# Exact Solution of a Constraint Optimization Problem for the Thermoelectric Figure of Merit

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## Abstract

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## 1. Introduction

## 2. Linear Functions k(T) = z(T)T

**Figure 1.**Relative performance increase R as function of the parameter ξ: left: ${R}_{\eta}$ $={\eta}_{sc}/{\eta}_{sc}^{\left({k}_{o}\right)}-1$ for TEG (${T}_{a}=600\phantom{\rule{3.33333pt}{0ex}}K,\phantom{\rule{3.33333pt}{0ex}}{T}_{s}=300\phantom{\rule{3.33333pt}{0ex}}K$) for ${k}_{o}=1$ (solid curve, optimal efficiency ${\eta}_{sc,opt}$ at ${\xi}_{opt}=4.2$) and ${k}_{o}=0.6$ (dashed curve, ${\eta}_{sc,opt}$ at ${\xi}_{opt}=11.9$, curve slowly decreasing for $\xi >{\xi}_{opt}$ as long as ${k}_{o}>0.5$); right: ${R}_{\phi}={\phi}_{sc}/{\phi}_{sc}^{\left({k}_{o}\right)}-1$ for TEC (${T}_{a}=270\phantom{\rule{3.33333pt}{0ex}}K,\phantom{\rule{3.33333pt}{0ex}}{T}_{s}=300\phantom{\rule{3.33333pt}{0ex}}K$) for ${k}_{o}=1$ (solid) and ${k}_{o}=0.6$ (dashed), optimal coefficient of performance ${\phi}_{sc,opt}=1.0002$ ${\phi}_{sc}^{\left({k}_{o}\right)}$ at ${\xi}_{opt}=1.055$ for both curves.

**Figure 2.**Optimal straight line ${k}_{opt}\left(T\right)=k(T,{\xi}_{opt})$ plotted with the optimal parameter ${\xi}_{opt}$ derived from Figure 1: left (TEG): ${\xi}_{opt}=4.2$ for ${k}_{o}=1$ (purple) and ${\xi}_{opt}=11.9$ for ${k}_{o}=0.6$ (blue); right (TEC): ${\xi}_{opt}\approx 1$ (from ${\xi}_{opt}=1.054$ for ${k}_{o}=0.1$ to ${\xi}_{opt}=1.062$ for ${k}_{o}=10$, with ${\xi}_{opt}=1.055$ for ${k}_{o}=0.6$ and ${k}_{o}=1$).

## 3. Isoperimetric Variational Problem

**Theorem 1.**

**Figure 4.**Graph of polynomials ${P}_{1}\left(x\right)=x{(x+1)}^{2}$ and ${P}_{2}\left(x\right)=x{(x-1)}^{2}$, see Equation (13).

**Theorem 2.**

- (i)
- In case of a TEG there is a constant ${\overline{k}}_{o}$ such that the following holds: If ${k}_{o}\ge {\overline{k}}_{o}$ there exists a unique $\mu ={\mu}^{*}$ such that the function ${k}_{{\mu}^{*}}$ defined by Equation (14) fulfills Equation (11a) as well as Condition (12). Hence, ${z}_{max}\left(T\right):={z}_{{\mu}^{*}}\left(T\right)$. The corresponding ${k}_{max}\left(T\right)={z}_{max}\left(T\right)T$ is nonnegative on the interval $[{T}_{1},{T}_{2}]$, strictly monotonically decreasing and convex. If $0<{k}_{o}<{\overline{k}}_{o}$ there is no constant μ such that the corresponding solution ${z}_{\mu}\left(T\right)$ of Equation (11a) is nonnegative for every $T\in [{T}_{1},{T}_{2}]$ and fulfills Equation (12). In this case there is no optimal profile.
- (ii)
- In case of a TEC for every ${k}_{o}>0$ there exist a unique $\mu ={\mu}^{*}$ and a unique function ${z}_{min}\left(T\right):={z}_{{\mu}^{*}}\left(T\right)$ which solve Equations (11b) and (12). The corresponding ${k}_{min}\left(T\right)={z}_{min}\left(T\right)T$ is nonnegative, strictly monotonically decreasing and convex.

- (i)
- Let ${k}_{\mu}$ be the (unique) solution of Equation (11a) for fixed $\mu >0$ given by Equation (14). We rewrite Equation (11a) by$$\sqrt{1+{k}_{\mu}\left(T\right)}{\left(\sqrt{1+{k}_{\mu}\left(T\right)}+1\right)}^{2}=\frac{\mu}{T}$$$$\frac{\mu}{{T}_{2}}=\sqrt{1+{k}_{\mu}\left({T}_{2}\right)}{\left(\sqrt{1+{k}_{\mu}\left({T}_{2}\right)}+1\right)}^{2}\ge 4.$$$$av\left(\mu \right):=\frac{1}{{T}_{2}-{T}_{1}}\phantom{\rule{0.166667em}{0ex}}{\int}_{{T}_{1}}^{{T}_{2}}{k}_{\mu}\left(T\right)\phantom{\rule{0.166667em}{0ex}}dT$$
- (ii)
- By the discussion above it is obvious that in the case of a TEC there is a unique and nonnegative solution ${k}_{\mu}\left(T\right)={z}_{\mu}\left(T\right)T$ of Equation (11b) for every fixed $\mu >0$. The representation$$\sqrt{1+{k}_{\mu}\left(T\right)}{\left(\sqrt{1+{k}_{\mu}\left(T\right)}-1\right)}^{2}=\frac{\mu}{T}$$

