2.1 Preliminary study

First we start by calculating the radius of these spheres. The condition on the composition in volume led to the relationship (Eq. 1) shown below.

This relationship gives us the radius of the equal-sized spheres that have been inserted.

2.1.1 Two-dimensional arrangement

The PCM spheres are centered on a same single insertion plane. To prevent the spheres from overlapping, the representative surface of their intersections with the insertion plane (cross section) must be smaller than the total surface of this plane (Figure 1 ). This relationship is expressed mathematically as follows (Eqs. 3a and b) :

Figure 1:

Configuration of spheres. (A) perpendicular plane frontal to the flux, (B) plane parallel to the flux, (C) tilted plane parallel to the flux, (D) tilted plane frontal to the flux.

Figure 2:

Three-dimensional periodic arrangement of the spheres in the composite.

(3a, b){nπrn2≤a2       for mediator planenπrn2≤2a2   for tilted plane

This condition is necessary but not sufficient. Although it provides information on the maximum number of spheres that can be inserted, it should be checked further whether the diameters of the spheres do not exceed the edge or the diagonal of the insertion plane cited above. This last condition cannot be predicted but must be checked case by case in accordance with the number of spheres.

Cubic matrix is of edge a=10 cm. The radius of the integrated spheres is a function of their number n. This number n is not infinitely possible, but its maximum value obviously varies with the percentage and according to the position of the plane of arrangement, either a mediator of one of the edges (Figure 1A and B), or it contains the diagonal of the entry face (Figure 1C and D). For the sake of comparison, n is limited to number four; maximum number available without overlapping spheres for the mediator planes of edges for these geometrical forms of the matrix and the PCM and the percentage of composition 20%.

2.2 Equation of heat transfer in the composite

Our system consists of a material of important thermal inertia, where a second material inserted can change its physical status in the temperature range studied. The composite material is subjected to a heat flux through one of its faces. The study of heat transfer of a physical problem requires the resolution of the energy equation which makes it possible to obtain the space-time evolution of the temperature and heat flux. In this work, the study is limited to the steady case and to the solid-solid PCM. The energy balance (Eq. 6) allows us to consider the equation of heat (Eq. 7) valid for each phase [18].

In each presumedly isotropic component i of the composite, this equation can be simplified in equation (Eq. 8) by considering that there is no generation of energy within the material while being interested in the steady state.

2.3 Boundary conditions

The conductivity of a material is an intrinsic characteristic of that material and the function of its temperature. Thus, it is necessary to limit the interval of temperature in the vicinity of that state change in order to consider constant conductivity. The latter is independent of the nature of the conditions imposed on the material surfaces, such as the flux, the temperature, or both. In our study, the following assumptions of simplification are considered: the thermophysical characteristics are constant; the entry face and the exit face of the cube are maintained at temperatures T₁ and T₂, respectively, and assumd to be uniform and constant; and the side faces are imposed on a null flux. Inside the composite material, a perfect contact between the two solids is assured.

2.4 Calculation of equivalent conductivity

Generally, the analytical calculation of the conductivity of composite materials is not possible because of its complex geometry. Hence, it must be calculated numerically. In this work, we are interested in the materials that undergo a solid-solid transition; therefore, the transfer is purely conductive. The difference between the thermal expansion coefficients of wood and polyurethane is very weak and of a few tens of μ°C⁻¹. In addition, the study interval ΔT (≈25°C) on both sides of the phase transition temperature is small, thus making the difference in expanded volumes negligible. Consequently, the interface thermal resistance is neglected. The heat transfer with outside through side surfaces is maintained null. The heat density flux is proportional to the thermal conductivity of the medium and the temperature gradient according to the Fourier analysis (9), which gives the expression of conductivity (10) below.

4.1 Two-dimensional case

With a percentage of 20%, a matrix of wood whose thermal conductivity is equal to λ_b=0.173 W/m°C and a PCM of polyurethane of thermal conductivity λ_p=0.032 W/m°C are examined. The latter is putrefied in the form of spheres arranged in only one plane of insertion (Figure 1). Two cases of planes are studied: frontal planes to the heat flux along axis (Oz); [plane (z=0) and plane (y=z)], and planes parallel to this flux [plane (y=0) and plane (x=y)]. We impose a uniform inlet temperature, constant and equal to 70°C and a temperature equal to 20°C to the face of exit. The heat transfer with outside through the side surfaces is supposed to be null φ=0. We handled a numerical simulation code to solve the heat transfer equation. Doing so enabled us to determine the evolution of the temperature and of the heat flux and thus deduce thermal conductivity.

