# Thermoelectric Cycle and the Second Law of Thermodynamics

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Clausius’ Original Version of the Second Law of Thermodynamics

_{2}F(t

_{1},t

_{2}), where F(t

_{1},t

_{2}) is a function of temperature t

_{1}and t

_{2}. By means of a fundamental principle, i.e., heat can never pass from a cold to a hot body without some other change, Clausius then demonstrated that the sum of the equivalence-values for all transformations in the reversible thermodynamic cycle was zero [5,7,8], as follows:

_{1}is the temperature of the heat reservoir and T

_{2}is the temperature of the heat sink.

## 3. Theorem of the Equivalence of Transformations for a Thermoelectric Cycle

## 4. Theorem of the Equivalence of Transformations for a Combined Power–Refrigeration Cycle

_{1}from T

_{1}to T

_{2}, Q

_{1}F(T

_{1},T

_{2}), and the transformation of Q

_{m}at T

_{1}to electrical work, −Q

_{m}f(T

_{1}) and the refrigeration cycle part contains the transformation of Q

_{2}from T

_{4}to T

_{3}, Q

_{2}F(T

_{4},T

_{3}) and the transformation of electrical work to Q

_{m}at T

_{3}, Q

_{m}f(T

_{3}). According to Equation (2), Q

_{m}F(T

_{1},T

_{3}) = Q

_{m}f(T

_{3}) − Q

_{m}f(T

_{1}). That is, the transformation of Q

_{m}at T

_{1}to electrical work in the power cycle part and the transformation of electrical work to Q

_{m}at T

_{3}can be combined into one transformation of Q

_{m}from T

_{1}to T

_{3}. Therefore, the reversible combined cycle can be regarded as containing three transformations, i.e., (1) the transformation of Q

_{1}from T

_{1}to T

_{2}; (2) the transformation of Q

_{m}from T

_{1}to T

_{3}; (3) the transformation of Q

_{2}from T

_{4}to T

_{3}. These three transformations cancel each other out; thus, the algebraic sum of the equivalent values of all transformations is zero:

_{1}from T

_{b,1}to T

_{r}; (2) the transformation of Q

_{m}from T

_{b,1}to T

_{r}; (3) the transformation of Q

_{2}from T

_{b,2}to T

_{r}, as shown in Figure 6a. Thus, the theorem of the equivalence of transformations is expressed as:

_{1}+ Q

_{m}) from T

_{b,1}to T

_{r}; and (2) the transformation of Q

_{2}from T

_{b,2}to T

_{r}, as shown in Figure 6b. Thus, the theorem of the equivalence of transformations is simplified as:

_{1}+ Q

_{m}) from T

_{b,1}to T

_{r}is positive and the transformation of Q

_{2}from T

_{b,2}to T

_{r}is negative. The occurrence of the negative transformation must be compensated by the positive transformation. In other words, heat could pass from the colder body to the ambient, but before that can happen, the body itself needs to have a higher temperature than the ambient as compensation.

_{V}is the body’s heat capacity at a constant volume. Solving for Equation (13) leads to the same result as that given by Schilling et al.:

_{b}(0), and the ambient temperature, T

_{r}, are given, Equation (14) is an implicit calculation formula related to the minimum temperature the body can reach, T

_{b,min}. According to Equation (14), when the initial temperature of the body is higher and the ambient temperature is lower, the minimum temperature that can be reached by the body is lower. Both a higher initial temperature of the body and a lower ambient temperature can create a larger positive transformation, which can compensate for the larger negative transformation corresponding to a larger reduction in the body temperature in the second stage.

_{1}+ Q

_{m}+ Q

_{2}) is all the heat that moves away from the body and also all the heat that enters the ambient. All the heat that moves away from the body can be expressed as $-{\displaystyle {\int}_{{T}_{\mathrm{b}}(0)}^{{T}_{\mathrm{b},\mathrm{min}}}{C}_{V}}\mathrm{d}{T}_{\mathrm{b}}$ and all the heat that enters the ambient can be expressed as $\int}_{{T}_{\mathrm{r}}(0)}^{{T}_{\mathrm{r},\mathrm{max}}}{C}_{V,\mathrm{r}}}\mathrm{d}{T}_{\mathrm{r$. Therefore, for the second case, the specific expression of Equation (12) is:

_{V}= C

_{V}

_{,r}. Combining Equation (17) with the energy conservation equation easily yields:

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schilling et al.’s thermoelectric experiments [1]. T

_{b}(0) denotes the initial temperature of the body, T

_{b,min}denotes the minimum temperature the body can reach, T

_{r}denotes the ambient temperature and τ denotes time. Π is the Peltier element, L is the electric inductance and I is the electric current.

**Figure 3.**Kelvin’s thermoelectric circuit [21].

**Figure 6.**Schematic of transformations in Schilling et al.’s combined cycle. (

**a**) Normal expression. (

**b**) Simplified expression.

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Xue, T.-W.; Guo, Z.-Y.
Thermoelectric Cycle and the Second Law of Thermodynamics. *Entropy* **2023**, *25*, 155.
https://doi.org/10.3390/e25010155

**AMA Style**

Xue T-W, Guo Z-Y.
Thermoelectric Cycle and the Second Law of Thermodynamics. *Entropy*. 2023; 25(1):155.
https://doi.org/10.3390/e25010155

**Chicago/Turabian Style**

Xue, Ti-Wei, and Zeng-Yuan Guo.
2023. "Thermoelectric Cycle and the Second Law of Thermodynamics" *Entropy* 25, no. 1: 155.
https://doi.org/10.3390/e25010155