# A Novel Response Factor-Based Method for In Situ Measurement of Wall Thermal Resistance

^{*}

## Abstract

**:**

## 1. Introduction

Measurement Methods | The Calculation Formula for Wall Thermal Resistance |
---|---|

Average method [5] (Standard HFM method) | $R=\frac{{\sum}_{j=1}^{n}\left({t}_{in,j}-{t}_{out,j}\right)}{{\sum}_{j=1}^{n}{q}_{j}}$ |

Dynamic analysis method [5,19] | ${q}_{\mathrm{i}}=\frac{1}{R}\left({\theta}_{\mathrm{Ii}}{-\theta}_{\mathrm{Ei}}\right){+K}_{1}\dot{{\theta}_{\mathrm{Ii}}}{-K}_{2}\dot{{\theta}_{\mathrm{Ei}}}+\sum _{\mathrm{n}}{p}_{\mathrm{n}}\sum _{\mathrm{j}=\mathrm{i}-\mathrm{p}}^{\mathrm{i}=1}\dot{{\theta}_{\mathrm{Ij}}}\left({1-\beta}_{\mathrm{n}}\right){\beta}_{\mathrm{n}}\left(\mathrm{i}-\mathrm{j}\right)+\sum _{\mathrm{n}}{Q}_{\mathrm{n}}\sum _{\mathrm{j}=\mathrm{i}-\mathrm{p}}^{\mathrm{i}=1}\dot{{\theta}_{\mathrm{Ej}}}\left({1-\beta}_{\mathrm{n}}\right){\beta}_{\mathrm{n}}\left(\mathrm{i}-\mathrm{j}\right)$ |

Shi et al. [14] | $R=\frac{{\sum}_{i=1}^{n}\left(\frac{\left|{T}_{si,\mathrm{max}}-{T}_{se,\mathrm{max}}\right|+\left|{T}_{si,\mathrm{min}}-{T}_{se,\mathrm{min}}\right|}{2}\right)/n}{{\sum}_{i=1}^{n}\left(\frac{\left|{q}_{si}\right|+\left|{q}_{se}\right|}{2}\right)/n}$ |

Rasooli et al. [15] | ${R}_{c-\mathrm{ave}}=\frac{\left({R}_{c-\mathrm{in}}+{R}_{c-\mathrm{out}}\right)}{2}=\left({\displaystyle \sum _{t=0}^{m}}\Delta {T}^{t}/{\displaystyle \sum _{t=0}^{m}}{\left({q}^{t}\right)}_{1}+{\displaystyle \sum _{t=0}^{m}}\Delta {T}^{t}/{\displaystyle \sum _{t=0}^{m}}{\left({q}^{t}\right)}_{2}\right)/2$ |

Park et al. [17] | ${U}_{\mathrm{HMF},\mathrm{ave}}=\frac{{\sum}^{}q}{{\sum}^{}\left({T}_{\mathrm{in}}-{T}_{\mathrm{out}}\right)},{U}_{\mathrm{TBM},\mathrm{ave}}=\frac{{\alpha}_{in}{\sum}^{}\left({T}_{s,\mathrm{in}}-{T}_{s,\mathrm{out}}\right)}{{\sum}^{}\left({T}_{in}-{T}_{out}\right)},$ ${U}_{\mathrm{TBM}\_\mathrm{abs},\mathrm{ave}}=\frac{{\alpha}_{\mathrm{in}}{\sum}^{}\left|{T}_{s,\mathrm{in}}-{T}_{s,\mathrm{out}}\right|}{{\sum}^{}\left|{T}_{\mathrm{in}}-{T}_{\mathrm{out}}\right|}$ |

Shen et al. [18] | $\left[Q,\Delta {T}_{1},-\Delta {T}_{2},dT\right]\left[\begin{array}{c}R\\ \begin{array}{c}X\\ \begin{array}{c}Y\\ -1\end{array}\end{array}\end{array}\right]+\left[{E}_{t},{E}_{q}\right]\left[\begin{array}{c}R\\ \begin{array}{c}X\\ \begin{array}{c}Y\\ -1\end{array}\end{array}\end{array}\right]=0$ |

