# Verification and Improving the Heat Transfer Model in Radiators in the Wide Change Operating Parameters

## Abstract

**:**

## 1. Introduction

- ${K}_{T},m$—radiator constants based on the test results.
- $\Delta {T}_{w}$—water temperature drop in the radiator

## 2. Description of the Existing State

## 3. NEHTMiR—New Extended Model of the Heat Transfer in Radiators

## 4. Purpuse and Scope of the Studies Carried Out

#### Test Methodology

- -
- temperature of the radiator supply water changed from 37.7 ${}^{\circ}$C to 68.8 ${}^{\circ}$C, the average value 51.2 ${}^{\circ}$C.
- -
- temperature of return from the radiator changed from 25.6 ${}^{\circ}$C to 60.8 ${}^{\circ}$C, the average value 40.0 ${}^{\circ}$C.
- -
- the radiator water mass flow varies from 0.0036 kg/s, 29.6% (nominal value) to 0.0256 kg/s, 212.1% (nominal value)—the average water mass flow 0.0121 kg/s
- -
- the radiator heat output varies from 122.9 W to 788.6 W—the average heat output 409.0 W.

- -
- temperature of the radiator supply water 70.2 ${}^{\circ}$C + −0.4 K,
- -
- return temperature from the radiator changed from 35.3 ${}^{\circ}$C to 63.4 ${}^{\circ}$C,
- -
- the radiator water mass flow varies changed from 0.0034 kg/s (39.0% nominal value) to 0.031 kg/s (356.0% nominal value)
- -
- the radiator heat output varies from 481 W to 924 W.

## 5. Result and Analysis

## 6. Calculations Examples

#### 6.1. Implementation of the “NEHTMiR”—Iron Elements Radiator T-1

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

#### The Discrepancy

#### 6.2. Implementation of the “NEHTMiR” for 2-Row Delonghi-22 Ribbed Radiator

**Example**

**6.**

**Example**

**7.**

**Example**

**8.**

**Example**

**9.**

#### The Discrepancy

#### 6.3. Implementation of the “$NEHTMi{R}_{md}$” for High Ribbed Convective Radiator type-8RR

**Example**

**10.**

**Example**

**11.**

**Example**

**12.**

**Example**

**13.**

#### The Discrepancy

## 7. Conclusions

- directly assessing the actual energy efficiency of the heating system in the heating season.
- improve the indications of the heat allocator due their proper programming in the Individual Heating Costs Allocation Systems [14].
- to establish the suitable conditions for $operational$ regulation of the Central Heating Installations especially in existing buildings after their refurbishment as well as for the District Heating Systems with $substantial$ oversized heat exchangers.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Abbreviations

DOAJ | Directory of open access journals |

TLA | Three letter acronym |

LD | Linear dichroism |

## References

- EN 442-1. Radiators and Convectors—Part 1: Technical Specification and Requirements; European Committee for Standardization: Brussels, Belgium, 2014. [Google Scholar]
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**Figure 1.**The discrepancy between the actual radiator power heat output and calculated according to the equation for the $TBOE$ system.

**Figure 3.**An example of a temperature distribution on the surface of heater T1 according to thermography measurements—measurement 2.

**Figure 11.**Comparison of measurement and calculations results of water temperature, radiator surface temperature and radiator heat out distribution—measurement 1: ${t}_{s}=70.0{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C, ${t}_{r}=63.4{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C, ${t}_{i}=21.1{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C, ${m}_{r}$ = 0.0307 kg/s $\left({m}_{rmax}\right)$.

**Figure 12.**Comparison of measurement and calculations results of water temperature, radiator surface temperature and radiator heat out distribution—measurement 4: ${t}_{s}=69.7{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C, ${t}_{r}=42.2{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C, ${t}_{i}=20.3{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C, ${m}_{r}$ = 0.0051 kg/s $\left({m}_{rmin}\right)$.

**Figure 13.**Profile of the radiator surface temperature temperature change, radiator T1. ${t}_{s}=70.2{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C, $qmx$ from $39\%$ to $356\%$ $qmo$ (nominal flow rate).

**Figure 14.**Profile of the radiator surface temperature temperature change, radiator T1. ${t}_{s}$ = $70.2{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C, $qmx$ from $39\%$ to $356\%$ $qmo$ (nominal flow rate).

**Figure 15.**Radiator T1 surface temperature distribution according thermography measurements, measurements 1, flow ${q}_{mx}$ = $0.0307$ kg/s (${q}_{max}$)—356% nominal value.

