# 3D Modeling of the Thermal Transfer through Precast Buildings Envelopes

^{*}

## Abstract

**:**

## 1. Introduction

_{2}emissions [1]. In Morocco, consumption represents about 25%, from which 18% concerns residential buildings while the rest is devoted to the tertiary sector [2].

^{6}, a Prandtl number of 0.71 and an aspect ratio of 1. Nonetheless, the method is considered to be more general and can be applied to any kind of geometry. Another interesting work has been performed by Cordoba et al. [15]. In this, the authors conducted a numerical study on a laminar, steady, incompressible, free convection with surface radiation in a two-dimensional open cavity. A heating source was imposed on the wall in parallel to the opening, while the rest of the walls were considered adiabatic. The governing equations of, respectively, energy, mass and momentum were resolved based on the usage of a finite volume approach, which was implemented in FORTRAN. Their study revolved around heat transfer characteristics, the orientation of the structure, and the influence of surface radiation on the fluid flow. Reported results showed that the heat transfer decreases when decreasing the cavity tilt angle for all used values of Rayleigh number and emissivity. They also mentioned that, at a certain tilt angle value, thermal radiation exchange between walls is much more significant. De Vahl [16] investigated the bi-dimensional circulation of air in an enclosed square cavity. Desirable movement was generated using a gradient of temperature imposed at both left and right walls as boundary conditions. Accordingly, De Vahl was able to prove that, in terms of vorticity transport and energy equations for Rayleigh numbers, up to 2 × 10

^{−5}, can converge. Additionally, he also reported the impact of the Prandtl number, which can play the role of a stabilizing factor, especially if it ranges from 10

^{−1}to 10

^{3}. Karatas and Derbently [17], in their study, have considered conjugated radiation and natural convection in a rectangular-shaped closed cavity with only one active vertical wall. The cooled surface is made of aluminum and plastic for the rest of the structure. However, since all walls are painted in white, the emissivity is considered to be constant.

^{3}and 10

^{6}, were used for a Newtonian fluid with a Prandtl number of 1. Results showed that the presence of these elements has an impact on the hydrodynamic and thermal behavior of the fluid in the cavity. Khatamifar et al. [19] presented numerical simulation results related to combined heat transfer and natural convection flow in a gradually heated square cavity for a large scope of the Rayleigh number (10

^{5}–10

^{9}). In fact, simulations were carried out using the finite volume method following three dimensionless partition thicknesses and positions. Results showed that the Nusselt number increases with the Rayleigh number but decreases with the thickness. Pandey et al., in their paper [20], presented a review of numerical and experimental research works linked to natural convection in enclosures with/out the presence of internal bodies. These latter were taken under different shapes in order to figure out their impact on buoyancy driven-flow among enclosures. The used methods mainly cover spectral element method, finite element approaches, Latice Boltzmann method (LBM), finite volume method and so on. Their paper also discussed the effect of multiple parameters, such as Rayleigh, Grashof, and Prandtl numbers, in order to make the optimal choice of design parameters according to the desired system requirements. Ouakarrouch et al. [21] presented a simulation englobing the conduction heat transfer and natural convection as well as radiative heat through two kinds of alveolar walls used in a recent building’s construction (3- and 6-hole numbers). Results showed that heat transfer is increased once radiative transfer is considered into calculations. Further, it was proved that conductive transfer accounts for almost the half of the heat flux in the block of 20 incorporating a thermally conducting vertical partition.

- Extensive experiments have been conducted using thermography analysis to detect thermal bridge placement among the EEBLAB.
- A transient numerical model is proposed for buildings’ thermal characterization, taking into account its dynamic behavior.
- A mechanical numerical model is presented to evaluate the distribution of mechanical stress towards buildings’ envelopes in response to the effect of their own weights.

## 2. Theoretical Background

#### 2.1. Thermal Model Equations

^{3}), C

_{p}is the specific heat capacity (J/kg·K), q is the flux density (W/m

^{2}), S is the surface of the medium (m

^{2}), f is a function which includes all thermal loading sources, k is the conductivity (W/m·K), ∆T is the temperature variation, and n is the normal vector to the surface:

^{−1}), g is the gravity (m/s

^{2}), L

_{c}is the characteristic length (m), $\rho $ is the density (kg/m

^{3}) and μ is the dynamic viscosity (Pa·s).

