# General Approach for Composite Thermoelectric Systems with Thermal Coupling: The Case of a Dual Thermoelectric Cooler

^{*}

## Abstract

**:**

## 1. Introduction

_{c}and coefficient of performance (COP). New techniques and thermoelectric materials have been proposed for designing new thermoelectric specific systems that allow the improvement of the device performance through the optimization of the thermal and electrical transport properties [1–3]. These TECS, known as solid-state devices, are used in many different applications, ranging from controlling the temperature of laser diodes, infrared detectors, superconductor applications, aerospace applications, electronic devices and food storage [4].

_{C}, and their behavior as a function of the thermal impedances. Finally, in Section 5, we present our conclusions, and future work is discussed briefly.

## 2. Dual Thermoelectric Cooling System

_{c}and T

_{h}with T

_{h}> T

_{c}. K

_{c}and K

_{h}are the thermal conductances of the heat exchangers at both sides, cold and hot, respectively. Each TEM is characterized by a Seebeck coefficient α

_{i}, thermal conductance K

_{i}and an electrical resistanceR

_{i}, where i can be one or two as appropriate. The TEM

_{1}and TEM

_{2}are formed by m and n thermocouples number, respectively. I is the electrical current through the TEMs.

## 3. Heat Balance Equations

_{h}is the heat rejected from the TEC system to the heat reservoir and Q

_{c}is the cooling capacity or the heat absorbed from the cooled object. Assuming that the heat flow between each stage and the heat reservoirs obey Newton’s law, we have:

_{1}and TEM

_{2},

_{ci}and Q

_{hi}(i = 1, 2), for each TEM are given by:

_{1}and TEM

_{2}, respectively. Equations (5)–(8) contain three terms, namely αIT, Peltier heat, I

^{2}R, the internal heat generated by the Joulean loss and K(T

_{1}− T

_{2}), conduction heat loss. Clearly, the properties of different thermoelectric semiconductor materials of each TEM are included in the above equations.

#### 3.1. Equivalent Dual TEC System

_{h},

_{c},

_{eqH}and α

_{eqC}are the effective Seebeck coefficients for the hot and cold side, respectively. These equivalent parameters not only combine all intrinsic thermoelectric properties of both modules, but also they are significantly influenced by external thermal conductances. In Section 4.4, we will show that the numerical behavior of the figure of merit for the whole system satisfies Bergman’s theorem. Notice that the equivalent heat balance equations of the system (9) and (13), depend only on the temperatures T

_{c}and T

_{h}. Furthermore, we will use these results and recover, as limit cases, the design parameters for TEC system, which have been previously studied by other authors; see Section 4.5.

#### 3.2. Dimensionless Equivalent Heat Balance Equations

_{eq}is the equivalent figure of merit and µ is a coupling factor. Notice that we recover the dimensionless quantities q

_{h}and q

_{c}, which have been considered as the entropy flow normalized by the intrinsic thermal conductances, mK

_{1}+ nK

_{2}, of the inhomogeneous TEC system.

_{eq}, Equation (23), can be written in terms of the number of pairs of each TEM as:

_{r}, are given by:

## 4. Results and Discussion

_{c}and the COP of the TEC system. In our calculations, we use values T

_{c}= 286K and T

_{h}= 296K for the temperatures of the cold and hot reservoirs, respectively.

#### 4.1. Cooling Capacity q_{c}: External Conductances Match

_{c}is reached as the ratio K

_{ch}decreases for a given value of φ when K

_{c}< K

_{h}. As the external conductance ratio decreases, the change in maximal values of the cooling capacity are determined by the electrical current φ. Moreover, the limit values of q

_{c}are determined by the ratio K

_{ch}. Thus, the condition K

_{c}< K

_{h}is bounded, because q

_{c}will reach limit values. Notice that it is possible to reach the same limit value of q

_{c}, using different values of electrical current, φ.

_{c}, in terms of any two thermoelectric parameters (φ, K

_{ch}, Z

_{r}) of the equivalent TEC system. For example, Figure 3 shows the behavior of q

_{c}as a function of the figure of merit ratio Z

_{r}and external conductances ratio, K

_{ch}, for different values of electrical current, φ.

