# Design of an Algorithm for Modeling Multiple Thermal Zones Using a Lumped-Parameter Model

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## Abstract

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## 1. Introduction

_{2}emissions, especially in cities and buildings, which, according to additional research, can consume up to $40\%$ of the annual energy production [7,8,9].

## 2. Mathematical Model

- The identification of geometrical and physical parameters of the thermal zones, including internal loads.
- RC circuit construction and the calculation of resistors and capacitors
- Define the equations of the dynamic system, using the theory of energy flow transfer.

#### 2.1. Case 1: A Single Thermal Zone (m = 1)

#### 2.2. Case 2: Two Thermal Zones (m = 2)

#### 2.3. Case 3: Three Thermal Zones ($m=$ 3)

#### 2.4. Case 4: Four Thermal Zones ($m=$ 4)

## 3. Algorithm Design

Algorithm 1 Algorithm to design differential equation systems for multiple thermal zones |

## 4. Experimental Development

#### Tuning Process

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

HVAC | Heat, ventilation and air conditioning | |

LPM | Lumped-parameter model | |

RC | Resistor and capacitor circuit | |

Nomenclature | ||

A | Superficial area | ${\mathrm{m}}^{2}$ |

$Ce$ | Specific heat | $\frac{\mathrm{kJ}}{\mathrm{kg}\xb7\mathrm{K}}$ |

$Cr$ | Air thermal capacity | $\frac{\mathrm{kJ}}{\mathrm{K}}$ |

$Cw$ | Walls’ thermal capacity | $\frac{\mathrm{kJ}}{\mathrm{K}}$ |

e | Error rate | % |

${F}_{objective}$ | Objective function | |

$hi$ | Internal heat convection | $\frac{\mathrm{kJ}}{\mathrm{h}\xb7\mathrm{m}\xb7\mathrm{K}}$ |

$he$ | External heat convection | $\frac{\mathrm{kJ}}{\mathrm{h}\xb7\mathrm{m}\xb7\mathrm{K}}$ |

$kt$ | Thermal conductivity | $\frac{\mathrm{kJ}}{\mathrm{h}\xb7{\mathrm{m}}^{2}\xb7\mathrm{K}}$ |

L | Thickness | m |

m | Thermal zones number | |

N | Number of equations | |

R | Conduction thermal resistance | $\frac{\mathrm{h}\xb7\mathrm{K}}{\mathrm{kJ}}$ |

${R}_{in}$ | Internal convection resistance | $\frac{\mathrm{h}\xb7\mathrm{K}}{\mathrm{kJ}}$ |

${R}_{ex}$ | External convection resistance | $\frac{\mathrm{h}\xb7\mathrm{K}}{\mathrm{kJ}}$ |

$Rw$ | Inner envelope walls | $\frac{\mathrm{h}\xb7\mathrm{K}}{\mathrm{kJ}}$ |

u | Contact matrix | |

T | Internal temperature | °C |

$Tw$ | Walls’ temperature | °C |

${T}_{exterior}$ | Environmental temperature | °C |

${T}_{ground}$ | Ground temperature | °C |

$\alpha $ | Heat transfer coefficient between zones | $\frac{\mathrm{kJ}}{\mathrm{h}\xb7\mathrm{K}}$ |

$\beta $ | Heat transfer coefficient with environmental conditions | $\frac{\mathrm{kJ}}{\mathrm{h}\xb7\mathrm{K}}$ |

$\varphi $ | Heat transfer coefficient with ground | $\frac{\mathrm{kJ}}{\mathrm{h}\xb7\mathrm{K}}$ |

$\rho $ | Density | $\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}$ |

## Appendix A

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**Figure 10.**Thermal zone 1 with internal heating load. (

