# Thermal Inertia Characterization of Multilayer Lightweight Walls: Numerical Analysis and Experimental Validation

^{*}

## Abstract

**:**

## 1. Introduction

^{2}K/W) of building enclosures is a well-known method to reduce heat loss. Thermal transmittance (U-value, W/m

^{2}K) is the inverse of the total resistance values of enclosure heat transfer (R

_{tot}). The U-value is the rate of heat transfer (in watts) through 1 m

^{2}of a building element and is determined by the difference in temperature across the wall. Most building standards [2] evaluate the U-value of the materials that form the enclosure, characterizing their steady-state thermal physical properties. However, this method has the disadvantage of not accounting for the heat storage capacity of materials when environmental conditions change. In recent years, it has been concluded that it is not possible to design thermally efficient buildings based on U-value alone. The determination of the transient thermal behavior of different construction solutions is receiving a great deal of interest in order to advance the concept of the thermal inertia of construction elements [3].

^{3}in wall assemblies with one layer of insulation. They also found that the placement of insulation into two layers improves the optimal decrement factor and time lag values using concretes with a density above 1200 kg/m

^{3}. The best results were obtained by placing the insulation on the external surface and the center of the enclosure.

^{2}instrumented hot-box climatic chamber, and the material was subjected to variable temperature and humidity conditions to obtain thermal inertia values according to the ISO standard rule [3]. We then created a finite element numerical model of a bi-dimensional wall section with holes, which was subjected to the temperature conditions established in the experimental test, both under steady- and transient-state conditions. We then compared the numerical and experimental findings and related them to standard thermal inertia results.

## 2. Materials and Methods

#### 2.1. Materials Used

^{2}wall with 0.34 m thickness made up of lightweight concrete blocks that were 0.30 m long, 0.25 m thick, and 0.15 m high. The block wall was coated with insulating mortar with a thickness of 0.015 m and 0.060 m. The wall layer composition is shown in Table 1.

#### 2.2. Experimental Tests

^{2}wall that was built and placed in a climatic air-conditioned system that controlled relative humidity (RH) and temperature conditions. The air-conditioned system was joined to a climatic box (hot-box) with a 1 m

^{3}inside volume, where the desired thermal conditions were provided. The wall tested was built and placed over a steel frame and joined to the hot-box, as shown in Figure 1d. The perimeter of the wall has been thermally insulated with low thermal conductivity foam in order to avoid lateral heat losses. Sensors at different locations were used to measure temperature, relative humidity, and thermal fluxes. Sixteen SHT75 capacitive sensors and one Sensirion EK-H3 datalogger were used to measure the temperature and relative humidity, while four Hukseflux HFP01 sensors and four LI-19 data-loggers were used to measure heat fluxes through the wall. The locations of the sensors are shown in Figure 2.

#### 2.3. Numerical Models

#### 2.3.1. Geometrical Model

#### 2.3.2. Numerical Mesh

#### 2.3.3. Boundary Conditions

^{2}K for an ambient temperature of 23 °C, which was the temperature experimentally measured in the indoor face of the wall during the laboratory tests. Second, finite element SURF151 was used to apply a film coefficient, including convection and radiation. This coefficient was calculated using the ISO 6946 standard, Annex D [2].

^{2}K) and ${h}_{a}$ is the film coefficient related to the convection (W/m

^{2}K).

_{m}(°C), using the following equation:

^{2}was applied to the wall’s outdoor surface (Layer ID 4). In the transient thermal analysis, a sine-wave function was applied using 10 temperature cycles of 24 h. These cycles ranged between 10 and 37.5 °C and followed experimental hot-box tests. The total duration of the sine curve was 839,400 s, which was divided into 78 load steps.

