# Optimization of Radiators, Underfloor and Ceiling Heater Towards the Definition of a Reference Ideal Heater for Energy Efficient Buildings

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## Abstract

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## 1. Introduction

## 2. Method

## 3. Results

#### 3.1. Panel radiator

- given the same area, the efficiency varies with height,
- an ideal convector with 100% convection performs worse than panel radiators,
- the 10-type can be identified as our ideal heater,
- the operative temperature is more dependent on the height than on the width,
- there exists an ideal width range for 21-type radiators.

#### 3.2. Underfloor Heating

#### 3.3. Ceiling Heater

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. View Factors and Operative Temperature Formulas

**Figure A1.**Operative temperatures for a 21-type, IDA ICE cross-check: arithmetic (crosses) vs. weighted average for different clothing surface temperatures.

**Figure A3.**Comparison of surface temperatures for a 21-type radiator, $h=0.9$ m. IDA ICE (crosses) vs. 4th order polynomial interpolation (diamonds).

**Table A1.**Operative temperature ${t}_{op}$ (${}^{\circ}$C) in function of panel width and height, 10-type.

Height (m) | 0.30 | $-0.0046{w}^{2}+0.0357w+19.833$ |

0.45 | $-0.0066{w}^{2}+0.0494w+19.839$ | |

0.60 | $-0.0069{w}^{2}+0.053w+19.851$ | |

0.90 | $-0.0111{w}^{2}+0.0786w+19.859$ | |

Width (m) | 0.60 | - |

0.80 | - | |

1.20 | $0.1079h+19.84$ | |

1.60 | $0.1268h+19.842$ | |

2.00 | $0.1404h+19.846$ | |

2.60 | $0.1489h+19.853$ | |

3.00 | $0.1403h+19.86$ |

**Table A2.**Operative temperature ${t}_{op}$ (${}^{\circ}$C) in function of panel width and height, 21-type.

Height (m) | 0.30 | $-0.0031{w}^{2}+0.0216w+19.766$ |

0.45 | $-0.0049{w}^{2}+0.0355w+19.773$ | |

0.60 | $-0.0054{w}^{2}+0.042w+19.778$ | |

0.90 | $-0.007{w}^{2}+0.0573w+19.788$ | |

Width (m) | 0.60 | $0.0593h+19.765$ |

0.80 | $0.0773h+19.761$ | |

1.20 | $0.0938h+19.763$ | |

1.60 | $0.1101h+19.764$ | |

2.00 | $0.125h+19.764$ | |

2.60 | $0.1408h+19.764$ | |

3.00 | $0.1509h+19.763$ |

$\mathbf{A}\left(\mathbf{h}\right)$ | $-0.1074{h}^{3}+0.1828{h}^{2}-0.1045h+0.0132$ |

$\mathbf{B}\left(\mathbf{h}\right)$ | $0.6012{h}^{3}-1.0361{h}^{2}+0.6114h-0.0707$ |

$\mathbf{C}\left(\mathbf{h}\right)$ | 19.846 |

$\mathbf{A}\left(\mathbf{w}\right)$ | $-0.0234{w}^{2}+0.1174w+3\times {10}^{-5}$ |

$\mathbf{B}\left(\mathbf{w}\right)$ | $0.0039{w}^{2}-0.0054w+19.841$ |

$\mathbf{A}\left(\mathbf{h}\right)$ | $-0.0061h-0.0017$ |

$\mathbf{B}\left(\mathbf{h}\right)$ | $0.3025{h}^{3}-0.5728{h}^{2}+0.3929h-0.0529$ |

$\mathbf{C}\left(\mathbf{h}\right)$ | $0.0741{h}^{3}-0.1444{h}^{2}+0.1233h+19.74$ |

$\mathbf{A}\left(\mathbf{w}\right)$ | $-0.0084{w}^{2}+0.0667w+0.0254$ |

$\mathbf{B}\left(\mathbf{w}\right)$ | 19.763 |

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**Figure 5.**Calculation of the mean radiant temperature from IDA ICE [26].

**Figure 6.**Calculation of the mean radiant temperature from ISO 7726 [25].

**Figure 7.**IDA ICE operative temperatures for a 10-type radiator and convector in function of the panel area.

**Figure 8.**IDA ICE operative temperatures for a 21-type radiator and convector in function of the panel area.

**Figure 9.**Operative temperatures for a 10-type panel radiator, with $h=0.3,\phantom{\rule{0.166667em}{0ex}}0.45,\phantom{\rule{0.166667em}{0ex}}0.6,\phantom{\rule{0.166667em}{0ex}}0.9$ m.

**Figure 10.**Operative temperatures for a 21-type panel radiator, with $h=\phantom{\rule{0.166667em}{0ex}}0.3,\phantom{\rule{0.166667em}{0ex}}0.45,\phantom{\rule{0.166667em}{0ex}}0.6,\phantom{\rule{0.166667em}{0ex}}0.9$ m.

**Figure 11.**First derivative of the analytical operative temperature for a 21-type radiator, $h=0.3$ m.

**Figure 12.**First derivative of the analytical operative temperature for a 21-type radiator, $h=0.9$ m.

**Table 1.**Projected area factors of a person [25].

Standing | Seated | |
---|---|---|

Up/down | 0.08 | 0.18 |

Left/right | 0.23 | 0.22 |

Front/back | 0.35 | 0.30 |

Emitter Type | Emitter Surf. | Floor | Ceiling | Sidewalls | Backwall | External Wall |
---|---|---|---|---|---|---|

10-type radiator | 33.83 | 19.78 | 19.80 | 19.72 | 19.74 | 18.59 |

21-type radiator | 32.04 | 19.61 | 19.71 | 19.60 | 19.62 | 18.53 |

UFH, square | 24.66 | 19.89 | 20.03 | 20.01 | 20.03 | 18.92 |

Ceiling heater, square | 31.78 | 20.00 | 19.90 | 19.95 | 19.96 | 18.87 |

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**MDPI and ACS Style**

Ferrantelli, A.; Võsa, K.-V.; Kurnitski, J. Optimization of Radiators, Underfloor and Ceiling Heater Towards the Definition of a Reference Ideal Heater for Energy Efficient Buildings. *Appl. Sci.* **2018**, *8*, 2477.
https://doi.org/10.3390/app8122477

**AMA Style**

Ferrantelli A, Võsa K-V, Kurnitski J. Optimization of Radiators, Underfloor and Ceiling Heater Towards the Definition of a Reference Ideal Heater for Energy Efficient Buildings. *Applied Sciences*. 2018; 8(12):2477.
https://doi.org/10.3390/app8122477

**Chicago/Turabian Style**

Ferrantelli, Andrea, Karl-Villem Võsa, and Jarek Kurnitski. 2018. "Optimization of Radiators, Underfloor and Ceiling Heater Towards the Definition of a Reference Ideal Heater for Energy Efficient Buildings" *Applied Sciences* 8, no. 12: 2477.
https://doi.org/10.3390/app8122477