Experimental and Numerical Validation of the One-Process Modeling Approach for the Hydration of K₂CO₃ Particles

Round 1

Reviewer 1 Report

In this work an experimental and numerical study of the hydration of potassium carbonate particles to potassium carbonate sesquihydrate is proposed. A bunch of thermal analyses results, tests, and mathematical fittings of curves is shown and commented. Despite the amount of work carried out, it is the feeling of this reviewer that the work is not convincing, not structured, providing small new information, missing data, conclusions not clearly supported by results. Briefly, thermal data are not explicitly shown, speculation about water diffusion in a single-layer are tricky compared to the measurement protocol, kinetic study is based on just two arbitrary points, potentially not representative, some assumptions are not correctly formulated, some results are redundant. Therefore, this reviewer is forced to reject the paper. Some specific motivations are given below.

 

1. In the Introduction the authors explain the concepts of heat battery and adsorption/absorption. The connection between the concepts is not clear because, for example, expert and non-expert readers wonder how it is technologically possible “Keeping the sorbent and sorbate separate allows thermal energy to be stored for an indefinite amount of time” (page 2, lines 39-40) being this process spontaneous, whereas it is not clarified if the target reaction is an absorption with/without a chemical reaction.

 

2. In vision of potential application of the results to a heat battery reactor, the claimed “insight in how to describe the hydration reaction of K2CO3 as thermal energy storage material on particle level” (page 2, lines 78-79), is tricky because from subsection 3.1 and 4.1 it is clear that the result is specific for a “single layer” of particles and of a specific size, since a similar previous study ref [16] “concluded that for their case the hydration limitation is water diffusion to the particle surface” (page 10, lines 335-336). Of course, this is obvious, because the conditions, morphology, size, are completely different, showing that morphology, sample distribution (inside the crucible, hence, inside the heat battery reactor for scaled-up systems) do matter, and influence the dynamic, but this is not systematically investigated.

 

3. Even if a specific section is dedicated to this, the sample preparation and assumptions made over particle morphology are debatable. It is not clear if the mechanical pulverization is hand-made or not, i.e., reproducible or not, as well as the sieving operation as the Authors state at page 7 lines 256-257; SEM images clearly show that the spherical assumption does not apply, the aggregates may have internal non-negligible porosity, hence all the calculations and speculations about nucleation, diffusion, etc., are weak. Figure 5c also does not help in supporting the claim because it is evident that the distribution in the crucible is not neat and homogeneous. One should hope that it is just a bad photo. These aspects are in contrast with the discussion about the single-layer and the water diffusion limitation, and their characterization appears incomplete.

 

4. Another unclear point in the experimental procedure is the intended use of gas flow rate because it is not clear to which water partial pressures the three flow rates correspond to (the term vapour pressure is referred correctly to water liquid/vapour at equilibrium, not to a gas concentration expressed as pressure, please use carefully these concepts this in the text), if the total flow rate to the equipment is constant or not, and if the sweeping gas is dry air or else. In fact, to reviewer’s knowledge, the used equipment has a vertical furnace design with an upward flow for gas, usually operated at 60-100 mL/min, far lower than those used by Authors, wondering if the turbulence could affect the water vapour diffusion in case. However, the amount of sample used is also debatable, because one can say that the amount used is not enough to have a reliable STA signal, especially if the stated amounts (1.5, 2, and 4 mg) are weights considered from the batch, because they could contain moisture or readily adsorb it during preparation, hence the weight of dry K2CO3 after the pre-dehydration step (Figure 3) could be lower, affecting the measurement. It is true that Aluminium crucibles help in detecting signals from STA, but considering the detection limit, and the packed bed effect, a study of the hydration time/mg of sample, and varying the flow rate or not, could have helped. Hydration time as function of weight is not enough. Further, the f(α) test (page 13, line 396) is conducted at another different weight, 1.3 mg. Is it a typos? How about the single-layer effect, the number of particle and all the speculation about? Can be considered coherent with the rest of the study?

 

5. The paper is full of explanations and equations about kinetic, factors, rates, functions, variables, etc., that, although of interest, appear too didactic, lengthy, exposed in a way that creates confusion rather than clarity, and should be condensed and reduced to the essential. RDS, nucleation, growth, n-steps, are mentioned, but not a clear mechanism or reaction equations are reported to understand the process and, more importantly, the thermodynamic equilibrium of the system is not mentioned. The whole investigation is based on just one temperature (40°C) and two water partial pressures (10 and 12 mbar), hence two “kinetic” (and arbitrary) points, if we exclude the points at 120°C and 40°C and 0.2 mbar, that cannot represent the generality of a reactive system.