#### 2.4. Design of Experiments-Based Optimization

## 3. Results

#### 3.1. Experimental Results

^{2}K); $\dot{q}$ is the heat flux (W/m

^{2}); $\Delta T$ is the temperature increase (K); ${R}_{se}$ is the thermal resistance of the outdoor wall surface for a horizontal heat flux (0.04 m

^{2}K/W, according to [2]); ${R}_{si}$ is the thermal resistance of the indoor wall surface for a horizontal heat flux (0.13 m

^{2}K/W, according to [2]); $R={R}_{GWP}+{R}_{LWGA}+{R}_{LWARB25}+{R}_{GWG}$ is the wall thermal resistance excluding the outdoor and indoor surface resistances (m

^{2}K/W); ${R}_{tot}$ is the total thermal resistance of the wall, including surface resistances (m

^{2}K/W); $e$ is the thickness of the wall (m); and is the equivalent thermal conductivity of the wall (W/mK).

_{tot}and λ

_{eq}) according to Equations (6)–(8) are shown in Table 3 (in which material thermal properties were taken from Table 2).

^{2}K). This may have been due to the non-uniform moisture content in the wall or the laboratory ambient conditions (60–70% RH). Therefore, the effective insulation of the lightweight mortar (LWMA) in the interior wall decreased more than expected.

_{i,max}is the maximum (peak) indoor surface temperature (°C); T

_{i,ave}is the average indoor surface temperature (°C); T

_{e,max}is the peak outdoor surface temperature (°C); T

_{e,min}is the average exterior outdoor temperature (°C); t

_{Ti,max}are the peak indoor temperature time points (h); and t

_{Te,max}are the peak outdoor temperature time points (h).

^{2}.

^{2}wall, place it in the hot-box and wait for at least two weeks for materials to stabilize before testing, so as not to bias results.

#### 3.2. Numerical Results

#### 3.2.1. Design of Experiment Analyses (DOE)

^{2}):

^{2}); ${A}^{num}=\frac{{\dot{q}}_{num}^{max}-{\dot{q}}_{num}^{min}}{2}$ is the numerical thermal flux amplitude (W/m

^{2}); ${\zeta}^{exp}={\dot{q}}_{exp}^{max}-{A}^{exp}$ is the experimental thermal flux phase (W/m

^{2}); ${\zeta}^{num}={\dot{q}}_{num}^{max}-{A}^{num}$, is the numerical thermal flux phase (W/m

^{2}); ${\dot{q}}_{exp}^{max}$ is the maximum experimental thermal flux value (see Figure 7, W/m

^{2}); ${\dot{q}}_{exp}^{min}$ is the minimum experimental thermal flux value (see Figure 7, W/m

^{2}); ${\dot{q}}_{num}^{max}$ is the maximum numerical thermal flux value (W/m

^{2}); and ${\dot{q}}_{num}^{min}$ is the minimum numerical thermal flux value (W/m

^{2}).

^{2}for the objective function according to Equation (11). The difference between the numerical and experimental results at this candidate point was approximately 12%.

#### 3.2.2. Steady-State Thermal Analysis

^{2}. On the outdoor face of the wall, the average temperature was 41.223 °C. According to Equation (6), the numerical wall thermal transmittance was 0.751 W/m

^{2}K.

#### 3.2.3. Transient Thermal Analysis

## 4. Experimental, Numerical and Empirical Comparison

#### 4.1. Steady-State Numerical and Experimental Comparison

#### 4.2. Transient-State Numerical and Experimental Comparison

^{2}while the experimental hot-box result had an amplitude of 6.225 W/m

^{2}. The difference between the numerical and experimental thermal flux phases was 0.67 W/m

^{2}. According to the objective function indicated in Equation (11), a difference of 0.54 W/m

^{2}was reached, representing an 8% difference with respect to the experimental thermal flux amplitude.

