# Operator-Based Nonlinear Control of Calorimetric System Actuated by Peltier Device

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## Abstract

**:**

## 1. Introduction

## 2. Presentation of the System and Control System Design

#### 2.1. The Structure of the Peltier Device

#### 2.2. Principles of the Calorimetric System Using the Peltier Device

#### 2.3. Modeling of the System

#### 2.4. Temperature Dependence of Thermal Conductivity

#### 2.5. Operator-Based Nonlinear Control Feedback System Design

#### 2.6. Experimental System

## 3. Results and Discussion

#### 3.1. Simulation Results

#### 3.2. Experimental Results

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

DUT | Device under test |

## References

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**Figure 1.**The structure of the Peltier device [20].

**Figure 2.**The structure of the calorimetric system using the Peltier device [5].

**Figure 3.**Temperature dependence of thermal conductivity [19].

**Figure 6.**Implementation of the experimental system [5].

**Figure 7.**Realized experimental system [5].

**Figure 8.**Temperature of ${T}_{\mathrm{in}}$, ${T}_{\mathrm{amb}}$, ${T}_{\mathrm{h}}$, ${T}_{\mathrm{c}}.$

**Figure 11.**Temperature of ${T}_{\mathrm{in}}$, ${T}_{\mathrm{amb}}$, ${T}_{\mathrm{h}}$, ${T}_{\mathrm{c}}$ (w/o operator theory).

**Figure 12.**Temperature of ${T}_{\mathrm{in}}$, ${T}_{\mathrm{amb}}$, ${T}_{\mathrm{h}}$, ${T}_{\mathrm{c}}$ (w/ operator theory).

Parameter | Definition | Value |
---|---|---|

${S}_{\mathrm{p}}$ | Seebeck coefficient | (V/K) |

$\lambda $ | Thermal conductivity | (W/mK) |

A | Area of the Peltier device | $({\mathrm{m}}^{2})$ |

d | Thickness of the Peltier device | (m) |

${r}_{\mathrm{p}}$ | Internal resistance of the Peltier device | $(\mathsf{\Omega})$ |

${T}_{\mathrm{h}}$ | Hot side temperature | (K) |

${T}_{\mathrm{c}}$ | Cold side temperature | (K) |

I | Current | (A) |

Parameter | Definition | Value |
---|---|---|

q | Heat flux | (K/m) |

$\frac{dT}{dx}$ | Temperature gradient inside solid | (K/m) |

$\lambda $ | Thermal conductivity | (W/mK) |

$\alpha $ | Heat transfer coefficient | $(\mathrm{W}/{\mathrm{m}}^{2}\mathrm{K})$ |

T | Temperature of solid | (K) |

${T}_{\mathrm{m}}$ | Temperature of fluid | (K) |

Parameter | Definition | Value |
---|---|---|

${S}_{\mathrm{p}}$ | Thermopower | 0.0411 V/K |

${r}_{\mathrm{p}}$ | Resistance of Peltier device | 1.51 $\mathsf{\Omega}$ |

${T}_{\mathrm{amb}}$ | Outside temperature | 21 ${}^{\circ}$C |

${R}_{\mathrm{in}}$ | Thermal resistance of internal air | 166.7 K/W |

${R}_{\mathrm{c}}$ | Thermal resistance of cold-side cooler | 0.4 K/W |

${R}_{\mathrm{h}}$ | Thermal resistance of hot-side cooler | 0.15 K/W |

${R}_{\mathrm{d}}$ | Thermal resistance of DUT | 3 K/W |

${R}_{\mathrm{w}}$ | Thermal resistance of chamber | 2 K/W |

${C}_{\mathrm{in}}$ | Heat capacity of internal air | 15.98 J/K |

${C}_{\mathrm{c}}$ | Heat capacity of cold-side cooler | 100 J/K |

${C}_{\mathrm{h}}$ | Heat capacity of hot-side cooler | 70 J/K |

${C}_{\mathrm{d}}$ | Heat capacity of DUT | 10 J/K |

${Q}_{\mathrm{in}}$ | Power dissipation of DUT | 15 W |

${Q}_{{\mathrm{F}}_{\mathrm{h}}}$ | Power dissipation of hot-side cooler | 1.25 W |

${Q}_{{\mathrm{F}}_{\mathrm{c}}}$ | Power dissipation of cold-side cooler | 1.25 W |

${A}_{\mathrm{p}}$ | Area of Peltier device | 1.6 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ ${\mathrm{m}}^{2}$ |

