## 1. Introduction

## 2. Method

## 3. Physical Dynamic Heating Test Experiments and Results

_{start}has been normalized to the same value of 10 °C for all six tests, with a corresponding adjustment for the outside temperature so that the temperature difference between inside and outside remained the same as measured. The cooling part of the tests started at the time the heaters were switched off. The building had been allowed to cool down to a free floating indoor temperature before the heating part of each test. The indoor air temperature was calculated as a volume-weighted average of all measured room temperatures. An occurrence of steady room temperature was used as a criterion for the end of the heating part of each test. The influence of solar radiation on the building response was reduced by means of thermal blinds. These were specially designed for this purpose, with a reflective outside surface and a thermally insulating fabric with edge seals.

Case 1: | tc = 44.4 ± 2.7 h HLC = 256.2 ± 24.6 W/K C = 41.1 ± 6.0 MJ/K | Case 2: | tc = 46.4 ± 0.4 h HLC = 268.5 ± 1.8 W/K C = 44.8 ± 0.4 MJ/K |

^{2}of floor area). If we assume that the Demonstration House is in the heavy category due to high density concrete blockwork in the internal layer of the external walls, and due to concrete slabs, then the calculated capacitance for 83.7 m

^{2}of floor area is 40.6 MJ/K. The effective capacitance obtained from the experimental data is between 41.1 ± 6.0 MJ/K (Table 4), and therefore within 1.3% relative error with reference to the theoretical value from the ASHRAE Handbook of Fundamentals [18].

## 4. Simulated Dynamic Heating Test Experiments and Results

Before retrofit: | tc = 21.93 ± 2.65 h HLC = 328.70 ± 33.05 W/K C = 25.64 ± 0.38 MJ/K | After retrofit: | tc = 78.39 ± 1.40 h HLC = 147.32 ± 12.52 W/K C = 41.59 ± 3.82 MJ/K |

^{2}K.

## 5. Discussion

## 6. Conclusions

## Funding

## Conflicts of Interest

## Nomenclature

ψ_{j} | linear thermal transmittance of the j-th thermal bridge in W/(mK) |

A_{i} | surface area of the corresponding i-th thermal element in m^{2} |

C | effective thermal capacitance in MJ/K |

E_{measured} | measured energy consumption in kWh |

E_{simulated} | simulated energy consumption in kWh |

HLC | overall heat loss coefficient; HLC = UA + H_{tb} + H_{v} (see [3], p. 92); referred to as “theoretical” if calculated manually using published information on material properties from reference tables |

H_{tb} | thermal bridging heat loss coefficient in W/K; ${H}_{tb}={\sum}_{j=1}^{j=M}{L}_{j}{\psi}_{j}$ (see [3], p. 89) |

H_{v} | ventilation and infiltration heat loss coefficient in W/K; H_{v} = nV/3 (see [3], p. 90) |

K | total number of thermal elements |

L_{j} | length of the j-th linear thermal bridge in m |

n | air tightness in volume changes per hour in 1/h |

N | number of data points |

Q | heating rate in W |

Q_{int} | internal heat gain in the building arising from heating or from casual gains |

Q_{sol} | heat gain from solar radiation |

RMSE | root-mean-squared error |

t | time in hours |

T_{a} | outside air temperature in °C |

tc = C/HLC | building time constant in hours |

T_{max} | maximum internal air temperature reached as a result of the heat input |

T_{r} | room air temperature at time t, either measured in the physical dynamic heating test or simulated with EnergyPlus [15] in the simulated dynamic heating test in °C |

T_{room} | difference between room temperature and the initial room temperature T_{t} − T_{start} |

T_{start} | starting internal air temperature at the time when heat input was switched on |

T_{t} | room air temperature at time t calculated using Equation (3) in °C |

UA | overall thermal transmittance-area product in W/K; $UA=\frac{{\sum}_{i=1}^{i=K}{U}_{i}{A}_{i}}{{\sum}_{i=1}^{i=K}{A}_{i}}$ (see [3], pp. 87–88) |

U_{i} | thermal transmittance of i-th thermal element (walls, roof, ground floor slab, windows, doors, etc.) in W/(m^{2}K) |

V | internal volume of the building in m^{3} |

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**Figure 2.**Scatter plot of calibration of the existing building used for the simulated dynamic heating test.