Laurenti et al. [19] | ${q}_{\mathrm{n}+1}=\frac{1}{R}\left({T}_{os,\mathrm{n}+1}-{T}_{is,\mathrm{n}+1}\right)+{\displaystyle \sum _{\mathrm{j}=0}^{2-\mathrm{p}}}{\beta}_{1,\mathrm{j}}\left({q}_{\mathrm{n}+\mathrm{j}}-{q}_{\mathrm{n}+1}\right)+{\displaystyle \sum _{\mathrm{j}=0}^{2-\mathrm{p}}}{\beta}_{2,\mathrm{j}}\left({T}_{is,\mathrm{n}+\mathrm{j}}-{T}_{is,\mathrm{n}+1}\right)+{\displaystyle \sum _{\mathrm{j}=0}^{2-\mathrm{p}}}{\beta}_{3,j}\left({T}_{os,\mathrm{n}+\mathrm{j}}-{T}_{os,\mathrm{n}+1}\right)$ |

Pentaur method from Flores Larsen et al. [20] | ${q}_{p}=\frac{{T}_{\mathrm{int},p}-{T}_{\mathrm{out},p}}{R}+{\displaystyle \sum _{n=1}^{k}}d{T}_{\mathrm{int},n}{A}_{n}+{\displaystyle \sum _{n=1}^{k}}d{T}_{\mathrm{out},n}{B}_{n}+{\displaystyle \sum _{n=1}^{k}}d{Q}_{n}{C}_{n}+dErr$ |

Ref. No. | Method | Estimation Errors (%) | Convergence Time (Days) | Test Condition | |||
---|---|---|---|---|---|---|---|

Period | Location | Wall Type | Data Source | ||||

[14] | Average method with storage corrections (RHS-HFM) | 3.86 | 2 | 1/27–1/29 (2 days) | Hot summer and cold winter areas in China | Clay hollow brick with cement mortar and insulation. | In situ measurement |

Double-side heat flow meter method (D-HFM) | 10.49 | >2 | |||||

Average method (use inside heat flux) | 3.13 | 2 | |||||

Average method (use outside heat flux) | 16.58 | >2 | |||||

[15] | Double-side heat flux methods | 1.4 | 5 | 2018/04 (16 days) | Netherlands | Unknown but insulation inside the construction. | In situ measurement |

Average method (use inside heat flux) | <5 | >16 | |||||

Average method (use outside heat flux) | <5 | >16 | |||||

[17] | Temperature based method (TBM) | 0.21–1.79 | NA | 2021/02–2021/04 | Korea | Concrete wall; insulated curtain wall structure with panel finishing. | In situ measurement |

Average method | 0.20–1.82 | NA | |||||

[19] | Proposed dynamic method | 0.1–4 | 1–12.5 | An entire year (24 datasets with 15 days for each set) | NA | Light, well-insulated wall; massive wall with insulation layers on both sides; moderately massive homogeneous wall. | Simulation |

[20] | Average method | 15.0 | 20 | Four seasons with 30 days for each season | Salta, Argentina | Compound wall (solid brick and hollow brick wall with insulation). | Simulation |

Average method with storage corrections | 5.0 | 6–18 | |||||

Fourier method | >17.0 | 8–24 | |||||

Pentaur method | <5.0 | 2 |

## 2. Methods

#### 2.1. Fundamentals of Wall Heat Conduction Process

^{3}, W/m·K; l is wall thickness in m; $x$ is spatial variable; and $\tau $ is time variable.

#### 2.2. A Truncation Model for Wall Thermal Resistance Calculation

#### 2.3. Convergence Criteria of In Situ Measurement

- Monotonicity: $r$ decreases with the increase of $n$ and $L$, i.e., $\frac{\partial r}{\partial n}<0,\frac{\partial r}{\partial L}0$;
- Tropism: the calculated value R approaches the true value ${R}_{0}$ when n and L tend to infinity, i.e., $r\to 0$ when $n\to \infty $ and $L\to \infty $.

^{−3}to determine that the partial derivatives are close to zero. It should be pointed out that the threshold value of 2 × 10

^{−3}is an empirical value obtained after many trials, and it is a trade-off between the convergence time and the accuracy of wall thermal resistance estimations. When the three equations are met, the measurement error of wall thermal resistance $r$ is small enough, and consequently, a set of optimal values $n$, $L$, and $R$ are found.