## 5. Conclusions

- (1)
- Experimental results
- -
- When the hot-box relative humidity is 70%, the measured thermal transmittance is 12% higher than at relative humidity of 30%. This demonstrates the importance of the moisture content in porous materials due to their hygroscopicity. In this sense, it is necessary to obtain the material thermal properties at specific humidity contents and account for the moisture transport inside multilayer walls. Moreover, the laboratory ambient conditions and the wall construction process can produce a non-uniform moisture content inside blocks and layers, affecting the thermal results.
- -
- Thermal transmittance in steady-state tests was higher than the empirical value calculated following the ISO 6946 standard [2], with differences ranging from 5 to 15%. Consequently, moisture content must be taken into account to obtain the thermal conductivity values of each layer of porous materials, in this case, Layer 3 (LWMA) and Layer 2 (LWBA25).
- -
- Transient thermal tests in different thermal flux directions showed significant differences in terms of time lag (1.25 h difference) and decrement factor (23% difference). Thermal inertia values were, therefore, strongly affected by the multilayer wall composition.
- -
- According to the thermal inertia tests, Layer 3 (LWMA) was appropriate to be exposed to the outdoor conditions. This material had the lowest density, thermal conductivity, thermal diffusivity and heat capacity in the multi-layer wall. With this configuration, a high time lag and low decrement factor were obtained, so the optimal indoor conditions were reached. Thus, materials with lower values in terms of density and thermal properties are recommended to be used at outdoor conditions.
- -
- Empirical dynamic thermal inertia parameters exhibited lower time lag and higher decrement factor values than the experimental results. In the empirical formulation, the decrement factor is given by the periodic thermal transmittance divided by the static thermal transmittance of layers with homogeneous materials. In this work, lightweight concrete blocks with cavities were not homogeneous. Their dynamic thermal behavior was not calculated accurately by the empirical formulation. Consequently, the ISO-13786 standard [3] provides more restrictive designs than the experimental ones, and further research is required in this field.

- (2)
- Numerical simulation
- -
- A bidimensional FEM model, including conduction, convection and radiation, was used to simulate both steady- and transient-state thermal conditions in a multi-layer wall. The average temperature and heat flux of the block in the middle of the wall was used to obtain numerical results for the thermal transmittance, time lag, and decrement factor.
- -
- Two DOE analyses were performed to determine the most influential parameters and obtain accurate material thermal properties similar to the experimental behavior.
- -
- Sensitivity analysis obtained from the first DOE under steady-state conditions showed that the three most important input parameters were the thermal conductivity of Layer 3 (LWMA), the thermal conductivity of Layer 2 (LWBA25), and the emissivity of the cavities in the block.
- -
- A multicriteria optimization based on second DOE analysis under transient thermal conditions was used to fit the experimental and numerical results. Thermal flux amplitude and phase Equation were used as an objective function according to Equation (11). From this optimization, the thermal conductivity of Layer 3 (LWMA) and Layer 2 (LWBA25) was obtained and used in FEM numerical models for transient- and steady-state analyses.

