the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 15 Jul 2024
Laboratory heat transport experiments reveal grain size and flow velocity dependent local thermal non-equilibrium effects
Abstract. Heat transport in porous media is crucial for gaining earth science process understanding and engineering applications such as geothermal system design. While heat transport models are commonly simplified by assuming local thermal equilibrium (LTE, solid and fluid phases are averaged), local thermal non-equilibrium (LTNE, solid and fluid phases are considered separately) heat transport has long been hypothesized and reports have emerged. However, experiments with realistic grain sizes and flow conditions are still lacking in the literature. To detect LTNE effects, we conducted comprehensive laboratory heat transport experiments at Darcy velocities ranging from 3 to 23 m d−1 and measured the temperature of fluid and solid phases separately for glass spheres with diameters of 5, 10, 15, 20, 25 and 30 mm. Four replicas of each size were embedded at discrete distances along the flow path in small glass beads to stabilize the flow field. Our sensors were meticulously calibrated and measurements were post-processed to reveal LTNE, expressed as the difference between solid and fluid temperature during the passing of a thermal step input. To gain insight into the heat transport properties and processes, we simulated our experimental results in 1D using commonly accepted analytical solutions for LTE and a numerical solution of LTNE equations. Our results demonstrate significant LTNE effects with increasing grain size and water flow velocity. Surprisingly, some temperature differences were negative indicating that the heat front propagates non-uniformly likely caused by spatial variations of the flow field. The fluid temperature modeled by the LTE analytical solution exhibited relatively good agreement with experimental fluid temperature only for grain sizes from 5 mm to 15 mm. However, for larger grain sizes (between 20 mm and 30 mm), the temperature difference between fluid and solid phases became too significant to be represented by an LTE model. Additionally, for larger grain sizes (≥ 20 mm), the LTNE model failed to predict the magnitude of LTNE for all tested flow velocities due to experimental conditions being inadequately represented by the 1D model with ideal step input. Future studies should employ more sophisticated numerical models to examine the heat transport processes and accurately analyze LTNE effects, considering non-uniform flow effects and multi-dimensional solution. This is essential to determine the validity limits of LTE conditions for heat transport in natural systems such as gravel aquifers with grain sizes larger than 20 mm.
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CC1: 'Comment on egusphere-2024-1949', Giacomo Medici, 06 Aug 2024
General comments
Good theoretical research that can be improved addressing the specific comments below.
Specific comments
Line 23 “Accurately describing heat transport in porous media has long been a focus in both engineering and science”. Insert recent review papers on heat transport in geological media since the sentence is not backed-up by references.
- Review of discrete fracture network characterization for geothermal energy extraction. Frontiers in Earth Science, 11, p.1328397
- Review of geothermal energy resources, development, and applications in China: Current status and prospects. Energy, 93, pp.466-483
Line 27. Specify low enthalpy geothermal systems?
Line 78. The aim is clear, but please specify the 3 to 4 objectives of your research by using numbers (e.g., i, ii and iii).
Line 153. “Porous media” provide more detail on the material that you used to create the porous material that approximate the geological media in your analogue.
Lines 308-424. Provide more detail on the validity of your analogue experiment that is at small scale. Much smaller than the aquifer. This point can be addressed here or in the introduction.
Lines 308-424. Provide more detail on the validity of your analogue research taking into account that porous aquifers (typically siliciclastic) are very heterogeneous. This point can be done here or in the introduction.
Figures
Figure 1a. Insert the spatial scale.
Figure 1d. Insert the spatial scale.
Figures 3, 5 and 8. Room to make the figures larger.
Citation: https://doi.org/10.5194/egusphere-2024-1949-CC1 - AC1: 'Reply on CC1', Haegyeong Lee, 27 Aug 2024
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RC1: 'Comment on egusphere-2024-1949', quanrong wang, 07 Aug 2024
To investigate the presence of local thermal non-equilibrium (LTNE) effects during heat flow in porous media, in the present work, laboratory experiments systematically investigated heat transport by exposing water flow to temperature step inputs. The experiments were conducted at Darcy velocities ranging from 3 to 23 m/d through porous media comprising idealized spherical grains with diameters between 5 and 30 mm. This work is new and interesting, I think this paper can do well in the future. I have a few specific comments that I will list below in order to improve the current manuscript.