_{c}includes several effects that are contained in different planes of Figure 3. For example, maximum values for cooling capacity q

_{c}are shown in the plane q

_{c}vs. Z

_{r}, at different values of electrical current φ, for small figure of merit ratios, Z

_{r}, and thermal conductances ratios, K

_{ch}. This result is more clearly shown in Figure 4.

_{ch}, and (2) the figure of merit ratio, Z

_{r}, on the cooling capacity q

_{c}at different working conditions. These effects are shown in Figures 11 and 12 of the Appendix.

_{c}of both φ and Z

_{r}, in Figure 5, for a given value of K

_{ch}.

_{c}in terms of φ and Z

_{r}for constant values of other parameters, namely Z

_{1}, T

_{h}, θ, K

_{ch}and D, which determine the working conditions of the composite TEC system.

#### 4.2. Coefficient of Performance (COP): External Conductances Match

_{ch}, Z

_{r}and φ, respectively.

_{ch}changes. This fact shows that it is not possible to optimize the system by just fixing the K

_{ch}, but also a limit for the external conductances K

_{ch}exists. Table 1 shows the behavior of limit values for COP as we change the ratio K

_{ch}.

_{c}and COP. A maximum value for q

_{c}is obtained for high values of φ; meanwhile, the maximum values of COP are obtained in low ranges of φ. In Section 4.3, we obtain the qualitative behavior of COP for a commercial composite TEM connected in series.

#### 4.3. Numerical Validation

_{ch}, δ and ρ used in our calculations, only for completeness reasons.

_{1}= α

_{2}= α

_{Luo}, R

_{1}= R

_{2}= R

_{Luo}and K

_{1}= K

_{2}= K

_{Luo}, for a composite TEM connected in series proposed by Luo [15] for obtaining Figure 9. The behavior of the COP, shown in Figure 9, is very near to that reported in the datasheet for the SP − 254 − 1.0−2.5 (series) TEM connected electrically in series, with the same thermoelectric properties for each component of the TEM [16].

#### 4.4. Role of the Equivalent Figure of Merit

_{eq}, in the q

_{c}and COP parameters in terms of the number of pairs, n and m, and the figure of merit of each component TEM, Z

_{1}and Z

_{2}.

_{1}= 0.0027, we obtain Figure 10, which shows the equivalent figure of merit of the system as a function of the number pairs of each TEM.

_{r}= 1, the equivalent figure of merit Z

_{eq}becomes independent of the number of pairs. On the other hand, if Z

_{r}≠ 1, the equivalent figure of merit approaches Z

_{r}= 1, as the number of pairs increases. Meanwhile, if the number of pairs ratio decreases, the value of Z

_{eq}tends to the figure of merit that satisfies the Z

_{r}. Notice that the value of Z

_{eq}is not greater that the values of the figure of merit for each TEM of the system, independently of the number, n or m, of the number of pairs. Thus, the result is according to the theorem of Bergman [17], which says that the equivalent figure of merit of the TEM, can only be lower than the highest Z of the more efficient TEM. The value of Z

_{eq}is affected by the figure of merit ratio Z

_{r}. When Z

_{r}is lower than the unit, or Z

_{2}> Z

_{1}, it is possible to increase the values of the equivalent figure of merit by having a lower number of pairs ratio n > m, but it is not possible to get a higher value than the Z

_{2}that satisfies the Z

_{r}. Our analysis is consistent with the obtained results for thermoelectric generator systems [8] because of the generality of Bergman Theorem.