**Figure 16.**Radiator T1 surface temperature distribution according thermography measurements, measurements 4, flow ${q}_{mx}$ = $0.0051$ kg/s—59% nominal value.

**Figure 17.**Iron Element Radiator T1 type—water temperature distribution according $simulation$$calculations$, ${t}_{s}$ = $89.7{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C, mass flow ${q}_{mx}$ from 24% to 153% nominal value.

**Figure 18.**Iron Element Radiator T1 type—surface temperature distribution according $simulation$$calculations$, ${t}_{s}$ = $89.7{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C, mass flow ${q}_{mx}$ from 24% to 153% nominal value.

**Figure 19.**$Delonghi$-22 radiator—surface temperature distribution according thermography measurements, ${t}_{s}$ = $89.6{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C, ${t}_{r}=69.3{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C, ${t}_{i}=19.5{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C, mass flow rate${q}_{m}$ = $0.0247$ kg/s—measurement 2.

**Figure 20.**$Delonghi$-22 radiator—surface temperature distribution according thermography measurements, ${t}_{s}=89.8{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C. ${t}_{r}=54.0{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C, ${t}_{i}=19.7{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C, $mass$$flow$$rate$${q}_{m}$ = 0.0120 kg/s—measurement 3.

**Figure 21.**Comparison of measurement and calculations results of the $Delonghi$-22 radiator effectiveness, measurement 1–4.

**Figure 23.**$Delonghi$-22 comparison of measurement and calculations results of water temperature, radiator surface temperature and radiator heat out distribution—measurement 1, ${t}_{s}=89.8{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C, ${t}_{r}=75.9{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C, ${t}_{i}=19.9{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C, ${q}_{mx}$ = $0.0379$ kg/s, (${q}_{m.max}).$

**Figure 24.**$Delonghi$-22 comparison of measurement and calculations results of water temperature, radiator surface temperature and radiator heat out distribution—measurement 4, ${t}_{s}=89.7{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C, ${t}_{r}=37.1{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C, ${t}_{i}=19.7{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C, ${q}_{mx}$ = 0.0059 kg/s, (${q}_{m.min})$.

**Figure 25.**$Delonghi$-22 water temperature change profile, ${t}_{s}=90.0{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C, ${t}_{i}=19.9{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C, ${q}_{mx}$ from $6.0\%$ to $150\%$, (${q}_{nom}$).

**Figure 26.**$Delonghi$-22 surface temperature change profile, ${t}_{s}=90.0{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C, ${t}_{i}=19.9{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C, ${q}_{mx}$ from $6.0\%$ to $150\%$, (${q}_{nom}$).

**Figure 29.**Comparison of the test and calculations results of the convection radiator type-8RR heat output change as a function of the mass flow rate, ${t}_{s}$ = $51.1{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C.

**Figure 30.**Comparison of test and calculations results of the convector radiator type-8RS heat output change as a function of the mass flow rate, ${t}_{s}=58.9{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C.

**Figure 31.**$Radiator$T-1, effectiveness according to the tests and simulations calculations by “$NEHTMi{R}_{md}$” in the mass flow range ${m}_{r}$ from 39% to 356% of the nominal value.

**Figure 32.**$Radiator$ Delonghi-type 22—effectiveness according to the tests and simulations calculations by “$NEHTMi{R}_{md}$” in the mass flow range ${m}_{r}$ from 20% to 150% of the nominal value.

**Figure 33.**$Convector$$Radiator$-$8RR$—effectiveness according to the tests and simulations calculations by “$NEHTMi{R}_{md}$” in the mass flow range ${m}_{r}$ from 23% to 150% of the nominal value.

**Table 1.**The comparison between the measurement and the calculations results Radiator T1. ${t}_{s}=70.0{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C.

No. | ${\mathit{t}}_{\mathit{s}}$ | ${\mathit{t}}_{\mathit{r}}$ | ${\mathit{t}}_{\mathit{i}}$ | qm | Qr ac | Qr EN | dQr EN | Qr md | dQr md | qmx/qmo |
---|---|---|---|---|---|---|---|---|---|---|

- | °C | °C | °C | kg/s | W | W | % | W | % | - |

1 | 70.6 | 63.4 | 21.1 | 0.0307 | 924 | 913 | 1.2 | 911 | 1.3 | 3.56 |

2 | 70.6 | 57.8 | 20.9 | 0.0155 | 832 | 847 | 1.8 | 837 | −0.6 | 1.79 |

3 | 70.5 | 50.1 | 20.7 | 0.0086 | 737 | 756 | 2.6 | 733 | 0.5 | 1.00 |

4 | 69.7 | 42.2 | 20.3 | 0.0051 | 588 | 662 | 12.7 | 602 | −2.5 | 0.59 |

5 | 69.4 | 35.3 | 19.9 | 0.0034 | 481 | 588 | 22.3 | 491 | −2.2 | 0.39 |

**Table 2.**The comparison between the measurement and the calculations results Radiator T1. ${t}_{s}$ = $70{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C.