Lines 68-74: Yes, there are very few experimental studies on LTNE in porous media, and in addition to the two studies mentioned by the authors, laboratory experimental studies have been conducted in the study of Shi et al. (2024) [DOI:10.1029/2024WR037382], and it is suggested that the authors refer to their experimental studies section for an introduction as well.
Line 163: “thermal conductivity” -> “thermal conductivities”
Line 165: “specific heat capacity” -> “specific heat capacities”
Line 173: “dispersion coefficient” should be changed to “thermal dispersion coefficient” as “dispersion coefficient” is a part of “thermal dispersion coefficient”.
Eq. (14): Heinze (2024) [DOI: 10.1016/j.earscirev.2024.104730] provides a detailed overview of this relationship, and I would suggest that the authors could make appropriate references to the work of Heinze (2024) here. Just a suggestion:)
Line 321: “In the study by Bandai et al. (2017), they revealed…”-> “Bandai et al. (2017) revealed…”.
Lines 412-413: Please express this sentence in two sentences.
Citation: https://doi.org/10.5194/egusphere-2024-1949-RC1 - AC2: 'Reply on RC1', Haegyeong Lee, 27 Aug 2024
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RC2: 'Comment on egusphere-2024-1949', Anonymous Referee #2, 12 Aug 2024
The manuscript presents novel heat transport/heat transfer experiments that address a long-standing research gap on the relevance of LTNE effects in aquifer systems. The manuscript is a highly valuable contribution to the research topic and provides great insights into the microscopic temperature distribution in large grain size sediments, building on previous experiments by some of the authors.
Abstract: It would be great if the abstract could include some more quantitative values and clarifications. Examples:
- "some temperature differences were negative" -> the reader doesn't know what positive/negative temperature differences mean at this point.
- "relatively good agreement", "became too significant" -> can these statements be quantified?
- I believe that the term "magnitude of LTNE" is not universally known, given the wide readership of HESS. Therefore, I propose to define the term if it is used in the abstract.Materials and methods
* The authors state at the beginning that "specialised experimental instrumentation" was used, but it remains unclear in the rest of the text what this "specialised instrumentation" is and how the current procedure was "adapted" from Gossler et al. (2019). The lessons learned from Gossler et al. (2019) are nicely highlighted in the following paragraphs.
* Why were all samples measured in one experimental setup and stacked on top of each other? Why not test all grain sizes separately? Did the authors reverse the order of the spheres to check for a possible influence?
* When filling the layers with the small glass beads, was it necessary to keep the outer PT100 free from contact with the smaller beads or did the authors rely on a (rapid) LTE between the small glass beads and water?Analysis/Results
* What effect did the "correction" steps (pp. 225-229) have on the results? What was the difference/spread between the sensors? Should this be shown in Fig. 3? The curves in Fig. 3 are almost indistinguishable visually. In particular, which would be of interest, it is not possible to see whether the difference between Ts and Tf is consistent but shifted across the probes. Was there a systematic relationship between the spread and grain size or flow velocity? I would encourage the authors to try a different way of visualising Figure 3, perhaps adjusting the range of the x-axis or visualising differences rather than absolute values.
* The wording in the text and figures does not seem to be consistent throughout the text. For example, Figure 2 refers to "calibrated temperature", which is not used in the text. Also, "inner" and "outer" Tf measurements are not explained in the text (Fig. 3) - could this be done in Fig. 1?