**Figure 4.**Genetic operations carried out on the chromosomes (see [16]).

**Figure 6.**Solar Demonstration House in Bournville, Birmingham (see [3]).

Test No. | Nominal Heater Power (kW) | Heat Input Source: U—Underfloor Heating; A—Auxiliary Electric Heating | Test Duration (hours) |
---|---|---|---|

1 | 9 | U | 122 |

2 | 9 | A | 167 |

3 | 3 | U | 480 |

4 | 18 | (9 kW A + 9 kW U) | 104 |

5 | 12 | 9 kW A + 3 kW U | 167 |

6 | 6 | A | 239 |

Test No. | Measured Heater Power (kW) | Power Absorbed by Storage Tanks (kW) | Infiltration Loss (kW) | Heat Gain (kW) | Comments |
---|---|---|---|---|---|

Case 1 | |||||

1 | 9.00 | 1.10 | 2.01 | 5.89 | Measured power was used for Test 3; air tightness in Tests 1–4 was n = 1.8 1/h; air tightness in Tests 5 and 6 was n = 0.68 1/h. |

2 | 9.01 | 0.27 | 3.07 | 5.67 | |

3 | 3.53 | 0.55 | 1.27 | 1.71 | |

4 | 17.32 | 2.75 | 4.05 | 10.52 | |

5 | 12.14 | 0.96 | 1.31 | 9.87 | |

6 | 6.29 | 0.33 | 1.02 | 4.94 | |

Case 2 | |||||

1 | 9.00 | 1.10 | 1.34 | 6.56 | Nominal power was used for Test 3; air tightness in Tests 1–4 was n = 1.2 1/h; air tightness in Tests 5 and 6 was n = 0.68 1/h. |

2 | 9.01 | 0.27 | 2.05 | 6.70 | |

3 | 3.00 | 0.55 | 0.85 | 1.60 | |

4 | 17.32 | 2.75 | 2.70 | 11.87 | |

5 | 12.14 | 0.96 | 1.31 | 9.87 | |

6 | 6.29 | 0.33 | 1.02 | 4.94 |

Test No. | Time Constant (hours) | Heat Loss Coefficient (W/K) | Effective Thermal Capacitance (MJ/K) | Comments |
---|---|---|---|---|

Case 1 | ||||

1 | 39.7 | 237.9 | 34.0 | Measured power was used for Test 3; air tightness in Tests 1–4 was n = 1.8 1/h; air tightness in Tests 5 and 6 was n = 0.68 1/h. |

2 | 43.1 | 219.0 | 34.0 | |

3 | 46.8 | 288.1 | 48.5 | |

4 | 32.0 | 232.2 | 26.7 | |

5 | 45.9 | 269.8 | 44.6 | |

6 | 46.5 | 266.0 | 44.5 | |

Case 2 | ||||

1 | 39.7 | 265.0 | 37.9 | Nominal power was used for Test 3; air tightness in Tests 1–4 was n = 1.2 1/h; air tightness in Tests 5 and 6 was n = 0.68 1/h. |

2 | 43.1 | 258.5 | 40.1 | |

3 | 46.8 | 269.8 | 45.4 | |

4 | 32.0 | 262.0 | 30.1 | |

5 | 45.9 | 269.8 | 44.6 | |

6 | 46.5 | 266.0 | 44.5 |

Description | Time Constant (hours) | Heat Loss Coefficient (W/K) | Effective Thermal Capacitance (MJ/K) |
---|---|---|---|