No. | ${\mathit{t}}_{\mathit{s}}$ | ${\mathit{t}}_{\mathit{r}}$ | ${\mathit{t}}_{\mathit{i}}$ | qm | Qr ac | EFr md | EFr ac | dEFr | tr md | dtr | Ts 0.75 | qmx/qmo |
---|---|---|---|---|---|---|---|---|---|---|---|---|

- | °C | °C | °C | kg/s | W | - | °C | % | °C | % | - | |

1 | 70.6 | 63.4 | 21.1 | 0.0307 | 924 | 0.1430 | 0.1450 | −1.3 | 63.5 | −1.3 | 66.1 | 3.56 |

2 | 70.6 | 57.8 | 20.9 | 0.0155 | 832 | 0.2601 | 0.2585 | 0.6 | 57.7 | 0.6 | 63.3 | 1.79 |

3 | 70.5 | 50.1 | 20.7 | 0.0086 | 737 | 0.4075 | 0.4097 | −0.5 | 50.2 | −0.5 | 60.3 | 1.00 |

4 | 69.7 | 42.2 | 20.3 | 0.0051 | 588 | 0.5709 | 0.5571 | 2.5 | 41.5 | 2.5 | 56.2 | 0.59 |

5 | 69.4 | 35.3 | 19.9 | 0.0034 | 481 | 0.7039 | 0.6885 | 2.2 | 34.6 | 2.2 | 50.4 | 0.39 |

**Table 3.**The comparison between the measurement and the calculations results Radiator Delonghi 22, ${t}_{s}$ = $89.8{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C.

No. | ${\mathit{t}}_{\mathit{s}}$ | ${\mathit{t}}_{\mathit{r}}$ | ${\mathit{t}}_{\mathit{i}}$ | qm | Qr ac | Qr EN | dQr EN | Qr md | dQr md | Ts 0.75 | qmx/qmo |
---|---|---|---|---|---|---|---|---|---|---|---|

- | °C | °C | °C | kg/s | W | W | % | W | % | C | - |

1 | 89.8 | 75.9 | 19.9 | 0.0379 | 2222 | 2277 | −2.46 | 2208 | 0.34 | 80.4 | 1.53 |

2 | 89.6 | 69.3 | 19.5 | 0.0247 | 2103 | 2135 | −1.52 | 2115 | −0.55 | 77.5 | 1.00 |

3 | 89.8 | 54.0 | 19.7 | 0.0120 | 1806 | 1779 | 1.46 | 1804 | 0.09 | 57.0 | 0.49 |

4 | 89.7 | 37.1 | 19.7 | 0.0059 | 1289 | 1407 | −9.16 | 1304 | −1.19 | 60.5 | 0.24 |

**Table 4.**The comparison between the measurement and the calculations results Radiator Delonghi 22, ${t}_{s}$ = $89.8{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C.

No. | ${\mathit{t}}_{\mathit{s}}$ | ${\mathit{t}}_{\mathit{r}}$ | ${\mathit{t}}_{\mathit{i}}$ | qm | Qr ac | EFr ac | EFr md | dEFr | tr md | dtr | Ts 0.75 | qmx/qmo |
---|---|---|---|---|---|---|---|---|---|---|---|---|

- | °C | °C | °C | kg/s | W | - | - | % | C | % | C | - |

1 | 89.8 | 75.9 | 19.9 | 0.0379 | 2222 | 0.1989 | 0.1989 | 0.0 | 75.9 | 0.0 | 80.4 | 1.53 |

2 | 89.6 | 69.3 | 19.5 | 0.0247 | 2103 | 0.2892 | 0.2915 | −0.8 | 69.2 | 0.2 | 77.5 | 1.00 |

3 | 89.8 | 54.0 | 19.7 | 0.0120 | 1806 | 0.5106 | 0.5108 | 0.0 | 54.0 | 0.0 | 57.0 | 0.49 |

4 | 89.7 | 37.1 | 19.7 | 0.0059 | 1289 | 0.7513 | 0.7602 | −1.2 | 36.5 | 1.7 | 60.5 | 0.24 |

**Table 5.**The comparison between the measurement results and the simulation calculations Convector Radiator type-8RR, ${t}_{s}$ = $51.1{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C.