- (3)
- Numerical, empirical and experimental comparison
- -
- Numerical and empirical thermal transmittance results of steady-state analyses were in good agreement, showing differences of less than 6%.
- -
- In the transient-state numerical analysis, the temperature amplitude on the indoor face of the wall was lower than the experimental amplitude. Thus, the numerical decrement factor was lower than the value obtained experimentally or empirically. More research is necessary to better understand this effect in the transient state. In this sense, numerical FEM models could be solved with a different time discretization (smaller time steps and lower heat tolerance). In addition, dynamic heat transfer phenomena inside cavities, specifically convection and radiation, must be deeply studied.
- -
- Numerical and experimental time lags showed good agreement, with a difference of less than 3% between values. However, results of empirical calculations presented lower values.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Li, W.; Yang, L.; Ji, Y.; Xu, P. Estimating demand response potential under coupled thermal inertia of building and air-conditioning system. Energy Build.
**2019**, 182, 19–29. [Google Scholar] [CrossRef] - ISO 6946:2017. Building Components and Building Elements—Thermal Resistance and Thermal Transmittance—Calculation Methods; ISO: Geneva, Switzerland, 2017. [Google Scholar]
- ISO 13786:2017. Thermal Performance of Building Components—Dynamic Thermal Characteristics—Calculation Methods; ISO: Geneva, Switzerland, 2017. [Google Scholar]
- Aste, N.; Angelotti, A.; Buzzetti, M. The influence of the external walls thermal inertia on the energy performance of well insulated buildings. Energy Build.
**2009**, 41, 1181–1187. [Google Scholar] [CrossRef] - Kossecka, E.; Kosny, J. Influence of insulation configuration on heating and cooling loads in a continuously used building. Energy Build.
**2002**, 34, 321–331. [Google Scholar] [CrossRef] - Asan, H.; Sancaktar, Y. Effects of Wall’s thermophysical properties on time lag and decrement factor. Energy Build.
**1998**, 28, 159–166. [Google Scholar] [CrossRef] - Asan, H. Effects of wall’s insulation thickness and position on time lag and decrement factor. Energy Build.
**1998**, 28, 299–305. [Google Scholar] [CrossRef] - Ozel, M.; Pihtili, K. Optimum location and distribution of insulation layers on building walls with various orientations. Build. Environ.
**2007**, 42, 3051–3059. [Google Scholar] [CrossRef] - Di Perna, C.; Stazi, F.; Casalena, A.U.; D’Orazio, M. Influence of the internal inertia of the building envelope on summertime comfort in buildings with high internal heat loads. Energy Build.
**2011**, 43, 200–206. [Google Scholar] [CrossRef] - Ng, S.-C.; Low, K.-S.; Tioh, N.-H. Thermal inertia of newspaper sandwiched aerated lightweight concrete wall panels: Experimental study. Energy Build.
**2011**, 43, 2956–2960. [Google Scholar] [CrossRef] - Jin, X.; Zhang, X.; Cao, Y.; Wang, G. Thermal performance evaluation of the wall using heat flux time lag and decrement factor. Energy Build.
**2012**, 47, 369–374. [Google Scholar] [CrossRef] - Kontoleon, K.; Theodosiou, T.; Tsikaloudaki, K. The influence of concrete density and conductivity on walls’ thermal inertia parameters under a variety of masonry and insulation placements. Appl. Energy
**2013**, 112, 325–337. [Google Scholar] [CrossRef] - Del Coz Díaz, J.; Nieto, P.G.; Biempica, C.B.; Gero, M.P. Analysis and optimization of the heat-insulating light concrete hollow brick walls design by the finite element method. Appl. Therm. Eng.
**2007**, 27, 1445–1456. [Google Scholar] [CrossRef] - Del Coz Díaz, J.; Nieto, P.G.; Rabanal, F.A.; Hernández, J.D. Non-linear thermal analysis of the efficiency of light concrete multi-holed bricks with large recesses by FEM. Appl. Math. Comput.
