# A Bayesian Dynamic Method to Estimate the Thermophysical Properties of Building Elements in All Seasons, Orientations and with Reduced Error

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Static Model of Heat Transfer: The Average Method

#### 2.2. Dynamic Model: Lumped-Thermal-Mass Models

#### 2.3. Bayesian Inference: Optimisation Phase for Thermophysical Parameter Estimation

#### 2.3.1. Likelihood Function

#### Independent and Identically Distributed Residuals

#### Discrete Cosine Transform

#### 2.3.2. Prior Probability Distributions on the Parameters of the Model

#### Uniform Priors

#### Log-Normal Priors

#### 2.4. Model Selection and Validation

#### 2.4.1. Model Comparison

#### 2.4.2. Cross-Validation

## 3. Experimental Data Collection and Analysis

#### 3.1. Case Studies

#### 3.2. Definition of Priors

#### Uniform Prior Distributions on the Parameters of the Model

#### Log-Normal Prior Distributions on the Parameters of the Model

#### 3.3. Stabilisation Criteria and Monitoring Campaign Length

#### 3.4. Quantification of Uncertainties on in-Situ Observations

#### 3.5. Quantification of Systematic Measurement Errors

## 4. Results and Discussion

#### 4.1. Thermophysical Performance of North-Facing Walls Exposed to High Temperature Differences

#### 4.1.1. Thermophysical Performance of the Solid Wall

#### 4.1.2. Thermophysical Performance of the Cavity Wall

#### 4.2. Reducing the Required Monitoring Length and Temperature Difference

#### 4.3. Thermophysical Performance of an East-Facing Wall

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Flow chart of the dynamic Bayesian method developed for the estimation of the thermophysical properties of building elements with maximum a posteriori optimisation. The Markov chain Monte Carlo approach works in a similar fashion, with the following differences: (1) at each iteration a new set of candidate parameters is proposed based on the Metropolis–Hastings algorithms; and (2) the evidence is based on the reciprocal importance sampling method.

**Figure 2.**One thermal mass (1TM) model with (

**top**) and without (

**bottom**) the explicit incorporation of solar radiation as source of heat in the system. The diagram shows the equivalent electrical circuit modelling the heat transfer and the monitoring equipment; when explicitly accounting for solar radiation, ${T}_{\mathrm{s}}$ represents the surface temperature of the element. Parameters of the model are the effective thermal mass $\left({C}_{1}\right)$ and its initial temperature $\left({T}_{{\mathrm{C}}_{1}}\right)$, and up to three lumped thermal resistances $\left({R}_{1},{R}_{2},{R}_{3}\right)$. Measured quantities are the internal $\left({T}_{\mathrm{int}}\right)$ and external $\left({T}_{\mathrm{ext}}\right)$ temperatures, the heat fluxes entering the internal $\left({Q}_{\mathrm{m},\mathrm{in}}\right)$ and leaving the external $\left({Q}_{\mathrm{m},\mathrm{out}}\right)$ surfaces, and the incident solar radiation $\left({Q}_{\mathrm{sun}}\right)$.

**Figure 3.**Internal (

**left**) and external (

**right**) view of the solid brick wall (SWall) case study. Circled in blue are the pair of sensors analysed.

**Figure 4.**Internal (

**left**) view of the north- and east-facing walls (with the north-facing one on the left side of the corner), external north-facing wall (

**middle**), and external east-facing wall (

**right**) for the cavity wall (CWall) case study. The pair of sensors analysed are circled in blue.

**Figure 5.**Corner plot of the distribution of the thermophysical properties of the SWall using the 2TM model. The turquoise crossed lines indicate the maximum a posteriori (MAP) estimation used as starting point for the Markov chain Monte Carlo (MCMC) walk.

**Figure 7.**Corner plot of the distribution of the thermophysical properties of the CWall_N using the 2TM model. The turquoise crossed lines indicate the maximum a posteriori (MAP) estimation used as starting point for the Markov chain Monte Carlo (MCMC) walk.

**Figure 9.**Difference between the length of the hypothetical monitoring campaign required by the 2TM model and the average method to stabilise for the SWall (

**left**) and the CWall_N (

**right**). The crosses mark a period of missing data, while the grey bars indicate the periods where only the 2TM model stabilised.