#### 4.5. Simplified Approaches: Previously Considered Cases

_{c}= 0, it is possible to get the full solution for the temperature ratio θ,

_{h}:

_{max}can be approximated through a power series expansion, leading to:

_{max}into Equation (34) yields:

_{c}>> (mK

_{1}+ nK

_{2}), the maximum dimensionless current and temperature ratio are respectively reduced to:

_{1}= α

_{2}= α, R

_{1}= R

_{2}= R and K

_{1}= K

_{2}= K, then the equivalent figure of merit becomes the ordinary one, and from Equations (37) and (38) we obtain,

## 5. Conclusions

_{(1},

_{2)}, thermal conductivity, K

_{(1},

_{2)}, the electrical resistance, R

_{(1},

_{2)}, of each TEM and external thermal conductances K

_{(}

_{c,h}

_{)}. The corresponding dimensionless heat balance equations may be very useful for the design of composite thermoelectric systems, because they show the relation of the thermoelectric parameters of the TEM components of the system. The main parameters of the TEC system are external thermal conductances of heat exchangers, K

_{ch}, and the figures of merit ratio, Z

_{r}. The obtained results from this approach show the effect of two or more thermoelectric parameters on the COP and q

_{c}of the system. In this work, it is shown that the maximum values for COP and q

_{c}are determined by the external thermal conductances ratio K

_{ch}, with the condition K

_{c}< K

_{h}. In general, we have shown (see Section 4) that our approach permits us to determine the optimal values of q

_{c}and COP for different working conditions determined. The obtained results in this work are useful for designing composite TEC systems with thermal coupling, K

_{ch}. We show the consistency of our approach obtaining the COP as a function of the input voltage for a commercial thermoelectric module connected electrically in series. Furthermore, we have derived results previously obtained by many authors as limit cases of our approach. Finally, we point out that the proposed approach in this work can be easily extended to include many TEMs with different thermoelectric properties, each one connected thermally in parallel and electrically in series.

## Acknowledgments

## A. Effect of the Thermoelectric Parameters in COP and Q_{c}

_{c}of the system. Notice that several results are included in the 3D figures discussed in the above sections. The figures in this Appendix show important limit intervals of the thermoelectric parameters.

#### A.1. Cooling Capacity q_{c}

_{c}in terms of thermal conductances, K

_{c}and K

_{h}, electrical current, φ, and the figure of merit, Z.

_{ch}= K

_{c}/K

_{h}, on the cooling capacity q

_{c}. At the same electrical current φ, we can obtain different maximum values for q

_{c}for a given value of Z

_{r}= Z

_{1}/Z

_{2}.

_{ch}, the behavior of cooling capacity q

_{c}vs. the electrical current is shown in Figure 12 when we change the figures of merit ratio Z

_{r}.

#### A.2. Coefficient of Performance

_{ch}, electrical current, φ, and figure of merit, Z, see Figures 13–15. Our results contained in the Appendix show important intervals of limit values for the thermoelectric parameters. For example, as Z

_{r}increases in the TEMs, clearly, for the same current, we have different cooling capacities. In Figure 12, we show that there are intervals for φ in which q

_{c}is zero for a given Z

_{r}value. If Z of any TEM increases, q

_{c}increases also. Figure 4 shows some intervals for maximum values of q

_{c}for this condition.

