Reversibilities and irreversibilities in thermoelectric energy conversion

Abstract: Similarly to Thomson, we consider the thermoelectric generator at open circuit as a classical heat engine. It is shown that, as long as the Thomson coefficient is nonzero, the operation generates entropy and is therefore irreversible. By expanding Thomson's approach we show that the voltage produced can be described by the usual Guy--Stodola equation for classical heat engines.

• The paper demonstrates that including the Thomson effect reveals inherent irreversible entropy generation even in open-circuit thermoelectric converters.
• It extends classical thermodynamic frameworks by linking thermoelectric voltage production to the Gouy-Stodola theorem and detailed entropy balance.
• The analysis implies that practical thermoelectric device efficiency is fundamentally limited by unavoidable Thomson-related energy losses.

Reversibility and Irreversibility in Thermoelectric Energy Conversion

Introduction

The paper critically examines the longstanding debate on the reversibility of thermoelectric energy conversion, with a particular emphasis on entropy generation in systems with a nonzero Thomson coefficient. By extending classical thermodynamic arguments, the author provides a formal link between thermoelectric voltage generation and the Gouy-Stodola theorem, situating thermoelectric conversion more firmly within the theoretical framework of classical heat engines. The analysis challenges the interpretation of thermoelectric processes as reversible at open circuit when the Thomson effect is non-negligible.

Thermoelectric Generator as a Classical Heat Engine

The work builds upon Thomson’s original model, conceptualizing the thermoelectric generator operating at open circuit as a heat engine cycling between two reservoirs at temperatures $T$ and $T - \Delta T$. In this setting, each carrier transports both heat and charge, enabling the device to convert a fraction of thermal energy into electrical work. However, a key element is the inclusion of the Thomson coefficient, which accounts for the additional heat evolved (or absorbed) owing to the temperature gradient within the conductor.

The analysis demonstrates that, although the open-circuit operation might naively appear reversible (yielding voltage proportional to Carnot efficiency), the requirement of a nonzero Thomson coefficient necessitates the rejection of "Thomson heat" into the lower-temperature reservoir. This process, unlike Peltier and Seebeck effects alone, cannot generally be rendered isentropic. Entropy balance calculations show that the heat rejected into the cold reservoir is not fully compensated by entropy reduction of the working substance, resulting in net entropy production and irreversibility.

Finite Temperature Differences and Entropy Generation

Moving beyond infinitesimal gradients, the paper generalizes the Thomson argument to finite temperature intervals between $T_h$ and $T_c$. Unlike a Carnot cycle, which operates between two reservoirs, the step-wise equilibration across many intermediate temperatures in a real conductor necessitates explicit tracking of entropy exchanges at each stage. The work shows that the total voltage produced corresponds to the net heat converted with Carnot efficiency, diminished by an irreversibility term explicitly connected to the Thomson coefficient.

Mathematically, the open circuit voltage is expressed in analogy to the Gouy-Stodola theorem: the available work equals the difference between heat absorption at the hot side and the product of cold reservoir temperature with total entropy generation. In effect, the Guy-Stodola equation for heat engines is rederived for thermoelectric converters, emphasizing that as long as the Thomson effect is present, irreversible energy dissipation—manifested as entropy generation—is unavoidable even at reversible operating conditions in the conventional sense.

Implications for Thermoelectric Theory

The findings directly contest the view that isolating Joule heating and macroscopic Fourier conduction allows thermoelectric conversion to be treated as reversible at open circuit. Entropy generation associated with the Thomson effect is inherent to the thermodynamic description and limits achievable efficiency below the Carnot bound. This places restrictions on both practical device performance and the theoretical understanding of coupled transport in thermoelectrics.

The paper's analysis aligns the treatment of thermoelectric generators with that of other coupled-transport electrochemical sources, such as batteries and fuel cells, where open-circuit analyses must account for additional sources of irreversibility not captured by Onsager's macroscopic framework. It parses the contributions to entropy generation and highlights that Thomson-related losses are supplemental to those arising from finite transport coefficients (e.g., Joule heating), reinforcing the need for comprehensive loss accounting in any efficiency analysis.

Prospects for Future Research

These insights prompt a reconsideration of device modeling in thermoelectric materials research, with potential relevance for materials design targeting the minimization of irreversible processes. Work extending this analysis to nonlocal, time-dependent, and nonequilibrium steady-state regimes may further elucidate the distinction between intrinsic and extrinsic irreversibilities, with likely relevance in nanostructured and low-dimensional thermoelectrics where the Thomson coefficient can be engineered. The theoretical results also suggest a need for careful experimental quantification of the Thomson effect and related entropy flows in high-performance thermoelectric systems.

Conclusion

By rigorously extending the classical thermodynamic analysis of thermoelectric energy conversion, this work establishes that any system exhibiting a nonzero Thomson coefficient generates entropy and is thus irreducible to a fully reversible heat engine, even at open circuit. The operational voltage adheres to the Guy-Stodola principle, comprising both reversible and irreversible contributions, which fundamentally limits the realizable efficiency. These findings necessitate a nuanced approach to the thermodynamic analysis of thermoelectric devices, with implications spanning theoretical treatments and practical optimization strategies.