**Figure 10.**Kernel density estimation of the relative discrepancy between the U-value estimates from the 2TM model and average method, for the SWall (solid line) and the CWall_N (dashed line).

**Figure 11.**U-value and relative systematic measurement error estimates for the 2TM model (black x-crosses) and average method (grey crosses) as a function of the coefficient of variation of the temperature differences, for the SWall (

**top**) and CWall_N (

**bottom**).

**Figure 12.**U-value and relative systematic measurement error estimates for the 2TM model as a function of the coefficient of variation of both the temperature differences (

**top**) and the incident solar radiation (

**bottom**) for the CWall_E.

**Table 1.**Thermophysical properties for the SWall for the average (AM) and dynamic (using the 1TM (1HF), 1TM (2HF), and the 2TM models) method. Only the statistical error is shown, and the number of significant figures was chosen to illustrate the level of the error.

Parameters | Literature | AM | 1TM (1 HF) | 1TM (2 HF) | 2TM | Units |
---|---|---|---|---|---|---|

${R}_{1}$ | $0.071\pm 0.001$ | $0.075\pm 0.001$ | $0.075\pm 0.001$ | $\mathrm{m}{}^{2}\mathrm{K}{\mathrm{W}}^{-1}$ | ||

${R}_{2}$ | $0.344\pm 0.002$ | $0.350\pm 0.003$ | $0.314\pm 0.003$ | $\mathrm{m}{}^{2}\mathrm{K}{\mathrm{W}}^{-1}$ | ||

${R}_{3}$ | $0.044\pm 0.002$ | $\mathrm{m}{}^{2}\mathrm{K}{\mathrm{W}}^{-1}$ | ||||

${C}_{1}$ | $\left[1.08,1.80\right]\times {10}^{5}$ | $2.25\pm 0.05\times {10}^{5}$ | $2.28\pm 0.06\times {10}^{5}$ | $2.24\pm 0.04\times {10}^{5}$ | $\mathrm{J}\mathrm{m}{}^{-2}\mathrm{K}{}^{-1}$ | |

${C}_{2}$ | $\left[1.01,1.92\right]\times {10}^{5}$ | $0.56\pm 0.03\times {10}^{5}$ | $\mathrm{J}\mathrm{m}{}^{-2}\mathrm{K}{}^{-1}$ | |||

${T}_{{\mathrm{C}}_{1}}^{0}$ | $18.24\pm 0.05$ | $18.27\pm 0.06$ | $18.21\pm 0.06$ | ${}^{\circ}\mathrm{C}$ | ||

${T}_{{\mathrm{C}}_{2}}^{0}$ | $12.66\pm 0.08$ | ${}^{\circ}\mathrm{C}$ | ||||

R-value | $\left[0.29,0.73\right]$ | $0.42$ | $0.415\pm 0.001$ | $0.425\pm 0.002$ | $0.433\pm 0.002$ | $\mathrm{m}{}^{2}\mathrm{K}{\mathrm{W}}^{-1}$ |

U-value | $\left[1.11,2.16\right]$ | $1.69$ | $1.747\pm 0.007$ | $1.717\pm 0.012$ | $1.694\pm 0.011$ | $\mathrm{W}\mathrm{m}{}^{-2}\mathrm{K}{}^{-1}$ |

**Table 2.**Thermophysical properties for the CWall_N for the average (AM) and dynamic (using the 1TM (1HF), 1TM (2HF), and the 2TM models) method. Only the statistical error is shown, and the number of significant figures was chosen to illustrate the level of the error.

Parameters | Literature | AM | 1TM (1 HF) | 1TM (2 HF) | 2TM | Units |
---|---|---|---|---|---|---|

${R}_{1}$ | $0.066\pm 0.001$ | $1.411\pm 0.004$ | $0.067\pm 0.001$ | $\mathrm{m}{}^{2}\mathrm{K}{\mathrm{W}}^{-1}$ | ||

${R}_{2}$ | $1.361\pm 0.007$ | $0.062\pm 0.001$ | $1.299\pm 0.008$ | $\mathrm{m}{}^{2}\mathrm{K}{\mathrm{W}}^{-1}$ | ||

${R}_{3}$ | $0.061\pm 0.001$ | $\mathrm{m}{}^{2}\mathrm{K}{\mathrm{W}}^{-1}$ | ||||