* The relevance of the negative temperature pulses is difficult to assess. l. 267f states: "was observed in some pairs of (...) measurements over all (...) flow velocities". What does this mean? Some replicates showed negative temperature pulses while others showed positive pulses? Because this is associated with the smallest grain sizes: Could this be an experimental problem, as the Pt100 sensor is the same size for all grain sizes, so its placement within the glass beads is more sensitive in smaller beads? Also, l. 270: "both sides of the grain" suggests that there is a discrepancy between the Pt100 sensors to the left and right of the glass beads. This would be interesting to see rather than Fig 5a,c,e which shows pretty much two lines on top of each other. Does this grain size discrepancy only occur for the small grains or is it only of less significance for the larger glass beads? Is this what you want to show in Fig. 3?Discussion
* Can the "agreement" between numerical models and experiments be quantified in some way to quantitatively demonstrate the "better fit"?
* The authors measure the temperature in the middle of the glass beads and refer to it as "solid temperature". However, within the glass beads there will be a radial thermal gradient. The "mean solid temperature" across the glass beads might be higher than the temperature measured observed by the authors. Hence, the heat transport within the larger grains takes longer time due to the larger distance from the outside of the glass bead to the temperature sensor. At the contact surface of the grain and the water, there still might be thermal equilibrium. While the subsequent theoretical considerations in terms of the influence of flow velocity go beyond the scope of this work, a short discussion on the scale effect could enrich the Discussion section.Citation: https://doi.org/10.5194/egusphere-2024-1949-RC2 - AC3: 'Reply on RC2', Haegyeong Lee, 27 Aug 2024
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RC3: 'Comment on egusphere-2024-1949', Toshiyuki Bandai, 13 Aug 2024
The paper “Laboratory heat transport experiments reveal grain size and flow velocity dependent local thermal non-equilibrium effects” conducted laboratory experiments to clarify thermal non-equilibrium between solid and liquid phases in saturated porous media under forced convection. This study provides valuable laboratory experimental data on the effects of grain size and flow rates on thermal non-equilibrium between solid and fluid phases of saturated porous media. The authors measured not only the fluid temperature but also the temperature of solid phase by specifically designed probes, which only a few studies have reported before.
Regardless of the value of the experimental data, I have a few concerns regarding data analysis and interpretation of the experimental data. I believe the manuscript could be improved by re-analyzing the same data without additional experiments. Therefore, I recommend a major revision for potential publication of this manuscript.
Major and minor points are summarized below (numbers indicate the line numbers).
Major points
- Choice of LTNE model
While the authors used the LTNE model (Eq. 10 and Eq. 11), this LTNE model may not be applicable to the experimental data. The LTNE model with spatially uniform parameters (e.g., h_sf and lambda_s, eff) assumes that the physical property of solid phase is uniform in the spatial domain. However, in the experimental setup, glass spherical particles with varying sizes were embedded in finer glass spheres of 1 mm diameter. Although some of the parameters can be justified to be spatially uniform even in this setting (e.g., thermal conductivity of solid and porosity), this setting violates the assumption of the LTNE model. For example, the heat transfer coefficient and surface area are functions of grain sizes. The LTNE model with spatially uniform parameters is applicable when the porous media is made up of uniform grain sizes (could be non-uniform up to the validity of representative elementary volume).
If the authors (or the editor and other reviewers) want to include the analysis with an LTNE model, I would recommend a LTNE model presented in Wakao and Kaguei, 1982, where energy equation for a solid spherical particle is coupled with energy equation for fluid phase. In this way, the authors can simulate LTNE of a spherical particle embedded in saturated porous media.
Wakao, N. and Kaguei, S. (1982): Heat and mass transfer in packed peds. Gordon and Breach Science Publishers, Inc, 364p.
- Calibration of temperature data
Regarding Line 225-229, it is more natural to normalize the measured temperature by the temperature difference between the initial and final temperature, as in Eq. 19, not by the final temperature. Doing a proper temperature calibration might provide temperature data, that is more compatible with the LTE model:
- In Figure 2, what caused the increase in the calibrated temperature at deeper depths before the arrival of the thermal front. This would not be affected by the replenishment of the water bath with tap water during the experiment. Is this affected by the laboratory air? In that case, why was the increase smaller for Figure (a), which was conducted for a longer time?
