Numerical Optimization of Multi-Stage Thermoelectric Cooling Systems Using Bi₂Te₃ for Enhanced Cryosurgical Applications

Comment 1:

The concept of cascading Peltier modules to reach cryogenic temperatures is well established (e.g., Aliabadi et al. 2014). While the detailed COMSOL study adds quantitative depth, the manuscript does not clearly delineate how it advances beyond existing numerical optimizations or delivers new physical insight.

       Response:

Thank you for your comments. In response, we have added the following text to the revised Materials and Methods (Section 2.1): This study goes beyond earlier research like Aliabadi et al. [23], which mainly looked at whether cascaded Peltier modules could work, by using detailed simulations to explore how they perform step by step when exposed to cold surgical temperatures. This enables us to identify optimal operating currents, geometry, and material interfaces through a more rigorous and spatially resolved 3D model that captures the intricate coupling of electrical and thermal transport under cryogenic constraints.

Comment 2:

Section 2.1 describes iterative sweeps of current and geometry, but mesh‐convergence tests, solver settings, and tolerances are not reported. Without these details, reproducibility and numerical accuracy remain uncertain. Please include a subsection on mesh refinement studies (element size vs. ΔT error), solver convergence criteria, and parameter‐sweep resolution.

      Response:

Thank you for your valuable comments. In response to your feedback, we have included the following text in the revised manuscripts of the Materials and Methods (Section 2.1): To ensure reliable simulation outcomes, a study on mesh refinement and solver configuration was conducted. The finite element mesh was refined until temperature variations (ΔT) across critical TE regions stabilized, confirming mesh independence. An efficient mesh was adopted that balanced accuracy and computational cost. COMSOL’s stationary solver was configured with a direct solver, and tolerances were adjusted for stable convergence in the multiphysics environment. A parametric sweep of input current and TE geometry was performed to capture key performance variations. This strategy enabled accurate modeling of coupled thermal and electrical behavior under cryogenic conditions.

Comment 3:

Figures 4–6 provide performance curves, but the underlying physics (e.g., Joule‐heating tradeoffs, Thomson effect omission) are only briefly mentioned. Please include a discussion to interpret how material properties (Seebeck coefficient temperature‐dependence, thermal conductivity variation) specifically shape the performance limits, and discuss practical implications for device packaging and heat‐sink design.

       Response:

Thank you for your valuable suggestion regarding the need for a more in-depth discussion of the underlying physics and practical implications. In response, we have expanded Sections 3.1.2, 3.1.3, and 3.1.4 of the Results to provide a more detailed analysis of how material properties affect performance limits and their real-world relevance.

Here are minor comments:

Comment 1:

Table 1: P-type Bi₂Te₃ properties are listed twice with identical labels “S (T)” etc. Clarify distinction or remove redundancy.

Response:

[1] Thank you for your observation on the material configuration. The duplicate entry of p-type Bi₂Te₃ in Table 1 was a typographical error and has been corrected. In response, we have added the following text to Section 2.3 of the revised manuscript: Bi₂Te₃ is categorized into p-type and n-type based on carrier type, exhibiting opposite Seebeck coefficients. The p-type shows a positive Seebeck coefficient, while the n-type displays a negative one. This distinction is essential for accurately modeling TE behavior in the TEC module.

Comment 2:

Figure Quality: Some colorbars lack units (e.g., electric potential in Figure 3(b)). Ensure all axes and legends are fully labeled and legible at publication resolution.

Response:

Thank you for pointing out the missing units and labels. We’ve updated all figures to include clear units, labeled axes, and readable legends in the revised manuscript.

English & Typos:

Comment 3:

Line 22 (“…as compact, energy‐efficient, and precise solutions…”): change “solutions” to “solution.”

Response:

Thank you for your grammatical suggestion. We have corrected “solutions” to “solution” in the revised manuscript.

Comment 4:

Line 30 (“…demonstrating the potential of TE devices…”): insert comma after “demonstrating.” A thorough language edit is recommended.

Response:

Thank you for your valuable comments. We have revised the entire manuscript, tried to improve sentence readability, and ensured grammatical accuracy.

Comment 5:

Verify reference [22] (Aliabadi et al. 2014) DOI/link format; some conference proceedings are cited without full page ranges or DOI consistency.

Response:

Thank you for noting the inconsistency in the reference formatting. Reference [22] has been corrected in the revised manuscript.

Comment 1:

The design of multistage thermoelectric cooler (Figure 1).