No. | ${\mathit{t}}_{\mathit{s}}$ | ${\mathit{t}}_{\mathit{r}}$ | ${\mathit{t}}_{\mathit{i}}$ | qm | Qr ac | Qr md | dQr md | EFr ac | EFr md | dEFr | tr | dtr | qmx/qmo |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

- | °C | °C | °C | kg/s | W | W | % | - | - | % | °C | % | - |

1 | 51.2 | 46.6 | 21.0 | 0.0303 | 583 | 573 | 1.7 | 0.152 | 0.14944 | 1.7 | 46.7 | −0.2 | 2.16 |

2 | 51.1 | 36.9 | 21.0 | 0.0075 | 443 | 432 | 2.5 | 0.4702 | 0.45838 | 2.5 | 37.3 | −1.0 | 0.53 |

3 | 51.1 | 33.2 | 20.9 | 0.0053 | 396 | 379 | 4.3 | 0.5918 | 0.56656 | 4.3 | 34.0 | −2.3 | 0.38 |

4 | 51.0 | 33.3 | 21.0 | 0.0047 | 352 | 358 | −1.9 | 0.5907 | 0.60201 | −1.9 | 32.9 | 1.0 | 0.34 |

**Table 6.**The comparison between the measurement results and the simulation calculations Convector Radiator type-8RS, ${t}_{s}$ = $58.9{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C.

No. | ${\mathit{t}}_{\mathit{s}}$ | ${\mathit{t}}_{\mathit{r}}$ | ${\mathit{t}}_{\mathit{i}}$ | qm | Qr ac | Qr md | dQr md | qmx/qmo |
---|---|---|---|---|---|---|---|---|

- | °C | °C | °C | kg/s | W | W | % | - |

1 | 58.6 | 32.1 | 21.0 | 0.0035 | 388 | 399.1 | −2.8 | 0.26 |

2 | 59.0 | 39.9 | 21.0 | 0.0064 | 511 | 531.9 | −4.1 | 0.47 |

3 | 59.0 | 46.9 | 20.9 | 0.0131 | 662 | 658.6 | 0.5 | 0.97 |

4 | 59.0 | 51.5 | 21.0 | 0.0226 | 709 | 720.9 | −1.7 | 1.68 |

**Table 7.**The comparison between the measurement results and the simulation calculations Convector Radiator type-8RS, ${t}_{s}$ = $58.9{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C, Part 2.

No. | ${\mathit{t}}_{\mathit{s}}$ | ${\mathit{t}}_{\mathit{r}}$ | ${\mathit{t}}_{\mathit{i}}$ | qm | EFr ac | EFr md | dEFr | tr md | dtr | qmx/qmo |
---|---|---|---|---|---|---|---|---|---|---|

- | °C | °C | °C | kg/s | - | - | % | °C | % | - |

1 | 58.6 | 32.1 | 20.8 | 0.0035 | 0.7011 | 0.71998 | −2.7 | 31.4 | 2.2 | 0.26 |

2 | 59.0 | 39.9 | 21.4 | 0.0064 | 0.509 | 0.53006 | −4.1 | 39.1 | 2.0 | 0.47 |

3 | 59.0 | 46.9 | 21.7 | 0.0131 | 0.3236 | 0.32204 | 0.5 | 47.0 | −0.1 | 0.97 |

4 | 59.0 | 51.5 | 22.0 | 0.0226 | 0.2024 | 0.20584 | −1.7 | 51.4 | 0.2 | 1.68 |

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**MDPI and ACS Style**

Dzierzgowski, M. Verification and Improving the Heat Transfer Model in Radiators in the Wide Change Operating Parameters. *Energies* **2021**, *14*, 6543.
https://doi.org/10.3390/en14206543

**AMA Style**

Dzierzgowski M. Verification and Improving the Heat Transfer Model in Radiators in the Wide Change Operating Parameters. *Energies*. 2021; 14(20):6543.
https://doi.org/10.3390/en14206543

**Chicago/Turabian Style**

Dzierzgowski, Mieczysław. 2021. "Verification and Improving the Heat Transfer Model in Radiators in the Wide Change Operating Parameters" *Energies* 14, no. 20: 6543.
https://doi.org/10.3390/en14206543