_{a}is the ambiance (internal or external) temperature (K), T is the temperature of the wall (K), and S represents the area of the wall (m

^{2}).

^{2}K), to the conductivity matrix, and the addition of a term Q

_{c}, representing the term h × T

_{a}, to the second member.

^{2}).

^{3}. This factor can be extracted by factorizing by the term (T

_{a}− T) in Equation (12).

- Considering a wall separating two different atmospheres, the total thermal resistance is the set of all the layers of materials or air that constitute the wall, in addition to surface-exchange resistances (resulting from the phenomenon of convection) (see Figure 2a). From this resistance, we can obtain the heat transfer coefficient, which represents the amount of heat transmitted through this wall in steady state, per unit time, per unit area and temperature difference of one-degree Kelvin. In other words, this transmittance coefficient represents the opposite of the total thermal resistance of the wall and is given in W/m
^{2}·K. - However, if the wall is made of several non-homogeneous materials, a very simplified calculation method can be used. It mainly consists of considering that the total thermal resistance is between upper and lower limits. These limits, for each case, could be computed by dividing the wall into different sections whose layers are homogeneous and then applying the normal rules of calculation.

^{2}·K) can be subdivided into two coefficients (upper and lower) and can be calculated as shown in Equations (14) and (15).

_{up}is the upper limit of the transmission heat coefficient. U

_{low}is the lower limit of the transmission heat coefficient. P

_{as}, P

_{bs}, P

_{cs}, P

_{1}, P

_{2}and P

_{3}are respectively the percentage of material constituting layer a

_{s}, b

_{s}, c

_{s}, 1, 2 and 3.

_{T}) is then expressed as follows:

_{w}[26].

_{g}is the visible perimeter of the glazing; S

_{g}, S

_{f}, S

_{p}and S

_{vg}are respectively the surfaces of the glazing, frame, panel and ventilation grid.

^{2}of window.

#### 2.2. Mechanical Model Equations

- The equilibrium equation (Equation (23)), which illustrates the static or dynamic equilibrium of all external and internal forces of the studied system [27]:

- The behavior law following three-dimensional Hook elasticity as shown in Equation (24).

- The hypothesis of small deformations by assuming that the transformation between the initial equilibrium state and the current equilibrium state is infinitesimal [28], i.e.,

_{T}, σ

_{d}, U and C are respectively the total strain tensor (no unit), the stress tensor (N/m

^{2}), the displacement field (m) and the elasticity modulus tensor (Pa).

## 3. Materials and Methods

#### 3.1. Numerical Set-Up

#### 3.1.1. Geometry

^{2}of occupied surface and 30 m

^{3}of volume. Table 2 gives more information about its dimensions. It is essentially made from galvanized steel and expanded polyurethane injected into the walls for insulation purposes. The roof is made of a stratification of 3 layers (inside-out): plaster, air, galvanized steel. However, the floor is insulated using chipboard.

#### 3.1.2. Structure Meshing

^{®}Core™i7-7700HQ [email protected] 2.8 GHz, 2801 MHz, 4 cores and 8 logical processors (Dell Technologies, Round Rock, TX, USA). The assigned number of elements is 19,768 with about 41,030 nodes. Using the Cast3m tool, a density of 0.08 has been imposed in order to acquire the previously mentioned number of elements. The chosen type of meshing elements was cubic, since the geometry is regular and has a parallelepiped shape.

#### 3.1.3. Parameters Identification

#### 3.1.4. Numerical Tools

- (a)
- Thermal Boundary and Interface Conditions:

- (b)
- Mechanical Boundary Conditions:

^{2}(LC

^{2}). This is because the EEBLAB’s walls have different surfaces, which may complicate the calculation process and takes more simulation time.

#### 3.2. Experimental Set-Up

^{®}Xeon

^{®}CPU E5-2650 0 @ 2GHz computer (Dell Technologies, Round Rock, TX, USA); 64 Gb of RAM, a GPU of 43 Go-NVIDIA, an operating system of 64 bits and a 64 processor.