**a**) Single thermal zone. (

**b**) Single thermal zone with internal load.

**Figure 11.**Thermal zones for cases 2, 3 and 4 with internal heating load. (

**a**) Case 1. (

**b**) Case 2. (

**c**) Case 3.

Geometrical Parameters | ||||
---|---|---|---|---|

x | y | z | ${L}_{d}$ | ${L}_{w}$ |

0.111 m | 0.111 m | 0.2 m | 0.005 m | 0.008 m |

Physical Parameters | ||||

$\rho $ | Ce | kt | ||

$150.7485\phantom{\rule{3.33333pt}{0ex}}\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}$ | $3.0512\phantom{\rule{3.33333pt}{0ex}}\frac{\mathrm{kJ}}{\mathrm{kg}\xb7\xb0\mathrm{C}}$ | $0.1882\phantom{\rule{3.33333pt}{0ex}}\frac{\mathrm{kJ}}{\mathrm{m}\xb7\mathrm{h}\xb7\xb0\mathrm{C}}$ |

$\mathit{hi}\phantom{\rule{3.33333pt}{0ex}}(\frac{\mathbf{kJ}}{\mathbf{h}\xb7\mathbf{m}\xb7\mathbf{K}})$ | $\mathit{he}\phantom{\rule{3.33333pt}{0ex}}(\frac{\mathbf{kJ}}{\mathbf{h}\xb7\mathbf{m}\xb7\mathbf{K}})$ | |||||||
---|---|---|---|---|---|---|---|---|

Zone | 1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 |

Case 1 | 86.9 | 0 | 0 | 0 | 12.36 | 0 | 0 | 0 |

Case 2 | 34.88 | 0.0022 | 0 | 0 | 12.29 | 58.77 | 0 | 0 |

Case 3 | 179.54 | 0.0047 | 0 | 0 | 2.75 | 199.9 | 179.99 | 0 |

Case 4 | 37.44 | 0.0037 | 0.0047 | 0.0047 | 0.0001 | 60.95 | 61.70 | 103.14 |

$\mathit{hi}\phantom{\rule{3.33333pt}{0ex}}(\frac{\mathbf{kJ}}{\mathbf{h}\xb7\mathbf{m}\xb7\mathbf{K}})$ | $\mathit{he}\phantom{\rule{3.33333pt}{0ex}}(\frac{\mathbf{kJ}}{\mathbf{h}\xb7\mathbf{m}\xb7\mathbf{K}})$ | |||||||
---|---|---|---|---|---|---|---|---|

Zone | 1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 |

Case 1 | 0.6576 | 0 | 0 | 0 | 0.0145 | 0 | 0 | 0 |

Case 2 | 0.0025 | 0.001 | 0 | 0 | 3.39 | 21.45 | 0 | 0 |

Case 3 | 0.0027 | 0.0018 | 0.00027 | 0 | 18.65 | 0.0008 | 0.8387 | 0 |

Case 4 | 0.0027 | 0.0008 | 0.0008 | 0.0008 | 0.0008 | 4.08 | 1.89 | 4.67 |

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## Share and Cite

**MDPI and ACS Style**

Fernández de Córdoba, P.; Montes, F.F.; Martínez, M.E.I.; Carmenate, J.G.; Selvas, R.; Taborda, J.
Design of an Algorithm for Modeling Multiple Thermal Zones Using a Lumped-Parameter Model. *Energies* **2023**, *16*, 2247.
https://doi.org/10.3390/en16052247

**AMA Style**

Fernández de Córdoba P, Montes FF, Martínez MEI, Carmenate JG, Selvas R, Taborda J.
Design of an Algorithm for Modeling Multiple Thermal Zones Using a Lumped-Parameter Model. *Energies*. 2023; 16(5):2247.
https://doi.org/10.3390/en16052247

**Chicago/Turabian Style**

Fernández de Córdoba, Pedro, Frank Florez Montes, Miguel E. Iglesias Martínez, Jose Guerra Carmenate, Romeo Selvas, and John Taborda.
2023. "Design of an Algorithm for Modeling Multiple Thermal Zones Using a Lumped-Parameter Model" *Energies* 16, no. 5: 2247.
https://doi.org/10.3390/en16052247