Case 1 | |||

Mean | 44.4 | 256.2 | 41.1 |

Standard deviation | 2.7 | 24.6 | 6.0 |

Case 2 | |||

Mean | 46.4 | 268.5 | 44.8 |

Standard deviation | 0.4 | 1.8 | 0.4 |

**Table 5.**Comparison of theoretical heat loss coefficients and those measured with physical dynamic heating tests.

Description | Result |
---|---|

Theoretical heat loss coefficient (W/K) | 172.9 |

Heat loss coefficient obtained from simulated dynamic heating tests (W/K) | 256.2 ± 24.6 |

discrepancy = (simulated − theoretical)/theoretical × 100 | |

Lower end of the range | 34% |

Average | 48% |

Upper end of the range | 62% |

**Table 6.**Envelope characteristics before and after the retrofit (source [19]).

Description | Before Retrofit | After Retrofit |
---|---|---|

U-Value W/(m^{2}·K) | ||

Theoretical | ||

External walls 270 mm | 1.48 | 0.11 |

External glazing | 1.60 | 0.79 |

External door | 2.56 | 0.78 |

Ground floor slab | 1.49 | 0.26 |

Roof 270 mm | 0.47 | 0.10 |

Measured | ||

House | Air tightness 1/h at 50 Pascal | |

A | 6.05 | 1.78 |

B | 10.74 | 1.78 |

Calibration Parameters | Before Retrofit | After Retrofit |
---|---|---|

Wall and roof construction pairs | N/A | Insulation thickness: 216–324 mm in steps of 13.5 mm |

Infiltration air changes per hour | 0.6–10.8 1/h in steps of 0.1 1/h | 0.6–6.0 1/h in steps of 0.1 1/h |

Internal set temperature | Living room: 16–21 °C in steps of 1 °C Other rooms: 15–21 °C in steps of 1 °C | All rooms: 16–21 °C in steps of 0.5 °C |

Lighting power density | 5–8 W/m^{2} in steps of 0.5 W/m^{2} | 2–8 W/m^{2} in steps of 0.5 W/m^{2} |

Miscellaneous gains power density | 5–8 W/m^{2} in steps of 0.5 W/m^{2} | 2–8 W/m^{2} in steps of 0.5 W/m^{2} |

Description | Before Retrofit | After Retrofit |
---|---|---|

Gas consumption error | 0.33% | 0.42% |

Model accuracy—gas consumption | 99.67% | 99.58% |

Electricity consumption error | 0.17% | 0.05% |

Model accuracy—electricity consumption | 99.83% | 99.95% |

Parameter | Post-Retrofit | Pre-Retrofit | Ratio Post-Retrofit/Pre-Retrofit | ||
---|---|---|---|---|---|

Mean | Standard Deviation | Mean | Standard Deviation | ||

tc (h) | 78.39 | 1.40 | 21.93 | 2.65 | 3.57 |

HLC (W/K) | 147.32 | 12.52 | 328.70 | 33.05 | 0.45 |

C (MJ/K) | 41.59 | 3.82 | 25.64 | 0.38 | 1.62 |

U (W/m^{2}K) ^{1} | 0.69 | 1.54 | 0.45 |

^{1}U-value is calculated by dividing the heat loss coefficient from this table by the manually calculated building surface area. This is therefore an overall heat transmittance that includes all building thermal elements and the effects of infiltration and thermal bridges.

**Table 10.**Comparison of theoretical heat loss coefficients and those measured with simulated dynamic heating tests.

Description | Post-Retrofit | Pre-Retrofit |
---|---|---|

Theoretical heat loss coefficient (W/K) | 97.89 | 318.56 |

Heat loss coefficient obtained from simulated dynamic heating tests (W/K) | 147.32 ± 12.52 | 328.70 ± 33.05 |

discrepancy = (simulated − theoretical)/theoretical × 100 | ||

Lower end of the range | 38% | −7% |

Average | 50% | 14% |

Upper end of the range | 63% | 3% |

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