## 4. Results and Discussions

#### 4.1. Mesh Model and Grid Independence Checking

^{−6}Pa to 3.26 × 10

^{−5}Pa) until reaching a ‘dens’ value of 0.06 (SMXX = 3.25 × 10

^{−5}Pa). As known, reducing mesh sizing leads to increasing the number of elements, which might slow the simulation and certainly requires a lot of memory. Hence, for optimization purposes, we chose to stick with the 0.08 value (19,768 elements with about 41,030 nodes).

#### 4.2. Thermal Numerical Results

^{2}K and a maximum value of 0.06 W/m

^{2}K. The evolution curves have almost the same qualitative and quantitative aspects.

#### 4.3. Thermal Experimental Results

^{2}·K. and a maximum of 0.06 W/m

^{2}·K. It is worth to mention that the transmittance coefficients of all surfaces evolve quantitatively as well as qualitatively in the same way, except for some negligible off-sets noticed at the right wall and roof.

#### 4.4. Thermal Results Confrontation

^{2}K and a maximum of 0.06 W/m

^{2}K. Results are shown to be in agreement, with some discrepancies in the order of 1%, 7%, 6% and 8% on the front wall, right wall, left wall and the roof, respectively. This can be explained by the fact that the problem is non-linear and needs more calculation precision (number of iterations, time step, meshing size, etc.) to obtain more accurate results, not to forget the disturbances that numerical sensors may have experienced.

#### 4.5. Mechanical Numerical Results

^{−7}m.

^{4}Pa and −5.74 × 10

^{2}Pa for normal and shear stress, respectively. However, a stress concentration is evidenced through the window and its frame since the construction materials are different from the rest of the envelope.