- Interpretation of inverse peaks
I am glad to see Figure 5, illustrating the difficulty of the experiments. In my unpublished data, I observed similar inverse peaks for smaller grains (dp = 3 mm). I attributed this to the placement of fluid temperature sensors. It is extremely hard to make sure the depth of solid and fluid temperature measurement is the same. Smaller the grains are, more difficult. When I failed to do this, I observed inverse peaks regardless of fluid flow rates. Non-uniform flow could also cause the inverse peaks, but I do not think we can eliminate the possibility of misplacement of fluid temperature sensors relative to the location of the solid temperature sensors in this experimental setup.
Minor points
41: “at the same temperature at their interface”: I believe this is true for LTNE approaches. The LTE approach assumes the temperature of the phases are the same within an REV, not just their interface.
99: Could you provide the information on the glue used (e.g., the name of the product)? Also, what is the volumetric heat capacity of the glue?
100: How did you place temperature sensors next to the surface of the glass spheres? Accurately placing sensors for fluid temperatures is important to avoid the “inverse peaks”.
111: Could you describe how you achieved water saturation of the porous media? Also, did you use any thermal insulation for the column? Minimizing air in porous media and heat loss from the column is essential to get experimental data that is compatible with the LTE model.
131: 26-34: “C” is missing.
Equation 1: “x is spatial coordinate” is missing.
Equation 6: This analytical solution is for normalized temperature, not actual temperature T.
183: The effective thermal conductivity of fluid includes the effect of thermal dispersion under advection. Bandai et al., 2023 was not accurate for this description. Line 206 is accurate.
200: 1.5 m (= L)?
Figure 2: I would suggest using the same color for the temperatures measured at the same depths for both phases.
Figure 8: There is a discrepancy between the models and the data at the end of the thermal front (in addition to the end of the breakthrough curve) for q = 17.2 m d-1. What caused this discrepancy? Heat loss from the column can be one reason, but if this was the case, the discrepancy would have been larger for dp = 25 mm, which was located at a deeper depth. Or, the location of the sensors for dp = 5 mm was not far enough from the input? But, if this was the case, the discrepancy would have been larger for q = 22.8 m d-1. Another reason may be the artifact of the temperature calibration procedure.
344: “limited to 0.04 K”: This is not accurate. This value is maximum normalized temperature difference in Figure 8 in Bandai et al., 2023, which can be converted to approximately 0.6 K because the temperature difference was about 15 K.
Line 379: Could you describe how you fitted the models to the experimental data to estimate the heat transfer parameters? It would be better to define a minimization problem to be solved and specify optimization algorithms used to solve it.
Citation: https://doi.org/10.5194/egusphere-2024-1949-RC3 -
AC4: 'Reply on RC3', Haegyeong Lee, 27 Aug 2024
Dear Toshiyuki Bandai,
Please find attached our detailed replies.
Best regards,
Haegyeong Lee
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RC4: 'Reply on AC4', Toshiyuki Bandai, 27 Aug 2024
Dear Haegyeong,
Thank you for your response. Just to clarify my last comment, I meant that it would be good if the manuscript could describe how (not "how well") you estimated the parameters. What kind of optimization method is used, and the definition of objective function etc.
Citation: https://doi.org/10.5194/egusphere-2024-1949-RC4 -
AC5: 'Reply on RC4', Haegyeong Lee, 03 Sep 2024
Dear Toshiyuki Bandai,
Thank you for your reply.
We will include an explanation of how we estimated these parameters in our revised manuscript.
Best regards,
Haegyeong Lee
Citation: https://doi.org/10.5194/egusphere-2024-1949-AC5
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AC5: 'Reply on RC4', Haegyeong Lee, 03 Sep 2024
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RC4: 'Reply on AC4', Toshiyuki Bandai, 27 Aug 2024
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