Most of the energy savings in the building sector come from the choice of the materials used and their microphysical properties. In the present study, through numerical simulations a link is established between the thermal performance of composite materials and their microstructures. First, a two-phase 3D composite structure is modeled, then the RSA (Random Sequential Addition) algorithm and a finite element method (FE) are applied to evaluate the effective thermal conductivity of these composites in the steady-state. In particular, building composites based on gypsum and clay, consolidated with peanut shell additives and/or cork are considered. The numerically determined thermal conductivities are compared with values experimentally calculated using the typical tools of modern metrology, and with available analytical models. The calculated thermal conductivities of the clay-based materials are 0.453 and 0.301 W.m^{−1}.K^{−1} with peanut shells and cork, respectively. Those of the gypsum-based materials are 0.245 and 0.165 W.m^{−1}.K^{−1} with peanut shells and cork, respectively. It is shown that, in addition to its dependence on the volume fraction of inclusions, the effective thermal conductivity is also influenced by other parameters such as the shape of inclusions and their distribution. The relative deviations, on average, do not exceed 6.8%, which provides evidence for the reliability of the used approach for random heterogeneous materials.

During the last decades, a considerable increase in energy consumption has been noticed, particularly in the field of building due to rapid economic development and population growth. A large number of scientists have conducted research on methods and techniques that can reduce the energy consumption of buildings. In this context, the improvement of thermal comfort is ensured especially for the inhabitants of dry climate regions by increasing the quality of thermal insulation of building materials historically used in these regions. Research [

The steps of this technique are shown in

The first step is to generate a 3D microstructure with randomly dispersed and oriented additives in the matrix, then boundary conditions are applied and the volume response is calculated using numerical techniques such as finite elements (FE) [

The composite material studied is represented by a geometry of granular structure in which the distribution requires a probabilistic approach. Ellipsoidal particles are created with a free choice of the orthogonal distances of the ellipsoid (a, b, c), then the coordinates of their center and the Euler angles (Ψ, θ, φ), which define the rotation of the ellipse, are generated randomly in a cubic matrix of volume V = L^{3}, using RSA developed under MATLAB. To ensure non-overlapping of inclusions, the distance d between the centers of the ellipsoids must be greater than the largest of the orthogonal distances (a, b, c), while the maximum of this distance is related to the predefined volume fraction (

The particles are randomly introduced one after the other until the desired volume fraction is obtained, while checking the following conditions:

If a new particle overlaps previously placed particle(s), then this attempt is rejected. Otherwise, the placement is accepted. To ensure non-overlapping of inclusions, the distance d between the centers of the ellipsoids must be greater than the largest of the orthogonal distances (a, b, c), while the maximum of this distance is related to the predefined volume fraction (

If the particle intersects a face of the matrix it will be extended by the opposite face to ensure the periodicity of the macroscopic material (

If, after a certain number of attempts, a new particle is not accepted, the algorithm stops the program.

This RSA method is widely used to numerically study physical, chemical and biological phenomena, and was it is introduced by Feder [_{1}) (_{2}) (

The studied composite material is a plaster matrix reinforced with peanut shells. Plaster or gypsum hemihydrate is considered a building material generally used in ceilings or can be used as an internal layer for walls [

After generating the structure, it is then transferred to the Abaqus software to perform a finite element calculation of the effective thermal conductivity of the studied composite.

The study of the thermal properties of materials begins by applying an external temperature gradient and then solving the heat equation.

Laplace’s equation: The effective thermal conductivity, in steady state, can be obtained from the thermal conductivities of the matrix and the inclusions by solving the following Laplace equation:

Heat flux conservation: Partial differential equations lead to the physical notion of conservative flow given by a gradient:

Applying the divergence-flux formula (Ostrogradsky’s theorem), it comes:

We deduce from this that:

To simplify the resolution of

The effective thermal conductivity in the x-direction, for example, can be obtained by the relationship:

A statistical approach on a representative number of realizations, based on the methodology of Kanit et al. [

After transferring the generated structure to a finite element software, a numerical simulation methodology was followed to model the thermal properties of the clay-based material:

A temperature gradient is imposed between two opposite sides, in a stationary state, which generates a heat flux directed towards low temperatures. The choice, in the calculation software, of a heat flow vector circulating unidirectionally between two opposite faces of the representative elementary volume requires that the other faces be adiabatic. The imposed boundary conditions are given by the following equations:

The RVE geometries created in the previous section have been meshed. A free meshing with tetrahedral elements leads generally to a fairly high density of elements at the matrix/inclusion interface and it can be easily automated. It is used to obtain a high-quality periodic discretization of matrix-inclusion microstructures with as few elements as possible (

The effect of the mesh density on the thermal conductivity values was studied to justify the choice of the mesh.