## 5. Conclusions and Perspectives

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Symbol | Designation |

ICT | Information and Communication Technology |

IoT | Internet of Things |

EEBLAB | Energy Efficient Building Laboratory |

Ω | Domain |

ρ | Density |

Cp | Heat capacity |

T | Temperature |

Q | Flux |

q | Density of flux |

k | Conductivity |

Lc | Characteristic length |

V | Volume |

h | Heat transfer coefficient |

Ta | Ambient temperature |

ε | Emissivity |

C | Capacity matrix |

K | Conductivity matrix |

Gr | Grashof number |

Pr | Prandtl number |

β | Coefficient of thermal expansion |

μ | Dynamic viscosity |

ν | Cinematic viscosity |

g | Gravity |

γ, m | Numerical Coefficient |

U | Transmittance coefficient |

R | Resistance |

ψiLi | Punctual thermal bridges |

×j | Linear thermal bridges |

N | Nodes |

Svg/Sg/Sf/Sp | Surface of Ventilation grid/glazing/frame/panel |

σ | Constant of Steffan-Boltzman |

C | Stiffness tensor |

Pg | Visible perimeter of glazing |

U | Displacement vector |

σd | Tensor of total deformations |

εT | Stress tensor |

VDD | Voltage at drain |

DQ | Bidirectional data bus |

GND | Ground |

## References

- IEA. Global Status Report for Buildings and Construction: Towards a Zero-Emissions, Efficient and Resilient Buildings and Construction Sector. Available online: https://www.unep.org/resources/publication/2019-global-status-report-buildings-and-construction-sector (accessed on 10 June 2021).
- IEA. Perspectives for Clean Energy Transition, the Critical Role of Buildings. Available online: https://www.iea.org/reports/the-critical-role-of-buildings (accessed on 26 April 2020).
- Pérez-Lombard, L.; Ortiz, J.; Pout, C. A review on buildings energy consumption information. Energy Build.
**2008**, 40, 394–398. [Google Scholar] [CrossRef] - IEA. Energy Policies Beyond IEA Countries, Morocco 2019. Available online: https://www.connaissancedesenergies.org/sites/default/files/pdf-actualites/Energy_Policies_beyond_IEA_Contries_Morocco.pdf (accessed on 26 April 2020).
- Aderee. Règlement Thermique de Construction au Maroc, Version Simplifiée, Morocco. Available online: http://architectesmeknestafilalet.ma/documentation_telechargements/Efficacit%C3%A9%20energetique/Reglement_thermique_de_construction_au_Maroc_-_Version_simplifiee.pdf (accessed on 26 April 2020).
- Lachhab, F.; Bakhouya, M.; Ouladsine, R.; Essaaidi, M. Energy-Efficient Buildings as Complex Socio-technical Systems: Approaches and Challenges. In Advances in Complex Societal, Environmental and Engineered Systems; Springer: Cham, Switzerland, 2017; pp. 247–265. [Google Scholar]
- Bakhouya, M.; NaitMalek, Y.; Elmouatamid, A.; Lachhab, F.; Berouine, A.; Boulmrharj, S.; Ouladsine, R.; Félix, V.; Zinedine, K.; Khaidar, M.; et al. Towards a context-driven platform using IoT and big data technologies for energy efficient buildings. In Proceedings of the 2017 3rd International Conference of Cloud Computing Technologies and Applications (CloudTech), Rabat, Morocco, 24–26 October 2017; pp. 1–5. [Google Scholar]
- Elmouatamid, A.; NaitMalek, Y.; Bakhouya, M.; Ouladsine, R.; Elkamoun, N.; Zine-Dine, K.; Khaidar, M. An energy management platform for micro-grid systems using Internet of Things and Big-data technologies. Proc. Inst. Mech. Eng. Part I J. Syst. Control. Eng.
**2019**, 233, 904–917. [Google Scholar] [CrossRef] - Boulmrharj, S.; NaitMalek, Y.; Elmouatamid, A.; Bakhouya, M.; Ouladsine, R.; Zine-Dine, K.; Khaidar, M.; Siniti, M. Battery Characterization and Dimensioning Approaches for Micro-Grid Systems. Energies
**2019**, 12, 1305. [Google Scholar] [CrossRef][Green Version] - Boulmrharj, S.; NaitMalek, Y.; El Mouatamid, A.; Ouladsine, R.; Bakhouya, M.; Ouldmoussa, M.; Zine-Dine, K.; Khaidar, M.; Abid, R. Towards a Battery Characterization Methodology for Performance Evaluation of Micro-Grid Systems. In Proceedings of the 2018 International Conference on Smart Energy Systems and Technologies (SEST), Sevilla, Spain, 10–12 September 2018; pp. 1–6. [Google Scholar]
- Candau, Y.; Piar, G. An application of spectral decomposition to model validation in the thermal analysis of buildings. Int. J. Heat Mass Transf.