**2012**, 218, 10040–10049. [Google Scholar] [CrossRef] - Gencel, O.; del Coz Díaz, J.J.; Sutcu, M.; Kocyigit, F.; Rabanal, F.P.Á.; Alonso-Martínez, M.; Barrera, G.M. Thermal Performance Optimization of Lightweight Concrete/EPS Layered Composite Building Blocks. Int. J. Thermophys.
**2021**, 42, 1–14. [Google Scholar] [CrossRef] - Arendt, K.; Krzaczek, M.; Florczuk, J. Numerical analysis by FEM and analytical study of the dynamic thermal behavior of hollow bricks with different cavity concentration. Int. J. Therm. Sci.
**2011**, 50, 1543–1553. [Google Scholar] [CrossRef] - Zhang, Y.; Du, K.; He, J.; Yang, L.; Li, Y.; Li, S. Impact factors analysis on the thermal performance of hollow block wall. Energy Build.
**2014**, 75, 330–341. [Google Scholar] [CrossRef] - Soret, G.; Vacca, P.; Tignard, J.; Hidalgo, J.; Maluk, C.; Aitchison, M.; Torero, J. Thermal inertia as an integrative parameter for building performance. J. Build. Eng.
**2021**, 33, 101623. [Google Scholar] [CrossRef] - Del Coz Díaz, J.; Álvarez-Rabanal, F.; Gencel, O.; Nieto, P.G.; Alonso-Martínez, M.; Navarro-Manso, A.; Prendes-Gero, B. Hygrothermal study of lightweight concrete hollow bricks: A new proposed experimental–numerical method. Energy Build.
**2014**, 70, 194–206. [Google Scholar] [CrossRef] - Del Coz Díaz, J.J.; Rabanal, F.P.Á.; Nieto, P.J.G.; Hernández, J.D.; Soria, B.R.; Pérez-Bella, J.M. Hygrothermal properties of lightweight concrete: Experiments and numerical fitting study. Constr. Build. Mater.
**2013**, 40, 543–555. [Google Scholar] [CrossRef] - ASTM D7984—16. Standard Test Method for Measurement of Thermal Effusivity of Fabrics Using a Modified Transient Plane Source (MTPS) Instrument; ASTM: West Conshohocken, PA, USA, 2016. [Google Scholar]
- Bull, J.W. (Ed.) Computer Analysis and Design of Masonry Structures; Saxe-Coburg Publications: Kippen, Scotland, 2016. [Google Scholar]
- UNE-EN 1745:2020. Masonry and Masonry Products—Methods for Determining Thermal Properties; AENOR: Madrid, Spain, 2020. [Google Scholar]
- Moaveni, S. Finite Element Analysis, Theory and Application with ANSYS; Prentice Hall: Hoboken, NJ, USA, 1999. [Google Scholar]
- ANSYS Mechanical APDL Theory Reference. 2013. Available online: https://doi.org/www.ansys.com (accessed on 1 January 2020).
- Ansys
^{®}Academic Research Mechanical, Release 19.2, Mechanical User’s Guide; ANSYS, Inc.: Canonsburg, PA, USA, 2019. - Madenci, E.; Guven, I. The Finite Element Method and Applications in Engineering Using Ansys
^{®}; Springer: Berlin, Germany, 2006. [Google Scholar] [CrossRef] - Thompson, M.; Thompson, J. ANSYS Mechanical APDL for Finite Element Analysis; Butterworth-Heinemann: New York, NY, USA, 2017. [Google Scholar]
- Antony, J. Design of Experiments for Engineers and Scientist; Butterworth-Heinemann: New York, NY, USA, 2003. [Google Scholar]
- Preece, D.A.; Montgomery, D.C. Design and Analysis of Experiments; John Wiley & Sons: Hoboken, NJ, USA, 1978. [Google Scholar]
- Williams, F.A. Simplified Theory for Ignition Times of Hypergolic Gelled Propellants. J. Propuls. Power
**2009**, 25, 1354–1357. [Google Scholar] [CrossRef] - Ulgen, K. Experimental and theoretical investigation of effects of wall’s thermophysical properties on time lag and decrement factor. Energy Build.
**2002**, 34, 273–278. [Google Scholar] [CrossRef] - Ökten, K.; Özdemir, M. Experimental Investigation of the Performance of Porous Insulation Materials Under Temporary Humidity. J. Thermophys. Heat Transf.
**2021**, 35, 179–186. [Google Scholar] [CrossRef] - Wang, Y.; Zhao, Z.; Liu, Y.; Wang, D.; Ma, C.; Liu, J. Comprehensive correction of thermal conductivity of moist porous building materials with static moisture distribution and moisture transfer. Energy
**2019**, 176, 103–118. [Google Scholar] [CrossRef]