${d}_{\mathrm{a}}$ | Thickness of ceramic plate | 1 mm |

${d}_{\mathrm{s}}$ | Thickness of semiconductor | 2 mm |

${\lambda}_{\mathrm{c}}$ | Thermal conductivity of alumina | 32 W/mK |

${K}_{{\mathrm{P}}_{\mathrm{T}}}$ | Proportional gain | 0.05 |

${K}_{{\mathrm{I}}_{\mathrm{T}}}$ | Integral gain | 0.0005 |

${K}_{{\mathrm{P}}_{\mathrm{i}}}$ | Proportional gain | 0.4398 |

${K}_{{\mathrm{I}}_{\mathrm{i}}}$ | Integral gain | 197.4 |

B | Design parameter | 0.992 |

Parameter | Definition | Value |
---|---|---|

N | Number of atoms in a crystal | 2 |

V | Volume of the unit lattice | 1.065 ×$\phantom{\rule{3.33333pt}{0ex}}{10}^{-28}$ ${\mathrm{m}}^{3}$ |

${c}_{\mathrm{m}}$ | Effective sound velocity | 1790 m/s |

${L}_{\mathrm{o}}$ | Lorentz number | 2.44 ×$\phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ W$\mathsf{\Omega}$/${\mathrm{K}}^{2}$ |

$\sigma $ | Electrical conductivity | 5.13 ×$\phantom{\rule{3.33333pt}{0ex}}{10}^{4}$ 1/$\mathsf{\Omega}$m |

c | Heat capacity of phonon | 25 J/mol$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$K |

a | Dimensionless parameter | 0.95 ×$\phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

b | Dimensionless parameter | 0.5 |

${k}_{\mathrm{B}}$ | Boltzmann’s constant | 1.38 ×$\phantom{\rule{3.33333pt}{0ex}}{10}^{-23}$ J/K |

ℏ | Dirac’s constant | 1.054 ×$\phantom{\rule{3.33333pt}{0ex}}{10}^{-34}$ J$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$s |

${\theta}_{\mathrm{D}}$ | Debye temperature | 660.9 K |

**Table 5.**Materials used for calorimetric system [5].

Peltier device | TEC1-12706: Cooling capacity 50 W | |

Dimensions: 40 mm × 40 mm × 3.9 mm | ||

Sensors | Temperature | Pt100: Accuracy ±0.01 ${}^{\circ}$C |

Current | CASR 6-NP: Accuracy 0.8% | |

Chamber | Expanded polystyrene: | |

Dimensions: 300 mm × 300 mm × 300 mm, Wall thickness: 30 mm | ||

DUT | Aluminum resistor: Resistance 3.3 $\mathsf{\Omega}$ | |

Dimensions: 148 mm × 210 mm × 0.5 mm | ||

Coolers | Hot side | TY-140: |

Dimensions: 152 mm × 140 mm × 26.5 mm | ||

Cold side | CC-Siberian-01: | |

Dimensions: 120 mm × 96 mm × 66 mm |

Parameter | Definition | Value |
---|---|---|

${T}_{\mathrm{e}}$ | Experiment time | 1500 s |

${Q}_{\mathrm{in}}$ | Power dissipation of DUT | 5, 10, 15 W |

${T}_{{\mathrm{S}}_{\mathrm{T}}}$ | Sampling time of temperature | 1 s |

${K}_{{\mathrm{P}}_{\mathrm{T}}}$ | Proportional gain (w/o operator theory) | 0.5 |

${K}_{{\mathrm{I}}_{\mathrm{T}}}$ | Integral gain (w/o operator theory) | 0.05 |

${K}_{{\mathrm{P}}_{\mathrm{T}}}$ | Proportional gain (w/ operator theory) | 0.05 |

${K}_{{\mathrm{I}}_{\mathrm{T}}}$ | Integral gain (w/ operator theory) | 0.0005 |

w/o Operator Theory | w/ Operator Theory | |
---|---|---|

w/o Temperature dependence of thermal conductivity | 700 s | 600 s |

w/ Temperature dependence of thermal conductivity | 600 s | 500 s |

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**MDPI and ACS Style**

Chikaraishi, R.; Deng, M.
Operator-Based Nonlinear Control of Calorimetric System Actuated by Peltier Device. *Machines* **2021**, *9*, 174.
https://doi.org/10.3390/machines9080174

**AMA Style**

Chikaraishi R, Deng M.
Operator-Based Nonlinear Control of Calorimetric System Actuated by Peltier Device. *Machines*. 2021; 9(8):174.
https://doi.org/10.3390/machines9080174

**Chicago/Turabian Style**

Chikaraishi, Ryo, and Mingcong Deng.
2021. "Operator-Based Nonlinear Control of Calorimetric System Actuated by Peltier Device" *Machines* 9, no. 8: 174.
https://doi.org/10.3390/machines9080174