The thermal conduction inside the material is governed by Fourier’s law. The heat flux obtained within the elementary volume is almost uniform as shown in ^{−2} which allows for the calculation of the equivalent thermal conductivity using

As an illustrative example, a temperature distribution within the elementary volume (

An example of the result of temperature variations in the direction of thermal diffusion is shown in

Many geometric realizations (several random draws of positions and orientations) for the same 3D microstructure (same volume fraction and same aspect ratio) were simulated. Then the statistical average and the relative deviation were calculated for each number of implementations. The increase in the number of realizations was stopped when there was the convergence of the thermal conductivity. The average thermal conductivity and its calculated dispersion are presented in

The composite material chosen in this study as well as others have already been studied experimentally and their thermophysical properties have been characterized using modern metrology methods [

The samples of the studied composites were experimentally prepared by a process of mixing the matrix (clay or plaster) and inclusions (cork or peanut shells). Water is added to the matrix paste until a normal consistency is obtained, then inclusions are introduced into the matrix. The mixing process is carried out and then the pastes are placed in parallelepipedic molds. After demolding and drying the samples are kept in plastic bags, from which samples were taken for experimental characterizations. _{1} and T_{0} at the center of the upper and lower faces of the sample and the temperature T_{2} at the lower face of the insulating foam. The role of the two aluminum blocks, which have a fairly high thermal conductivity, is to reach the steady state after a reasonable time.

Where

The combination of these equations leads to the expression of the thermal conductivity of our sample:

The calculated values of the equivalent thermal conductivity are also compared to those obtained by the different analytical models reported in the literature.

The expressions for the equivalent thermal conductivity for each of the analytical models are given by the following relations [

Series model

Parallel model

Beck model

Maxwell model

Woodside model

with:

For all components of the composite materials studied, the thermal conductivities are:

^{−1}.K^{−1}, ^{−1}.K^{−1}, ^{−1}.K^{−1} and ^{−1}.K^{−1} [

Experimental value (W.m^{−1}.K^{−1}) |
Simulated value (W.m^{−1}.K^{−1}) |
Analytical model values (W.m^{−1}.K^{−1}) |
||||||
---|---|---|---|---|---|---|---|---|

Series | Beck | Woodsid | Maxwell | Parallel | ||||

Thermal conductivity | Plaster + peanut shells | 0.233 | 0.245 | 0.187 | 0.222 | 0.237 | 0.255 | 0.266 |

Clay + peanut shells | 0.435 | 0.453 | 0.237 | 0.354 | 0.420 | 0.498 | 0.530 | |

Plaster + granular cork | 0.150 | 0.165 | 0.133 | 0.182 | 0.197 | 0.231 | 0.249 | |

Clay + granular corck | 0.280 | 0.301 | 0.123 | 0.190 | 0.207 | 0.263 | 0.292 | |

Relative deviation | Plaster + peanut shells | 5.1% | 19.7% | 4.7% | 1.7% | 9.4% | 14.2% | |

Clay + peanut shells | 4.2% | 45.5% | 18.6% | 3.4% | 14.4% | 21.8% | ||

Plaster + granular cork | 10% | 11.3% | 21.3% | 31.3% | 54% | 66% | ||

Clay + granular corck | 7.5% | 56% | 32.1% | 26% | 6% | 4.6% |

From data in

The noticed discrepancy between numerical and experimental results shows the performance of the computational code used and its validation for random heterogeneous materials.

The four simulated values of the thermal conductivity are higher than those obtained experimentally. This is because the adopted numerical method does not consider the microporosity of the inclusions of these materials.

The reduction rate of the thermal conductivity of the clay/granular cork and clay/peanut shells composites compared to pure clay are respectively: 34.56% and 1.52%.

The reduction rate of the thermal conductivity of the plaster/granular cork and plaster/peanut shells composites compared to the pure plaster are respectively: 45%, 18% and 18%, 6%.

It appears that the geometric Woodside model is closest to representing the samples containing the peanut shells, while the cork-stabilized composites tend to be represented by the serial model (case of the gypsum/cork composite) or by the parallel model (case of the clay/cork composite) depending on the geometric mean of the random distributions of the additives in each of the clay and plaster matrices. The histogram shown in

We have presented a finite element based numerical methodology for predicting the effective thermal conductivity of random heterogeneous materials. We are particularly interested in ecological building materials based on gypsum and clay consolidated with peanut shell additives on the one hand and cork on the other hand. The calculated thermal conductivities are compared with experimental results from the literature and with existing analytical models. The method is shown to be very promising as a tool for predicting the effective properties of building materials, and also for the design of new materials used to increase the energy efficiency of buildings. The observed deviations from the experimental values can be explained by the fact that the inclusions do not have the same shapes and that the numerical study does not consider the microporous structure of the inclusions. In retrospect, we propose the generation of 3D structures with random distributions and orientations of inclusions of non-identical sizes. We also propose, in furtherance of this work, a study of the effect of thermal stress that can be involved in these materials especially in regions characterized by a dry climate with rapid daily variations in temperature variations. We will also be conducting a study on the optimization of different physical parameters such as acoustic properties and especially mechanical properties that are the basis of the stability of buildings.

The authors received no specific funding for this study.

The authors declare that they have no conflicts of interest to report regarding the present study.