 

6. The data are also confusing and a bit redundant. The work is entirely based on STA, but no real thermal data are shown, just the post elaboration for the “kinetic” study. Instead of weight losses and heat fluxes, space is given to secondary data like Figure 3, Figure 7, and Figure 9 (for this a table or even a value in the main text would have been enough). Moreover, the info in Figure 9 could have been visualized in Figure 8, resulting in a figure identical to Figure 10, because they are the same experiment! At page 13 line 406 “This means that the 2 mbar jump in vapor pressure can be applied at arbitrary α” i.e. exactly what was done in the previous test and reported in Figure 9. It is difficult to see any significant difference. Moreover, it is not reported what happens below 0.37, so probably the “jump” couldn’t be applied arbitrarily.

 

7. This protocol of “jump” is non-sense, not pertinent to a “kinetic” study but more to a perturbation study from an equilibrium condition. In fact, considering the thermodynamic plot, like for example in ref [16] among others, the point at 40°C and 10 mbar is clearly in a region where reaction (1) is completely shifted to the right, and sesquihydrate form is obtained. Considering Figure 3, when water partial pressure is increased from 0.2 to 10 mbar at constant temperature of 40°C, that is the real reaction, once steady state is reached (10 mbar is more than the equilibrium value). Further increasing water partial pressure from 10 to 12 mbar, at steady state, should be useless, please check. Changing water partial pressure “rapidly” (how rapid this is obtained is also another questionable point) from 10 to 12 mbar while the system is still evolving, apparently does not have much sense, it just speeds up a bit the reaching of the final state.

 

8. It is not clear the modelling, e.g., the analytical form of the equation used. In equation (2), the Arrhenius term k(T) is not studied (temperature is not changed), the pressure term, according to the values provided, is about 0.8 or 1, negligible or at least it is not clear how is considered in the calculations, the term f(α) is not analytically determined, or from equation (7) on it is not clear where it is gone, the equations are confusing, some of them began implicit, then in integral form. In subsection 4.4.1, numerical fitting of equation (8), with a non-analytical term, the “areic growth rate” probably a constant, is performed, resulting in an obvious good fit because it refers just on a single condition of T and P, with a free parameter to “optimize” the fit. Again, in subsection 4.4.2, a fitting of the same data is performed, but it is not clear the complete equation used because equation (14) and (15) are mentioned but the connection is unclear, the parameter of nucleation rate is not explicit in the equations, a free parameter, i.e. diffusion, is left arbitrary to “optimize” the fit. For this parameter, with instantaneous nucleation, in the caption of Figure 13, a value is provided, whereas on page 17 line 458 another value is written, whereas at line 459 a constitutive equation, probably the link between the models, is abruptly reported, and stated to give the same value for the two fittings, but values to use are unclear and readers cannot quickly check. Briefly, this is confusion, very difficult to accept, not inviting and not convincing the readers.

Author Response

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Reviewer 2 Report

The paper presents the experimental and numerical analysis of K2CO3 hydration reaction from kinetics point of view. The paper is well organised and written and the results are scientifically sound. In order for the paper to be published, the following comments should be addressed:

• the authors have proved that the hydration rate is basically a function of α. Accordingly, for the applications foreseen (thermal energy storage), this means it is a function of discharge operating conditions. Please discuss, even if only qualitatively, the conditions under which the discharge rage can be maximised or at least improved.
• it is true that the method proposed by the authors has not been applied to K2CO3 yet, but give some information on how the results obtained relate to the other salt hydrates cases analysed in the literature.
• How would cycling affect the dynamics of the material reaction? Please comment on this.

Author Response

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Reviewer 3 Report

This manuscript by Beving et al describes the numerical simulation of the kinetics of hydration of potassium carbonate in the viewpoint of its application to heat batteries. The authors assumed a generalized Arrhenius-type equation and tested several models to reproduce the observed results, especially including the “jump experiment”. The topic of heat storage materials is getting increasing attention and this study deserves a genuine contribution to the progress of this area. I would recommend this manuscript for publication in the present form.

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

This reviewer is positively impressed by the explanations and comments from Authors. Although it is still difficult to understand the protocols used in this field, the reviewer understands that this was rigorous and reproducible, in line with the recent trends in the related literature, so the work is of valuable novelty. The revisions in the manuscript have addressed the comments and improved the clarity and readability of the work.

This reviewer would also like to propose the criticism as a suggestion: it would be interesting for future works to do a similar study at different temperatures to determine k(T), and at other values of water partial pressures (close to equilibrium and below equilibrium), to have more info about the kinetic as well as the hydration behavior of future heat batteries operated not at ideal conditions.