#### 2.4. Implementation Process

- (1)
- Take the initial n value as 3 and the initial $L$ value as 8.
- (2)
- Start the initial estimation of wall thermal resistance when obtaining the length of the data $T\ge 3{n}_{0}+2=11$.
- (3)
- With the increase in measurement time and data accumulation, try to increase the values of n and $L$, and update the corresponding estimation of wall thermal resistance $R\left(n,L\right)$ according to Equation (8).
- (4)
- When the criteria in the form of Equations (16)–(18) is met, output the estimated thermal resistance value $R\left(n,L\right)$ at that time and end the measurement.

## 3. Theoretical Validation

#### 3.1. Selection of Walls

#### 3.2. Validation of Truncated Models

#### 3.2.1. Truncation Error of Response Factors

#### 3.2.2. Residual Error of Least Squares

^{2}), which indicates the whole deviation degree of the estimated values relative to the observed values. The closer R

^{2}is to 1, the smaller the deviation degree is. The calculation formula of R

^{2}is as follows:

^{2}of all walls varies significantly when n = 20, the R

^{2}of all walls increases with n and eventually approaches 1. When n = 40, the ${R}^{2}$ value of all walls except for Walls 3 and 4 is greater than 0.99. When n = 50, all wall ${R}^{2}$ values are greater than 0.99, and the model fitting effect is good. The heat flux calculated by the truncated model is consistent with the measured value. This result may be explained by the fact that when n = 20, the calculation has not fully converged, resulting in a significant residual error. When n increases to 50, the calculation converges, and the sum of coefficients is close to the wall heat transfer coefficient so the error approaches zero.

## 4. Experiment Validation

#### 4.1. Performance Indicators

#### 4.2. Simulation and Measurement Datasets

#### 4.3. Results

## 5. Conclusions

- The data analysis method was established, including the truncation model for the wall thermal resistance calculation derived from the classical theory of wall thermal response factors, the convergence criteria based on theoretical and empirical analysis, and the implementation process for applying the method to the in situ measurement.
- The threshold value of 2 × 10
^{−3}in the convergence criteria for determining if the measurement and calculation process converges is an empirical value obtained after many trials. It is a trade-off between the convergence time and the accuracy of wall thermal resistance estimations. - The truncation model is theoretically validated from the perspectives of the truncation error of response factors and the residual error of least squares. Seven typical types of walls were selected based on Chinese building energy-saving standards. The results show that when the number of truncation terms, $n$, is sufficient for different types of walls, a truncated form of wall response factors can replace the infinite form for wall thermal resistance calculation, and the truncation and residual errors can be ignored.
- The feasibility and accuracy of the proposed method are validated by one simulation dataset and one measurement dataset. The results show that the accuracy of the proposed method is similar to the standard average method and has a deviation of 3.86% on average. In terms of the convergence time, compared to the average method, the proposed method can effectively shorten the measurement period by about 36.4% on average.
- The preliminary validation results indicate that the proposed method and its convergence criteria apply to actual measurements while achieving a fast but accurate estimation result. Compared with other dynamic HFM methods, the proposed method has the significant advantages of simple principle and clear physical meaning.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**The first 50 response factors of Wall 1. (

**a**) Theoretical values of each response factor. (

**b**) The sum and truncation error.

**Figure 4.**Variation tendency of the sum of different wall coefficients with the truncation number n. (

**a**) $\sum {A}_{j}$. (

**b**) $\sum {B}_{j}$.

**Figure 6.**Variation of surface temperatures and heat flux over time. (

**a**) Simulation dataset. (

**b**) Measurement dataset.

**Figure 7.**The performance indicators of the proposed method and the average method. (

**a**) Estimation errors for the simulation dataset. (

**b**) Convergence time for the simulation dataset. (

**c**) Estimation errors for the measurement dataset. (

**d**) Convergence time for the measurement dataset.

**Figure 8.**The performance statistics of the proposed method and the average method. (

**a**) Distribution of estimation errors. (

**b**) Distribution of convergence time. (

**c**) Average values of estimation errors. (

**d**) Average values of convergence time.

No. | Wall Type | Material (From Inside to Outside) | Thickness [mm] | Theoretical Thermal Resistance [m ^{2}/(K·W)] |
---|---|---|---|---|

