🔬 applied physics

Reversibilities and irreversibilities in thermoelectric energy conversion

This paper demonstrates that a thermoelectric generator operating at open circuit is an irreversible classical heat engine that produces entropy whenever the Thomson coefficient is nonzero, and it derives the resulting voltage using the Guy-Stodola equation.

Original authors: Tom Markvart

Published 2026-07-08

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CC BY 4.0

Original authors: Tom Markvart

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Idea: Is the Thermoelectric Generator Perfect?

Imagine you have a special machine (a thermoelectric generator) that sits between a hot side and a cold side. Its job is to turn that temperature difference into electricity (voltage). For a long time, scientists debated whether this machine could be considered "perfect" or "reversible" when it isn't actually running a current (open circuit).

The author, Tom Markvart, argues that it is not perfect. Even when the machine is just sitting there doing nothing but sitting between hot and cold, it is creating "mess" (entropy) and losing potential energy, provided a specific property of the material (called the Thomson coefficient) is not zero.

The Analogy: The Water Wheel and the Leaky Bucket

To understand the paper, let's use an analogy of a water wheel trying to generate power.

1. The "Perfect" Scenario (The Old View)
Imagine a water wheel that is so efficient that for every drop of water falling from a high reservoir to a low one, it captures 100% of the potential energy to turn the wheel. In physics, this is called a "reversible" process. If you run the machine backward, it would work perfectly in reverse with no energy lost.

For decades, scientists thought thermoelectric generators worked like this if you ignored the heat that leaks through the material and the heat caused by electrical resistance (Joule heating). They thought that at the "open circuit" (when the machine is just waiting to be used), it was a perfect, reversible engine.

2. The "Leaky Bucket" Reality (The New View)
Markvart says, "Wait a minute." He looks at the machine using the logic of a classic heat engine (like a steam engine).

He imagines the heat traveling from the hot side to the cold side not as a smooth slide, but as a series of tiny steps.

The Peltier Effect (The Bucket): Think of the electrons as buckets carrying heat. As they move from hot to cold, they drop some heat.
The Thomson Effect (The Leak): Here is the catch. As the buckets move through the temperature change, the material itself causes the buckets to leak a little bit of heat along the way, not just at the end.

Markvart shows that if this "leak" (the Thomson coefficient) exists, the process creates entropy. In our analogy, entropy is like "mess" or "disorder." Once you create mess, you can't un-create it. The process becomes irreversible.

The "Lost Work" Equation

The paper derives a famous equation from classical thermodynamics called the Gouy-Stodola equation.

In a perfect world: You take the heat from the hot side, multiply it by the efficiency of the temperature difference (Carnot efficiency), and you get your maximum possible voltage.
In Markvart's world: You get that maximum voltage minus the "lost work."

Think of it like this:

You have a budget of $100 (the heat energy).
In a perfect engine, you spend $100 to get $100 worth of work.
In this thermoelectric engine, because of the "leak" (Thomson effect), you spend $100 but only get $90 worth of work. The missing $10 is the "lost work" caused by the entropy generated as the heat moves through the material.

The Conclusion: Why It Matters (According to the Paper)

The paper concludes that thermoelectric conversion is inherently irreversible as long as the Thomson coefficient is not zero.

It's not just about running the machine: Even when the machine is "open circuit" (not generating current, just sitting there), the mere act of having a temperature difference across the material generates entropy.
It's not just about resistance: This is a different kind of loss than the usual "Joule heating" (which happens when electricity flows and creates heat). This is a fundamental loss built into the thermodynamics of the material itself.
The Limit: Because of this irreversibility, you can never reach the theoretical maximum efficiency (Carnot efficiency) of a heat engine, even in this "perfect" open-circuit state.

Summary in One Sentence

The paper proves that a thermoelectric generator is like a water wheel with a slow, unavoidable leak; even when it's not turning, the leak creates enough "mess" (entropy) to prove the process is imperfect and irreversible, meaning you can never get the absolute maximum amount of energy out of the temperature difference.

