Reversibilities and irreversibilities in thermoelectric energy conversion

Referee: §3, Eqs. (7)-(9): The central irreversibility claim hinges on collapsing the continuous temperature gradient into a two-reservoir heat-engine model. The resulting entropy generation is proportional to τ, whereas in the Onsager framework the local entropy production rate at open circuit (J_e = 0) is σ = κ(∇T)²/T², independent of τ. The manuscript does not identify what physical mechanism produces this extra τ-dependent entropy, nor why the Onsager framework fails to capture it.

Authors: The referee correctly identifies the central tension of the paper, and we agree that the manuscript does not currently address it with sufficient clarity. We offer the following clarification, which we will incorporate into the revised manuscript, while being candid about what remains unresolved. The classical thermodynamic analysis in the paper asks a different question from the Onsager framework. The Onsager framework computes the local entropy production rate associated with transport processes (heat conduction, Joule heating, etc.) at a given point in the material. At open circuit (J_e = 0), this local production is indeed σ = κ(∇T)²/T², independent of τ, as the referee states. The classical analysis, by contrast, performs a global entropy balance over the entire device treated as a single heat engine operating between two reservoirs. The entropy generation ΔS_i = ∫_{Tc}^{Th} (1/Tc − 1/T) τ dT arises because, when τ ≠ 0, the Peltier coefficient π(T) = ε(T)·T varies with temperature in a way that prevents the global entropy balance from closing. Physically, the mechanism is the temperature dependence of the Seebeck coefficient: when ε is not constant, the thermoelectric cannot be decomposed into a series of infinitesimal reversible Carnot engines, because the entropy per electron ε(T) changes along the gradient. The 'extra' τ-dependent entropy is therefore not a local transport effect but a consequence of the global thermodynamic accounting when a continuous gradient is mapped onto a two-reservoir model. We acknowledge, however, that the referee raises a legitimate and sharp question: if this entropy generation is physically real, it should be reconcilable with the Onsager framework, which is the standard and complete description of irreversible thermodynamics in the revision: no

Referee: §3, Eq. (8): The heat rejected to the cold reservoir Q = π(Th) - π(Tc) - qV is defined within the heat-engine analogy, but at open circuit no electron transport occurs, so no Peltier or Thomson heat is physically transported between reservoirs. The author should clarify what physical quantity Q represents at open circuit and how it relates to measurable heat flow.

Authors: This is a fair point and we will revise the manuscript to clarify it. The quantity Q in Eq. (8) is a thermodynamic (virtual) quantity, not an actual heat flow measured at open circuit. It represents the heat that would be rejected to the cold reservoir per electron if the thermoelectric converter operated as a classical heat engine absorbing heat π(T_h) at the hot reservoir and producing work qV. This is analogous to the open-circuit analysis of electrochemical cells (batteries, fuel cells), where one computes thermodynamic quantities (e.g., the Nernst voltage) from the Gibbs free energy change even though no current flows at open circuit. The analysis determines the thermodynamic limit and the entropy balance associated with the conversion process, not an actual measurable heat flux. We will add a paragraph to §3 making this analogy explicit and stating clearly that Q is a thermodynamic bookkeeping quantity within the heat-engine framework, not a directly measurable heat flow at open circuit. We will also note that when current does flow (finite load), the actual heat flows include additional Onsager-type contributions (Joule heating, thermal conduction) that are separate from the thermodynamic Q defined here. revision: yes

Referee: §3, Eq. (10): The 'Carnot work' reference π(Th)(1 - Tc/Th) assumes all heat is absorbed at Th and rejected at Tc, which does not describe the thermoelectric's operation across a continuous gradient. The author should address whether this decomposition has physical content beyond a bookkeeping identity, given that the Seebeck voltage qV = ∫ ε dT (Eq. 7) is already well-established and does not require the two-reservoir framing.

Authors: The referee is correct that Eq. (7) alone gives the Seebeck voltage without requiring the two-reservoir framing, and that the Gouy-Stodola decomposition in Eq. (10) is, in a mathematical sense, a rearrangement of established results. The physical content of the decomposition, if it has any, lies in the interpretation: it identifies how much of the thermoelectric voltage can be associated with reversible Carnot conversion and how much is 'lost' due to the τ-dependent entropy generation. Whether this interpretation carries physical significance beyond bookkeeping depends on whether the entropy generation ΔS_i represents a real thermodynamic loss or an artifact of the two-reservoir model. This is precisely the issue raised in the first major comment, and we acknowledge that the manuscript does not currently establish the physical significance of the decomposition independently of that question. In the revision, we will (a) state explicitly that Eq. (10) is a rearrangement of Eq. (7) combined with the entropy balance, (b) clarify that its physical significance is contingent on the interpretation of ΔS_i as discussed in our response to the first comment, and (c) avoid overstating the decomposition as a falsifiable prediction until the relationship to the Onsager framework is more firmly established. We agree that the current manuscript language ('falsifiable, quantitative prediction') overstates what the decomposition demonstrates on its own. revision: partial