1 | Brick wall | Cement mortar | 20 | 0.764 |

Red clay brick | 310 | |||

Cement mortar | 20 | |||

2 | Concrete wall with thin interior insulation | Cement mortar | 20 | 0.764 |

Extruded polystyrene board (XPS) | 20 | |||

Steel reinforced concrete | 200 | |||

Cement mortar | 20 | |||

3 | Concrete wall with standard interior insulation | Cement mortar | 20 | 2.703 |

XPS | 84 | |||

Steel reinforced concrete | 200 | |||

Cement mortar | 20 | |||

4 | Concrete wall with thick interior insulation | Cement mortar | 20 | 5.612 |

XPS | 180 | |||

Steel reinforced concrete | 200 | |||

Cement mortar | 20 | |||

5 | Concrete wall with thin exterior insulation | Cement mortar | 20 | 0.764 |

Steel reinforced concrete | 200 | |||

XPS | 20 | |||

Cement mortar | 20 | |||

6 | Concrete wall with standard exterior insulation | Cement mortar | 20 | 2.703 |

Steel reinforced concrete | 200 | |||

XPS | 84 | |||

Cement mortar | 20 | |||

7 | Concrete wall with thick exterior insulation | Cement mortar | 20 | 5.612 |

Steel reinforced concrete | 200 | |||

XPS | 180 | |||

Cement mortar | 20 |

Material | Thermal Conductivity [w/(m·K)] | Density [kg/m ^{3}] | Specific Heat [J/(kg·k)] |
---|---|---|---|

Cement mortar | 0.93 | 1800 | 1050 |

Red clay brick | 0.43 | 1668 | 754 |

Extruded polystyrene board (XPS) | 0.033 | 29 | 1791.06 |

Steel reinforced concrete | 1.74 | 2500 | 920 |

Method | The Calculation Formula for Wall Thermal Resistance | Termination Criteria |
---|---|---|

Proposed method | Equations (6) and (8) | Equations (16)–(18). As the measurement period gradually increases, try increasing $n$ and $L$ to estimate the thermal resistance $R\left(n,L\right)$. The calculation stops at the first occurrence of $R\left(n,L\right)$ that meets the above conditions. |

Average method [6] | $R=\frac{{\sum}_{j=1}^{n}\left({t}_{in,j}-{t}_{out,j}\right)}{{\sum}_{j=1}^{n}{q}_{j}}$ | (1) The thermal resistance obtained at the time does not deviate by more than ±5% from the value obtained 24 h prior to it: $\left|\frac{{R}_{{D}_{T}}-{R}_{{D}_{T}-24\mathrm{h}}}{{R}_{{D}_{T}-24\mathrm{h}}}\right|\le 5\%$ (2) The thermal resistance obtained from the first INT(2 × D _{T}/3) days does not deviate by more than ±5% from the values obtained from the last period of the same days, where D_{T} is the measurement duration (in days) from the start, and INT is a rounding function for the integer part:$\left|\frac{{R}_{INT\left(2\times \frac{{D}_{T}}{3}\right),first}-{R}_{INT\left(2\times \frac{{D}_{T}}{3}\right),last}}{{R}_{INT\left(2\times \frac{{D}_{T}}{3}\right),last}}\right|\le 5\%$ |

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## Share and Cite

**MDPI and ACS Style**

Wang, C.; Fu, X.; Tao, X.; Li, X.; An, J.
A Novel Response Factor-Based Method for In Situ Measurement of Wall Thermal Resistance. *Buildings* **2023**, *13*, 1986.
https://doi.org/10.3390/buildings13081986

**AMA Style**

Wang C, Fu X, Tao X, Li X, An J.
A Novel Response Factor-Based Method for In Situ Measurement of Wall Thermal Resistance. *Buildings*. 2023; 13(8):1986.
https://doi.org/10.3390/buildings13081986

**Chicago/Turabian Style**

Wang, Chuang, Xiao Fu, Xiaoran Tao, Xiaoyan Li, and Jingjing An.
2023. "A Novel Response Factor-Based Method for In Situ Measurement of Wall Thermal Resistance" *Buildings* 13, no. 8: 1986.
https://doi.org/10.3390/buildings13081986