Technical Summary: Reversibilities and Irreversibilities in Thermoelectric Energy Conversion

Problem Statement
The paper addresses the long-standing theoretical debate regarding the reversibility of thermoelectric energy conversion. While a consensus exists that thermoelectric conversion can be viewed as reversible when considered independently of Joule heating and thermal conduction, the author questions whether this holds true when the Thomson coefficient is non-zero. Specifically, the paper investigates whether a thermoelectric generator operating at open circuit (negligible current) functions as a truly reversible classical heat engine or if it inherently generates entropy, thereby rendering the process irreversible.

Methodology
The author employs an extension of Lord Kelvin's (Thomson's) classical thermodynamic approach, originally formulated for infinitesimal temperature differences, to a finite temperature difference ( $T_h - T_c$ ).

Modeling: The thermoelectric converter is treated as a classical heat engine operating between a high-temperature reservoir ( $T_h$ ) and a low-temperature reservoir ( $T_c$ ).
Entropy Balance: The analysis focuses on the entropy balance of the system. It tracks the entropy carried by electrons (associated with Peltier heat, $\pi$ ) and the entropy changes in the reservoirs.
Generalization: The derivation moves from a differential case (small $\delta T$ ) involving a single heat rejection step to an integral case involving finite temperature differences. This requires integrating the Peltier coefficient ( $\pi$ ) and the Thomson coefficient ( $\tau$ ) across the temperature gradient.
Thermodynamic Relations: The derivation utilizes Kelvin's two relations of thermoelectricity, linking the Seebeck coefficient ( $\varepsilon$ ), Peltier coefficient ( $\pi$ ), and Thomson coefficient ( $\tau$ ).

Key Results

Entropy Generation: The analysis demonstrates that for a substance with a non-zero Thomson coefficient, the operation of a thermoelectric generator at open circuit generates entropy.
Irreversibility: Consequently, the process is inherently irreversible. The author shows that simply extending the reversible infinitesimal model to a finite temperature difference does not preserve reversibility because the heat rejection occurs at a single temperature ( $T_c$ ) rather than through a continuum of intermediate reservoirs.
Voltage Formulation: The generated voltage ( $V$ $V$ ) is derived as the difference between the ideal work output (based on Carnot efficiency) and the "lost work" due to irreversible entropy generation.
- The total entropy generation ( $S_i$ ) is expressed as:
  $S_i = \int_{T_c}^{T_h} \frac{\tau}{T} dT - \frac{1}{T_c} \int_{T_c}^{T_h} \tau dT$
- The resulting voltage equation is:
  $qV = \pi(T_h) - \pi(T_c) - T_c S_i$
Gouy-Stodola Analogy: The final voltage equation is identified as an analogue of the Gouy-Stodola equation for classical heat engines. It explicitly separates the reversible work component from the irreversible losses caused by entropy generation.

Significance and Claims
The paper claims to generalize previous results by showing that even at open circuit—where Joule heating and thermal conduction are typically excluded from reversibility discussions—the presence of a non-zero Thomson coefficient implies irreversibility.

Theoretical Contribution: The work provides a classical thermodynamic derivation (distinct from Onsager's irreversible thermodynamics) that confirms entropy generation in thermoelectric materials with non-zero Thomson coefficients.
Efficiency Limit: The author concludes that because of this inherent entropy generation, the Carnot efficiency cannot be reached even at open circuit.
Scope of Losses: The paper clarifies that these losses are distinct from, and additional to, the irreversibilities described by Onsager's formulation (which arise from finite-time operation and currents). The losses discussed here are intrinsic to the static, open-circuit thermodynamic state of the material itself.

In summary, the paper argues that the idealization of thermoelectric conversion as a reversible process is incomplete; when viewed through the lens of finite temperature differences and the Thomson effect, the process is fundamentally irreversible, and the output voltage is reduced by the associated entropy generation.