# Estimation of the Heating Time of Small-Scale Buildings Using Dynamic Models

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## Abstract

**:**

## 1. Introduction

## 2. System Description

## 3. The Model Development

## 4. Results

#### Energy Cost

## 5. Discussion

- Even though the building is not officially ventilated, it may receive a small amount of air flow due to the opening of the building door, causing some amount of heat loss. Estimation of this heat loss becomes a tough task, because of the factors that influence it, such as the wind and how frequently the door is opened. Infiltration through the porous structures is beyond a certain lower limit for the mentioned test building, as it is constructed as a passive house.
- The model is affected by the Sun on sunny days and wind on windy days [21]. A constant solar radiation is included in the model depending on the sunrise and sunset times for each day. According to the simulation performed for Case 4, it was observed that increased solar irradiation has reduced the heating time by 1.8 h, while reducing the energy consumption. Solar irradiation has a prominent impact on the heating time of a building, as it supports raising the inside temperature without any power usage. Hence, for better results, the utilization of real solar irradiation measurements will be more powerful. The wind increases the heat transfer coefficient and, hence, the heat losses. However, the effect of wind is not addressed in the model.
- The accuracy of the measured input variables, such as outside temperature, can cause deviations.
- The calibrated model parameters may not be the optimum values.

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**Plan of the test building. Measurements are in mm [18].

**Figure 4.**Algorithm for calculating the heating time and the energy consumption. This is implemented using the MATLAB ode23s solver.

**Figure 9.**Variation of the internal and external air temperatures for Case 4. The effect of the change in solar irradiation is also presented.

**Figure 11.**Predicted and measured total energy consumption of the building for each simulated period.

Symbols | Description |
---|---|

A | Surface area (m${}^{2}$) |

${\widehat{c}}_{p}$ | Specific heat capacity (J/(kgK)) |

E | Total energy consumption (J) |

M | Molar mass of air (kg/mol) |

$\dot{Q}$ | Heat flow rate (W) |

${\dot{Q}}_{Loss}$ | Heat loss from each building |

envelope component (W) | |

R | Gas constant (J/(mol K)) |

T | Temperature (K) |

x | Thickness (m) |

t | Time (s) |

U | Overall heat transfer coefficient (W(m${}^{2}$K)) |

V | Volume (m${}^{3}$) |

α | Thermal diffusivity (m${}^{2}$/s) |

ρ | Density (kg/m${}^{3}$) |

Subscripts | Description |

r | Roof |

f | Floor |

i | Inside the building unit |

o | Outside the building unit |

$mw$ | Massiv wall |

$sw$ | Standard wall |

$bw$ | Bærekraftig wall |

Superscripts | Description |

s | Surface |

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

${\alpha}_{mw}$ | 3.4 × 10${}^{-7}$ m${}^{2}$/s |

${\alpha}_{sw}$ | 7.9 × 10${}^{-7}$ m${}^{2}$/s |

${\alpha}_{bw}$ | 2.8 × 10${}^{-6}$ m${}^{2}$/s |

${\alpha}_{f}$ | 3.6 × 10${}^{-6}$ m${}^{2}$/s |

${\alpha}_{r}$ | 5.6 × 10${}^{-6}$ m${}^{2}$/s |

U${}_{mw}$ | 0.8 W/(m${}^{2}$K) |

U${}_{sw}$ | 0.8 W/(m${}^{2}$K) |

U${}_{bw}$ | 0.1 W/(m${}^{2}$K) |

U${}_{f}$ | 0.8 W/(m${}^{2}$K) |

U${}_{r}$ | 0.6–1.2 W/(m${}^{2}$K) |

U${}_{window}$ | 1.2 W/(m${}^{2}$K) |

U${}_{door}$ | 1.2 W/(m${}^{2}$K) |

Case | Period | T${}_{\mathbf{setpoint}}$ | T${}_{\mathbf{inside}\mathbf{\_}\mathbf{start}}$ | T${}_{\mathbf{outside}\mathbf{\_}\mathbf{start}}$ | Average Inside Relative Humidity | Solar Irradiation | |
---|---|---|---|---|---|---|---|

From | To | ||||||

1 | 12 June 2015 | 13 June 2015 | |||||

11:30 | 19:00 | 38.3 ${}^{\circ}$C | 24.8 ${}^{\circ}$C | 23.8 ${}^{\circ}$C | 38.9% | 300 W/m${}^{2}$ | |

2 | 28 September 2015 | 29 September 2015 | |||||

18:00 | 18:00 | 29 ${}^{\circ}$C | 21 ${}^{\circ}$C | 13.8 ${}^{\circ}$C | 55.5% | 150 W/m${}^{2}$ | |

3 | 12 November 2015 | 14 November 2015 | |||||

16:20 | 23:50 | 24 ${}^{\circ}$C | 14.2 ${}^{\circ}$C | 2 ${}^{\circ}$C | 56.9% | 150 W/m${}^{2}$ | |

4 | 23 November 2015 | 25 November 2015 | |||||

10:55 | 02:55 | 17 ${}^{\circ}$C | 6.5 ${}^{\circ}$C | −4.5 ${}^{\circ}$C | 59.4% | 20 W/m${}^{2}$ |

Time Period | Rate (Nok/kWh) |
---|---|

01:00–05:30 | 0.65 |

05:30–09:00 | 1 |

09:00–14:30 | 0.85 |

14:30–20:00 | 1 |

20:00–01:00 | 0.75 |

Scenario | Time Period | Energy (kWh) | Cost (Nok) | |
---|---|---|---|---|

1 | a | 0.54 | 0.36 | 2.23 |

b | 2.25 | 1.87 | ||

2 | c | 2.25 | 1.47 | 2.25 |

d | 0.78 | 0.78 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Perera, D.W.U.; Skeie, N.-O.
Estimation of the Heating Time of Small-Scale Buildings Using Dynamic Models. *Buildings* **2016**, *6*, 10.
https://doi.org/10.3390/buildings6010010

**AMA Style**

Perera DWU, Skeie N-O.
Estimation of the Heating Time of Small-Scale Buildings Using Dynamic Models. *Buildings*. 2016; 6(1):10.
https://doi.org/10.3390/buildings6010010

**Chicago/Turabian Style**

Perera, Degurunnehalage Wathsala Upamali, and Nils-Olav Skeie.
2016. "Estimation of the Heating Time of Small-Scale Buildings Using Dynamic Models" *Buildings* 6, no. 1: 10.
https://doi.org/10.3390/buildings6010010