${C}_{1}$ | $\left[0.51,0.90\right]\times {10}^{5}$ | $0.82\pm 0.01\times {10}^{5}$ | $0.94\pm 0.01\times {10}^{5}$ | $0.80\pm 0.01\times {10}^{5}$ | $\mathrm{J}\mathrm{m}{}^{-2}\mathrm{K}{}^{-1}$ | |

${C}_{2}$ | $\left[0.96,2.08\right]\times {10}^{5}$ | $0.92\pm 0.01$ | $\mathrm{J}\mathrm{m}{}^{-2}\mathrm{K}{}^{-1}$ | |||

${T}_{{\mathrm{C}}_{1}}^{0}$ | $17.56\pm 0.03$ | $11.39\pm 0.05$ | $17.52\pm 0.03$ | ${}^{\circ}\mathrm{C}$ | ||

${T}_{{\mathrm{C}}_{2}}^{0}$ | $11.41\pm 0.05$ | ${}^{\circ}\mathrm{C}$ | ||||

R-value | $\left[2.36,3.01\right]$ | $1.47$ | $1.427\pm 0.007$ | $1.473\pm 0.004$ | $1.427\pm 0.008$ | $\mathrm{m}{}^{2}\mathrm{K}{\mathrm{W}}^{-1}$ |

U-value | $\left[0.32,0.40\right]$ | $0.61$ | $0.631\pm 0.003$ | $0.613\pm 0.002$ | $0.631\pm 0.004$ | $\mathrm{W}\mathrm{m}{}^{-2}\mathrm{K}{}^{-1}$ |

**Table 3.**Minimum, maximum, mean, and standard deviation of U-value and relative systematic measurement error estimates for the SWall and the CWall_N, using the average and dynamic (2TM model) method and hypothetical monitoring campaigns of different length.

Method | Min | Max | Mean | St Dev | Units | |
---|---|---|---|---|---|---|

SWall | AM | 1.28 | 1.92 | 1.71 | 0.14 | $\mathrm{W}\mathrm{m}{}^{-2}\mathrm{K}$ |

AM error | 14 | 50 | 22 | 8 | % | |

2TM | 1.43 | 1.87 | 1.72 | 0.08 | $\mathrm{W}\mathrm{m}{}^{-2}\mathrm{K}$ | |

2TM error | 8 | 32 | 15 | 6 | % | |

CWall_N | AM | 0.59 | 1.00 | 0.71 | 0.08 | $\mathrm{W}\mathrm{m}{}^{-2}\mathrm{K}$ |

AM error | 13 | 21 | 16 | 3 | % | |

2TM | 0.64 | 0.82 | 0.70 | 0.05 | $\mathrm{W}\mathrm{m}{}^{-2}\mathrm{K}$ | |

2TM error | 7 | 14 | 10 | 2 | % |

**Table 4.**Minimum, maximum, mean, and standard deviation of U-value and relative systematic measurement error estimates for the CWall_E, using the 2TM model and hypothetical monitoring campaigns of different length.

Model | Min | Max | Mean | St Dev | Units |
---|---|---|---|---|---|

2TM | 0.68 | 0.92 | 0.77 | 0.05 | $\mathrm{W}\mathrm{m}{}^{-2}\mathrm{K}{}^{-1}$ |

2TM error | 5 | 37 | 16 | 9 | % |

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## Share and Cite

**MDPI and ACS Style**

Gori, V.; Biddulph, P.; Elwell, C.A. A Bayesian Dynamic Method to Estimate the Thermophysical Properties of Building Elements in All Seasons, Orientations and with Reduced Error. *Energies* **2018**, *11*, 802.
https://doi.org/10.3390/en11040802

**AMA Style**

Gori V, Biddulph P, Elwell CA. A Bayesian Dynamic Method to Estimate the Thermophysical Properties of Building Elements in All Seasons, Orientations and with Reduced Error. *Energies*. 2018; 11(4):802.
https://doi.org/10.3390/en11040802

**Chicago/Turabian Style**

Gori, Virginia, Phillip Biddulph, and Clifford A. Elwell. 2018. "A Bayesian Dynamic Method to Estimate the Thermophysical Properties of Building Elements in All Seasons, Orientations and with Reduced Error" *Energies* 11, no. 4: 802.
https://doi.org/10.3390/en11040802