**1993**, 36, 645–650. [Google Scholar] [CrossRef] - Gargari, C.; Bibbiani, C.; Fantozzi, F.; Campiotti, C.A. Simulation of the Thermal Behaviour of a Building Retrofitted with a Green Roof: Optimization of Energy Efficiency with Reference to Italian Climatic Zones. Agric. Agric. Sci. Procedia
**2016**, 8, 628–636. [Google Scholar] [CrossRef] - Houda, M.; Djamel, A.; Fayçal, L. An Assessment of Thermal Comfort and Users’ “Perceptions” in Office Buildings—Case of Arid Areas with Hot and Dry Climate. Energy Procedia
**2015**, 74, 243–250. [Google Scholar] [CrossRef][Green Version] - Markatos, N.; Pericleous, K. Laminar and turbulent natural convection in an enclosed cavity. Int. J. Heat Mass Transf.
**1984**, 27, 755–772. [Google Scholar] [CrossRef] - Córdoba, P.A.; Silin, N.; Dari, E.A. Natural convection in a cubical cavity filled with a fluid showing temperature-dependent viscosity. Int. J. Therm. Sci.
**2015**, 98, 255–265. [Google Scholar] [CrossRef] - Davis, G.D.V. Natural convection of air in a square cavity: A bench mark numerical solution. Int. J. Numer. Methods Fluids
**1983**, 3, 249–264. [Google Scholar] [CrossRef] - Karatas, H.; Derbentli, T. Natural convection and radiation in rectangular cavities with one active vertical wall. Int. J. Therm. Sci.
**2018**, 123, 129–139. [Google Scholar] [CrossRef] - Yousaf, M.; Usman, S. Natural convection heat transfer in a square cavity with sinusoidal roughness elements. Int. J. Heat Mass Transf.
**2015**, 90, 180–190. [Google Scholar] [CrossRef] - Khatamifar, M.; Lin, W.; Armfield, S.; Holmes, D.; Kirkpatrick, M. Conjugate natural convection heat transfer in a partitioned differentially-heated square cavity. Int. Commun. Heat Mass Transf.
**2017**, 81, 92–103. [Google Scholar] [CrossRef] - Pandey, S.; Park, Y.G.; Ha, M.Y. An exhaustive review of studies on natural convection in enclosures with and without internal bodies of various shapes. Int. J. Heat Mass Transf.
**2019**, 138, 762–795. [Google Scholar] [CrossRef] - Ouakarrouch, M.; El Azhary, K.; Laaroussi, N.; Garoum, M.; Feiz, A. Three-dimensional numerical simulation of conduction, natural convection, and radiation through alveolar building walls. Case Stud. Constr. Mater.
**2019**, 11, e00249. [Google Scholar] [CrossRef] - Bianchi, F.; Pisello, A.L.; Baldinelli, G.; Asdrubali, F. Infrared Thermography Assessment of Thermal Bridges in Building Envelope: Experimental Validation in a Test Room Setup. Sustainability
**2014**, 6, 7107–7120. [Google Scholar] [CrossRef][Green Version] - Berrabah, S.; Moussa, M.O.; Bakhouya, M. Towards a thermo-mechanical characterization approach of buildings’ envelope. Energy Rep.
**2020**, 6, 240–245. [Google Scholar] [CrossRef] - Jannot, Y. Cours Transferts Thermiques 2ème 2 Année Ecole des Mines Nancy 2012. Available online: http://www.thermique55.com/principal/thermique.pdf (accessed on 26 April 2020).
- Sidebotham, G. Nusselt Number Correlations. In Heat Transfer Modeling; Springer: Berlin/Heidelberg, Germany, 2015; pp. 351–375. [Google Scholar]
- Energieplus le Site. Available online: https://energieplus-lesite.be/theories/enveloppe9/coefficient-de-transmission-thermique/coefficient-de-transmission-thermique-d-une-fenetre-uw-ou-d-une-porte/#c20932597 (accessed on 26 April 2020).
- Kyosev, Y. The finite element method (FEM) and its application to textile technology. In Simulation in Textile Technology; Elsevier: Amsterdam, The Netherlands, 2012; pp. 172–221, 222e. [Google Scholar]
- Salencon, J.; Bechtel, S. Handbook of Continuum Mechanics: General Concepts, Thermoelasticity. Appl. Mech. Rev.
**2002**, 55, B43–B44. [Google Scholar] [CrossRef] - Matweb. Materials Property Data. Available online: http://www.matweb.com/ (accessed on 28 April 2020).
- Africa Weather Files. Available online: http://climate.onebuilding.org/WMO_Region_1_Africa/MAR_Morocco/index.html (accessed on 10 June 2021).
- Carnet du Maker. Available online: https://www.carnetdumaker.net/articles/mesurer-une-temperature-avec-un-capteur-1-wire-ds18b20-et-une-carte-arduino-genuino/#le-capteur-ds18b20 (accessed on 30 May 2021).
- Malek, Y.N.; Kharbouch, A.; El Khoukhi, H.; Bakhouya, M.; De Florio, V.; El Ouadghiri, D.; Latre, S.; Blondia, C. On the use of IoT and Big Data Technologies for Real-time Monitoring and Data Processing. Procedia Comput. Sci.
**2017**, 113, 429–434. [Google Scholar] [CrossRef]