## Author Contributions

## Conflicts of Interest

## Nomenclature

COP | Coefficient of performance |

D | Number of pair ratio |

I | Electric current through thermoelements (A) |

K | Thermal conductance of thermoelement |

K_{c} | Thermal conductance of cold-end heat exchanger
$(\frac{W}{K})$ |

K_{ch} | Thermal conductance of heat exchangers ratio |

K_{h} | Thermal conductance of hot-end heat exchanger
$(\frac{W}{K})$ |

m | Total number of thermoelements at first TEM |

n | Total number of thermoelements at second TEM |

q_{c} | Dimensionless cooling capacity |

q_{h} | Dimensionless heat rejection |

Q_{c} | Cooling capacity of the TEC system (W) |

Q_{h} | Heat rejection of the TEC system (W) |

R | Electric resistance of thermoelement (Ω) |

T_{1} | Hot end temperature of TEMs (K) |

T_{2} | Cold end temperature of TEMs (K) |

T_{c} | Temperature of the cold reservoir (K) |

T_{h} | Temperature of the hot reservoir (K) |

V | Input voltage (V) |

Z | Figure of merit
$(\frac{1}{K})$ |

Z_{r} | Figures of merit ratio |

Z_{eq} | Equivalent figure of merit |

Greek Letters | |
---|---|

α | Seebeck coefficient of thermoelement
$(\frac{V}{K})$ |

δ | Cold conductance ratio |

∆T | Temperature across the TEMs (K) |

ρ | Hot conductance ratio |

θ | Temperature of heat reservoirs ratio |

φ | Dimensionless current |

µ | Coupling factor |

Subscripts | |
---|---|

1 | First TEM |

2 | Second TEM |

eqH | Equivalent for hot side |

eqC | Equivalent for cold side |

ch | Heat exchangers conductances ratio |

Lou | Thermoelectric value obtained through Z. Luo’s method |

r | Ratio |

max | Maximum |

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**Figure 2.**Cooling capacity vs. the external conductance ratio with different dimensionless currents.

**Figure 3.**Cooling capacity vs. the external conductance ratio with different figure of merit ratios and dimensionless current.

**Figure 7.**COP vs. the external conductances ratio with different figures of merit and dimensionless current.

**Figure 9.**COP vs. the input voltage corresponding to an SP − 254 − 1.0 − 2.5 (series) TEC module connected in series.

**Figure 10.**Equivalent figure of merit vs. the number of pairs ratio with different figure of merit ratios.

**Figure 11.**Cooling capacity vs. dimensionless current with different external thermal conductances ratios.

K_{ch} | COP | ∆COP |
---|---|---|

0 | 1.9866 | 0.045 |

0.05 | 1.9416 | 0.0437 |

0.10 | 1.8979 | 0.0424 |

0.15 | 1.8555 | 0.0412 |

0.20 | 1.8143 | 0.0400 |

0.25 | 1.7743 | 0.0389 |

0.30 | 1.7354 | 0.0378 |

0.35 | 1.6976 | 0.0368 |

0.40 | 1.6608 | 0.0358 |

0.45 | 1.6250 | 0.0349 |

0.50 | 1.5901 | 0.0339 |

0.55 | 1.5562 | 0.0331 |

0.60 | 1.5231 | 0.0322 |

0.65 | 1.4909 | 0.0314 |

0.70 | 1.4595 | 0.0306 |

0.75 | 1.4289 | 0.0298 |

0.80 | 1.3991 | 0.0292 |

0.85 | 1.3699 | 0.0284 |

0.90 | 1.3415 | 0.0278 |

0.95 | 1.3137 | 0.0271 |

K_{ch} | 0.3 | 0.5 | 1 | 2 |
---|---|---|---|---|

δ | 0.3522 | 0.3522 | 0.3522 | 0.3522 |

ρ | 0.10566 | 0.1761 | 0.3522 | 0.7044 |

Z_{r} | α_{2} | R_{2} | K_{2} | Z_{2} |
---|---|---|---|---|

0.1 | 0.0170 | 0.341 | 0.0313 | 0.0273 |

0.5 | 0.0381 | 1.705 | 0.1565 | 0.0054 |

1 | 0.054 | 3.41 | 0.313 | 0.0027 |

2 | 0.0766 | 6.82 | 0.626 | 0.0013 |

3 | 0.0935 | 10.23 | 0.939 | 0.0009 |

© 2015 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Flores-Niño, C.Y.; Olivares-Robles, M.A.; Loboda, I. General Approach for Composite Thermoelectric Systems with Thermal Coupling: The Case of a Dual Thermoelectric Cooler. *Entropy* **2015**, *17*, 3787-3805.
https://doi.org/10.3390/e17063787

**AMA Style**

Flores-Niño CY, Olivares-Robles MA, Loboda I. General Approach for Composite Thermoelectric Systems with Thermal Coupling: The Case of a Dual Thermoelectric Cooler. *Entropy*. 2015; 17(6):3787-3805.
https://doi.org/10.3390/e17063787

**Chicago/Turabian Style**

Flores-Niño, Cuautli Yanehowi, Miguel Angel Olivares-Robles, and Igor Loboda. 2015. "General Approach for Composite Thermoelectric Systems with Thermal Coupling: The Case of a Dual Thermoelectric Cooler" *Entropy* 17, no. 6: 3787-3805.
https://doi.org/10.3390/e17063787