Response:

Thank you for your valuable comments. In response to your feedback, we have included the following text in the revised manuscripts of the Materials and Methods (Section 2.3):  The theoretical study presents a structured, physically integrated MS TEC design optimized for IR detectors, emphasizing heat flow direction, material interfaces, and stage connectivity. It illustrates both the system level integration and thermocouple arrangement. In contrast, our numerical model focuses on device level optimization, employing simulation-based configuration of thermoelement dimensions, stage count, and material parameters within COMSOL.

Comment 2:

The methodic of calculations of parameters for the multistage TE cooler

Response:

In response to your feedback, we have included the following text in the revised manuscript's Materials and Methods (Section 2.3): The theoretical work utilizes optimal control theory to simultaneously optimize material inhomogeneity and stage-wise parameters under defined thermal constraints. This analytical framework enables a system-wide maximization of COP. Our study, however, adopts a numerical approach using finite element analysis, conducting parametric sweeps over input current and geometric variables, and refining the mesh to ensure accurate temperature and performance predictions.

Comment 3:

Results of the outside characteristics of the multistage TE cooler (maximum COP).

Response:

In response to your feedback, we have included the following text in the revised manuscripts of the Results (Section 3.1.6):  The theoretical results show that reducing contact resistance and using insulating plates and functionally graded materials can boost COP by 1.5–2.5 times compared to commercial modules [43]. Our numerical results also reflect improved performance under optimized geometry and operating conditions, though possibly without explicitly modeling functionally graded materials or precise contact resistances.

Comment 4:

Conclusions.

Response:

In response to your feedback, we have included the following text in the revised manuscripts of the Results (Section 3.2.5):  Both studies conclude that minimizing contact and thermal interface resistances is essential for enhancing TEC efficiency. While the theoretical model provides a system-level blueprint for designing high-performance TECs using optimal control, our numerical analysis offers practical insights into device optimization, making both approaches complementary in advancing TEC design for cryogenic and IR applications.

Comment 1:

The author using COMSOL to simulate the efficiency of thermoelectric cooling systems. However, Bi₂Te₃ is a toxic material and raise significant safety concern if implanted in the human body. For non-implanted cryosurgical applications, liquid nitrogen is already an effective and cost-efficient option. Therefore, I don’t believe Bi₂Te₃ is an appropriate material for cryosurgery. While Bi₂Te₃ has nearly maximum thermoelectric efficiency near the room temperature. It is important to note that the human body remains at room temperature. Combing n-type Bi₂Te₃ with p-type Sb₂Te₃ will yield higher thermoelectric efficiency, as demonstrated in “https://doi.org/10.1002/aelm.201800904”. Hence, using a combination of these two materials will likely result in better performance than using Bi₂Te₃ alone.

Response:

Thank you for your valuable comments. In response to your feedback, we have included the following text in the revised manuscripts of the Introduction section:  We used Bi₂Te₃ in our COMSOL simulations, focusing on external TE cooling systems where toxicity is not a concern. While liquid nitrogen is economical, TE cooling offers precise control, compactness, and ease of integration. Pairing n-type Bi₂Te₃ with p-type Sb₂Te₃ enhances performance. As noted by Witting et al. [16], Bi₂Te₃-based alloys achieve high ZT near room temperature due to favorable electronic and thermal properties. Alloying with Sb₂Te₃ and Bi₂Se₃ further improves efficiency. Bi₂Te₃ was chosen for its proven performance and extensive experimental and modeling data.

Comment 2:

Regarding conductive materials, sliver offers better thermal and electrical conductivity than Cu and should be considered as the primary candidate. The author using Bi₂Te₃ for both p-type and n-type materials, but in Table 1, it shows the author using p-type Bi₂Te₃for both sided. It doesn’t work to put the same kinds of materials at both sides. The p-type and n-type Bi₂Te₃ are different materials, and it depends on the doping process or defect in materials.

Response:

Thank you for your valuable comments. In response to your feedback, we have included the following text in the revised manuscripts of the Materials and Methods (Section 2.3): Silver (Ag) has superior thermal and electrical conductivity compared to Cu, potentially boosting TEC performance. Nevertheless, Cu was chosen for its affordability, availability, and ease of integration with conventional TEC fabrication methods.

We appreciate your observation regarding the material configuration. The initial duplication of p-type Bi₂Te₃ in Table 1 was a typographical error, and we have corrected it in the revised manuscript.

Comment 3:

The text in figures is too small and difficult to read, and many figures appear to be low quality and most of figures are screenshot. Please visualize these figures in a better way. Specifically, in Figure 3(a), 7 9, the temperature changes are not clearly visible. Please try to present these data in a better way.

Response:

Thank you for your thoughtful feedback on the figures’ quality and readability. In response, we have carefully revised and enhanced all figures to improve their clarity and ensure they accurately convey the scientific content.