**Figure 1.**Presentation of the building: (

**a**) its worldwide consumption among other sectors (

**b**) as a complex structure.

**Figure 2.**Heat transfer through walls: (

**a**) Wall with heterogeneous layers; (

**b**) Wall with non-homogeneous materials.

**Figure 6.**Illustration of thermal phenomena and boundary/interface conditions: (

**a**) boundary conditions; (

**b**) Interface conditions.

**Figure 8.**Weather data for the city of Sala EL Jadida during March 2019 [30].

**Figure 11.**Experimental Set-Up (

**a**) Thermal equipment (

**b**) Some examples of numerical sensors placement.

**Figure 12.**Time evolution of numerical estimated temperatures: (

**a**) External temperature; (

**b**) Internal temperature.

**Table 1.**Correlations for the calculus of Nusselt number [25].

Geometry | Gr × Pr | $\mathit{\gamma}$ | M |
---|---|---|---|

Vertical plate | 10^{4}–10^{9} | 0.59 | 1/4 |

10^{9}–10^{13} | 0.021 | 2/5 | |

Horizontal tube | 10^{4}–2.12 × 10^{7} | 0.53 | 0.25 |

2.12 × 10^{7}–10^{12} | 0.13 | 0.33 |

EEBLAB’s Dimensions (Height, Length, Width) | 4 m × 3 m × 2.2 m |
---|---|

Window’s dimensions | 1.2 m × 1 m |

Door’s dimensions | 0.85 m × 2.1 m |

**Table 3.**Thermal properties of the construction and insulation materials of the EEBLab [29].

Heat Capacity [J/kg·K] | Density [kg/m^{3}] | Conductivity [W/m∙ K] | |
---|---|---|---|

GS | 470 | 7800 | 52 |

Chipboard | 2100 | 170 | 0.042 |

PB | 1300 | 25 | 0.022 |

Glass | 800 | 2530 | 0.93 |

Al | 900 | 2690 | 210 |

Pl | 1000 | 40 | 0.024 |

**Table 4.**Mechanical properties of the construction and insulation materials of the EEBLAB [29].

Characteristics | Materials | ||||
---|---|---|---|---|---|

GSI | CB | PB | Al | Gl | |

Young Modulus (GPA) | 200 | 12 | 2 | 68 | 69 |

Poisson ration | 0.29 | 0.2 | 0.5 | 0.34 | 0.22 |

Density (kg/m^{3}) | 7800 | 7500 | 1250 | 2690 | 2530 |

Expansion coefficient (xE-06) | 11 | 40 | 78 | 23 | 9 |

Air Characteristics | Internal Ambiance | External Ambiance | |
---|---|---|---|

T (Temperature) | 297.8 | 291.5 | K |

G (Gravity) | 9.81 | 9.81 | N/kg |

$\rho $ (Density) | 1.187 | 1.217 | kg·m^{−3} |

μ (Dynamic viscosity) | 1.84 × 10^{−5} | 1.81 × 10^{−5} | kg·m^{−1}·s^{−1} |

ν (Cinematic viscosity) | 1.54 × 10^{−5} | 1.46 × 10^{−5} | m^{2}·s^{−1} |

β (Coefficient of thermal expansion) | 0.003358 | 0.003431 | K^{−1} |

C_{p} (Specific heat) | 1006 | 1005 | J·kg^{−1}·K^{−1} |

k (Conductivity) | 0.026 | 0.0255 | W·m^{−1}·K^{−1} |

Density of Meshing | SMXX (Shear Stress in Pa) |
---|---|

0.03 | |

0.04 | |

0.05 | |

0.06 | |

0.07 | |

0.08 | |

0.09 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Berrabah, S.; Moussa, M.O.; Bakhouya, M. 3D Modeling of the Thermal Transfer through Precast Buildings Envelopes. *Energies* **2021**, *14*, 3751.
https://doi.org/10.3390/en14133751

**AMA Style**

Berrabah S, Moussa MO, Bakhouya M. 3D Modeling of the Thermal Transfer through Precast Buildings Envelopes. *Energies*. 2021; 14(13):3751.
https://doi.org/10.3390/en14133751

**Chicago/Turabian Style**

Berrabah, Soukayna, Mohamed Ould Moussa, and Mohamed Bakhouya. 2021. "3D Modeling of the Thermal Transfer through Precast Buildings Envelopes" *Energies* 14, no. 13: 3751.
https://doi.org/10.3390/en14133751