**Remark 1.**

- 1.
- The observations in Section 2 on linear functions reflect the general result. Certain monotonically decreasing straight lines yield a better performance than the increasing ones. Moreover, as discussed in Section 2, also in the case of linear functions $k\left(T\right)$ there is a critical value ${\overline{k}}_{o}>0$ of ${k}_{o}$ for TEG, where we have no optimal linear function below of it. For a TEC such a critical ${\overline{k}}_{o}$ does not occur. There we have an optimal performance in the class of linear function for every ${k}_{o}>0$.
- 2.
- It is obvious that also ${z}_{opt}$ will be strictly monotonically decreasing since ${k}_{opt}\left(T\right)={z}_{opt}\left(T\right)T$ has this property. Even more, ${z}_{opt}$ will be a convex function. This can be justified by the following calculation using strict convexity of ${k}_{opt}\left(T\right)={z}_{opt}\left(T\right)T$:$$0<{k}_{opt}^{\u2033}\left(T\right)={\left({z}_{opt}\left(T\right)T\right)}^{\u2033}={\left({z}_{opt}^{\prime}\left(T\right)T+{z}_{opt}\left(T\right)\right)}^{\prime}=2{z}_{opt}^{\prime}\left(T\right)+{z}_{opt}^{\u2033}\left(T\right)T$$

TEG | ${k}_{o}=1$ | ${k}_{o}=0.6$ | ||
---|---|---|---|---|

${\eta}_{\text{sc}}$ | ${\eta}_{\text{sc}}/{\eta}_{\text{sc}}^{\left({k}_{o}\right)}$ | ${\eta}_{\text{sc}}$ | ${\eta}_{\text{sc}}/{\eta}_{\text{sc}}^{\left({k}_{o}\right)}$ | |

constant function $k\left(T\right)={k}_{o}$ | 0.112126 | 1.00000 | 0.077873 | 1.00000 |

linear function $k\left(T\right)=k(T,{\xi}_{opt})$ | 0.114786 | 1.02372 | 0.080752 | 1.03697 |

optimal function $k\left(T\right)={k}_{max}\left(T\right)$ | 0.114855 | 1.02434 | 0.080829 | 1.03796 |

**Figure 6.**Optimal monotonic functions ${k}_{min}\left(T\right)$ (red) compared with the best straight line $k(T,{\xi}_{opt})$ (blue) from Figure 2 plotted with the optimal parameter ${\xi}_{opt}=1.055$ derived from Figure 1. left: ${k}_{o}=1$; right: ${k}_{o}=0.6$. Please note the scaling of the y-axis.

TEC | ${k}_{o}=1$ | ${k}_{o}=0.6$ | ||
---|---|---|---|---|

${\phi}_{\text{sc}}$ | ${\phi}_{\text{sc}}/{\phi}_{\text{sc}}^{\left({k}_{o}\right)}$ | ${\phi}_{\text{sc}}$ | ${\phi}_{\text{sc}}/{\phi}_{\text{sc}}^{\left({k}_{o}\right)}$ | |

constant function $k\left(T\right)={k}_{o}$ | 1.17929125 | 1.0000000 | 0.68419337 | 1.0000000 |

linear function $k\left(T\right)=k(T,{\xi}_{opt})$ | 1.17955485 | 1.0002235 | 0.68438545 | 1.0002803 |

optimal function $k\left(T\right)={k}_{min}\left(T\right)$ | 1.17955497 | 1.0002236 | 0.68438554 | 1.0002804 |

## 4. Discussion and Conclusions

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## Appendix

**Lemma A1.**

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**MDPI and ACS Style**

Seifert, W.; Pluschke, V.
Exact Solution of a Constraint Optimization Problem for the Thermoelectric Figure of Merit. *Materials* **2012**, *5*, 528-539.
https://doi.org/10.3390/ma5030528

**AMA Style**

Seifert W, Pluschke V.
Exact Solution of a Constraint Optimization Problem for the Thermoelectric Figure of Merit. *Materials*. 2012; 5(3):528-539.
https://doi.org/10.3390/ma5030528

**Chicago/Turabian Style**

Seifert, Wolfgang, and Volker Pluschke.
2012. "Exact Solution of a Constraint Optimization Problem for the Thermoelectric Figure of Merit" *Materials* 5, no. 3: 528-539.
https://doi.org/10.3390/ma5030528