurements in the result section. If there is something increasing or decreasing, please state by how much or the type of influence. Also state the reason for the nature of the influence. 

Response 9: We appreciate the reviewer’s suggestion. However, we believe that the current presentation of the results already conveys the observed trends clearly and effectively for the reader. The qualitative descriptions are supported by figures and plots that illustrate the magnitude and direction of changes. For further context, please refer to our responses to Comments 8 and 13, where we address related points on clarity and data representation. 

 

Comment 10: What is the range of pressure and temperature to achieve equilibrium assumptions? 

Response 10: During the experiments, the flow controller ensured that the CO2 that desublimates from the gas phase, is replaced by new CO2, and therefore that the composition, partial pressure and total pressure do not change. The temperature is kept constant by pre-cooling the feed gas and continuously cooling the vessel. For the model, temperature, pressure and composition of the bulk phase are assumed to be constant as well.  

 

Comment 11: Clarify the transition between eq4&5. 

Response 11: To clarify this point we have added the following sentence at page 16, line 362: “the effective mass transfer coefficient can then be calculated by substituting equation 4 into the expression for the effective mass transfer coefficient in equation 3, obtaining:” 

 

Comment 12: Justify the choice of uniform frost temperature assumption. 

Response 12: The frost temperature is not uniform, there exists a temperature gradient inside the frost layer going from the gas-solid interface to the cold plate interface. 

 

Comment 13: Similarly to the previous the simulation results lack quantitative assessment. If something increases or decrease state by how much %. 

Response 13: We believe that observations and the corresponding analysis have been described sufficiently, as for all observations, a references is made to graphs that contain the quantitative information. We believe that repeating quantitative information in the text will clutter the analysis of the results. 

 

Comment 14: “cold plate temperature had a larger effect” discuss why it is expected. 

Response 14: We have clarified this point further by adding the following sentence to page 20, line 441: “…as the cold plate temperature directly affects the driving force for the heat flux, …” 

 

Comment 15: Discuss the model limitations. 

Response 15: A paragraph discussing the limitations of the model has been added at page 20, line 419: “Although the developed frost growth model captures the main trends observed experimentally, several simplifications introduce limitations. The model was derived based on the central region of the frost layer, so wall and edge effects are not considered. Additionally, it assumes uniform density and thermal conductivity throughout the frost layer, whereas real frost can be porous and heterogeneous when it is formed [refs]. This simplification may affect accuracy, particularly during the initial stage of frost formation on an empty surface, when significant morphological changes are expected. However, as the process progresses and the frost layer thickens, these effects are likely to diminish, meaning that for modeling the entire cycle, the impact of this assumption is expected to be minimal.” 

 

Comment 16: Where Peclet or Biot dimensionless number calculate? 

Response 16: We agree with Reviewer 2 that the Biot number can give an extra indication of the importance of conduction and convection heat transport contributions. Therefore, we have added the following sentence at page 22, line 496: “This also becomes visible when calculating the Biot number, giving values well below 0.01, indicating that the conduction of heat through the solid CO2 layer is significant faster than external heat transfer.” 

 

Comment 17: What was the Reynold number value? 

Response 17: The flow behavior of the system depends completely on the rotation speed of the stirrer. The rotation speed of the stirrer was chosen such that (near) ideal mixing behavior was achieved, this was verified by comparing experimentally observed results to the behavior in case of a ideally stirred tank reactor. Just to clarify it even more, we have also added an indication of the Reynolds number at page 5 line 160. 

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

Despite the reviwer accepting the current form of the manuscript, The reviwer encourages the authors to focus on the quantitative aspect as it provide clear indication of the process. Detailed quantitative analysis would enhance the manuscript rather than cluttering it.

Author Response

Comment: Despite the reviewer accepting the current form of the manuscript, The reviewer encourages the authors to focus on the quantitative aspect as it provide clear indication of the process. Detailed quantitative analysis would enhance the manuscript rather than cluttering it. 

We appreciate the reviewer’s thoughtful suggestion regarding the inclusion of additional quantitative results. While we believe that the manuscript is already clear, particularly because many of the conclusions are qualitative in nature, we understand the value that further quantitative detail can offer. To address the reviewer’s recommendation, we have expanded the discussion and conclusion sections by incorporating additional quantitative results relevant to the cryogenic packed bed. The following paragraph has been added and amended on page 23, line 553: “The results of the model showed that the frost layer reaches a stable and maximum thickness of 3.24∙10-5 m in less than 5 seconds, with a particle temperature of -105.5 °C. The value of the Biot number (<0.01) for the frost layer was very low, indicating that heat transfer in the frost layer was negligible. Also internal heat transfer in the particles is negligible, because the Fourier number reaches a value of 0.4 after at 2.2 seconds. So, it was shown that under packed bed conditions, internal heat transfer does not play a significant role, neither in the solid frost layer (Biotlayer<0.01), nor in the solid particle (Fourierparticle<1, at 2.2 s),…”. 

In addition we have made a further analysis on the accuracy of the prediction of the model, we have added the following sentence to the manuscript at page 20, line 435: “the Mean Absolute Error (MAE) of the predicted mass deposition rate was calculated to be 6.459∙10-4”. We have also added the following sentence at page 23, line 549: “The model predicted the mass deposition rate of CO2 with an MAE of 6.459∙10-4.”. 

We would like to thank the reviewer for the invested time and effort to improve the quality of the manuscript. 

Kind regards, 

Authors