**Figure 3.**Geometrical model and material layer identification: (

**1**) GWG, (

**2**) LWBA25, (

**3**) LWMA and (

**4**) GWP.

**Figure 5.**Transient-state test results showing the decrement factor and time lag in thermal flux Direction 1.

**Figure 6.**Transient-state test results showing the decrement factor and time lag in thermal flux Direction 2.

**Figure 7.**Transient experimental results showing temperature and heat fluxes in thermal flux Direction 2.

**Figure 8.**DOE results: (

**a**) Sensitivity analysis of parameters and response surfaces of (

**b**) LWAR25 and LWMA thermal conductivities vs. thermal transmittance and (

**c**) GWG and GWP thermal conductivities vs. thermal transmittance.

**Figure 11.**Numerical analysis in thermal flux Direction 2, showing the transient results of the decrement factor and time lag at the central location of the wall.

Material | Layer ID | Acronym | Thickness (m) | Wall Section |
---|---|---|---|---|

Gypsum-Weber grueso ^{®} | 1 | GWG | 0.015 | |

Lightweight brick Airblock 25 ^{®} | 2 | LWBA25 | 0.250 | |

Lightweight mortar-Aislone ^{®} | 3 | LWMA | 0.060 | |

Gypsum-Weber pral clima ^{®} | 4 | GWP | 0.015 |

Material | Layer ID | Thickness (m) | Thermal Conductivity (W/mK) | Density (kg/m^{3}) | Thermal Diffusivity (mm^{2}/s) | Heat Capacity (J/kgK) |
---|---|---|---|---|---|---|

GWG | 1 | 0.015 | 0.6781 | 1600 | 0.4106 | 1093.86 |

LWBA25 ^{1} | 2 | 0.250 | 0.3350 | 1106 | ||

LWMA | 3 | 0.060 | 0.0812 | 285 | 0.3178 | 983.42 |

GWP | 4 | 0.015 | 0.7087 | 1500 | 0.4240 | 1261.08 |

^{1}Thermal conductivity of LWBA25 material was given by the manufacturer.

**Table 3.**Steady-state thermal results. The thermal properties from Table 2 have been applied here.

Case | $\mathbf{\Delta}\mathit{T}\text{}\left(\mathbf{K}\right)$ | $\dot{\mathit{q}}$ (W/m^{2})
| $\mathit{U}$ (W/m^{2}K)
| ${\mathit{R}}_{\mathit{t}\mathit{o}\mathit{t}}\text{}({\mathbf{m}}^{2}\mathbf{K}/\mathbf{W})$ | ${\mathit{\lambda}}_{\mathit{e}\mathit{q}}\text{}(\mathbf{W}/\mathbf{mK})$ |
---|---|---|---|---|---|

RH 30% | 24.510 | 15.20 | 0.620 | 1.613 | 0.241 |

RH 70% | 27.940 | 19.28 | 0.690 | 1.449 | 0.256 |

Empirical [2] | 0.589 | 1.875 | 0.203 |

Case | Φ (h)
| $\mathit{f}$ (-) |
---|---|---|

Direction 1 | 13.58 | 0.0313 |

Direction 2 | 14.83 | 0.0241 |

Empirical (s/ISO 13786) | 10.86 | 0.1249 |

Material | Initial Value | Optimized Value |
---|---|---|

Layer 3—LWMA | 0.0812 | 0.1265 |

Layer 2—LWBA25 | 0.335 | 0.380 |

Case | U (W/m^{2}K) | R_{tot} (m^{2}K/W) | l_{eq} (W/mK) |
---|---|---|---|

Hot-box test | 0.690 | 1.449 | 0.256 |

Numerical FEM | 0.751 | 1.331 | 0.301 |

Empirical [2] | 0.743 | 1.345 | 0.289 |

Case | Φ (h) | f (-) |
---|---|---|

Hot-box test | 14.83 | 0.0241 |

Numerical FEM | 13.28 | 0.0072 |

Empirical [3] | 11.86 | 0.1605 |

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**MDPI and ACS Style**

del Coz-Díaz, J.J.; Álvarez-Rabanal, F.P.; Alonso-Martínez, M.; Martínez-Martínez, J.E.
Thermal Inertia Characterization of Multilayer Lightweight Walls: Numerical Analysis and Experimental Validation. *Appl. Sci.* **2021**, *11*, 5008.
https://doi.org/10.3390/app11115008

**AMA Style**

del Coz-Díaz JJ, Álvarez-Rabanal FP, Alonso-Martínez M, Martínez-Martínez JE.
Thermal Inertia Characterization of Multilayer Lightweight Walls: Numerical Analysis and Experimental Validation. *Applied Sciences*. 2021; 11(11):5008.
https://doi.org/10.3390/app11115008

**Chicago/Turabian Style**

del Coz-Díaz, Juan José, Felipe Pedro Álvarez-Rabanal, Mar Alonso-Martínez, and Juan Enrique Martínez-Martínez.
2021. "Thermal Inertia Characterization of Multilayer Lightweight Walls: Numerical Analysis and Experimental Validation" *Applied Sciences* 11, no. 11: 5008.
https://doi.org/10.3390/app11115008