Figure 4 describes the variation of thermal conductivity versus the number of spheres. This figure shows that thermal conductivity is a decreasing function with the number of spheres in the case of the frontal plane, whereas it is increasing in the case of the parallel plane. This figure also shows that the distribution of the PCM spheres according to a plane of insertion is tilted exposed in a side way to the heat flux, thus having the advantage of improving the equivalent thermal conductivity of the composite. In comparison, this conductivity decreases quickly when the distribution is according to a mediator plane of edges exposed perpendicular to the heat flux.

Figure 4:

Thermal conductivity versus the number of spheres.

Thermal conductivity is more important in the case of flux parallel and precisely when the plane of insertion is tilted – the only case where it exceeds the arithmetic conductivity λ_arith of composite material, which is equal to 0.1448 W/m°C (Eq. 11). This arrangement is thus the most effective and most useful in maintaining a material composite of conductivity close to its matrix while preserving its equivalent heat capacity.

These results remain valid as long as the PCM has a thermal conductivity that is weaker than that of the matrix. This situation is the most widespread in nature, while the weakness of the thermal conductivity of the PCMs materials is one of the handicaps of their industrial use [19], [20], [21], [22], [23], [24], [25]. Figure 5 shows that thermal conductivity and the heat flux evolve identically towards the rise of the number of spheres. These curves also indicate the anisotropy of the composite material obtained; consequently, crossing flux is no longer the same one and now varies according to the choice of the entry face.

Figure 5:

Thermal flux versus the number of spheres.

The borderline case of the distributions of spheres, which is a plate of PCM inserted in the matrix into the level of the plane of arrangement, is studied (Figure 6). The numerical results are shown in Table 1. Otherwise, by electric analogy, the thermal conductivity of this composite material is analytically obtained (Eqs. 12–14). They correspond to the cases A, B and C, respectively, as illustrated in Figure 6:

Figure 6:

Different configurations of the PCM plate. (A) perpendicular plane frontal to the flux, (B) plane parallel to the flux, (C) tilted plane parallel to the flux, (D) tilted plane frontal to the flux.

with

which is simplified as follows:

For the situation (D), the analytical calculation of λ based on the electric analogy is no longer valid, and the concept of association in derivation and series is lost. As the medium is not homogeneous neither on the level of all materials nor on the level of each crossed section, the applicable formula is as follows:

The calculation of this situation (D) lead us to the following expression:

with

This last result is valid for c<a/2 (i.e. f_v<0.75). If c>a/2, the calculation of this situation (D) leads us to the following modified expression:

These thermal conductivities, which are analytically calculated, are very close to the numerical results with relative errors less than 6% made by numerical modeling (Table 1). The limits of these functions (in kind of materials and in volume fractions), which are obtained analytically, are checked for each situation quoted above.

It is noted that for the situations (B) and (C) of Figure 6 (analoguous with the parallel resistances) λ is the same regardless of the configuration of the plate of PCM. Moreover, it is maximal and equal to arithmetic conductivity λ_arith (Eqs. 13 and 16 and Table 1). This is coherent with the result found for the discontinuous distribution of the PCM spheres, as illustrated in Figure 1.

For calculation with the Fluent code, in order to pass from the case of the frontal flux to the parallel flux case, the position of the insertion planes is preserved whereas the flux is carried by axis (Oy) instead of axis (Oz). We represent below, for example, the contours of the flux in the case of PCM distributed in four spheres. Figure 7 shows that the propagation of heat is more important when the flux is parallel (Figure 1A and B) than in the case of a frontal flux (Figure 1C and D) compared with the distribution plane of the PCM spheres, thus reflecting obviously on the thermal conductivity in a proportional way. We also note that the flux density is not uniform and is more intense in the area devoid of the PCM spheres.

Figure 7:

Flux density contours of composite wood/polyurethane on the levels of the entry and exit faces. Tilted insertion plane (x=z) case crossed by: (A) frontal heat flux along (Oz), (B) parallel heat flux along (Oy); mediator plane (z=0) case crossed by: (C) frontal heat flux along (Oz), (D) parallel heat flux along (Oy).

4.2 Three-dimensional case

Here, we consider a composite material consisting of a periodic array of identical spheres of radius r_n and conductivity λ₂ embedded in a uniform matrix of conductivity λ₁, in which no thermal resistance exists at the interface between the two phases.

Let

be the thermal resistance of the composite material. Below, R1∗is the thermal resistance of the matrix of thickness L1∗, and R2∗ is the thermal resistance of the part of the composite material of thickness 2r_n (Figure 8), including the PCM sphere.

Figure 8:

Elementary cell.

Let us suppose

Thus,

This calculation remains valid when the volume fraction does not exceed the limiting value corresponding to 2r_n≤a and results in f_v≤0.52.

Finally, we arrive at

Given that A is independent of n, and R1∗ and R2∗ are proportional to n^1/3, whereas a^* varies as a function of n^−1/3, therefore, λ is independent of n. Thus, for conductivity, the distribution of PCM either in a single sphere or in a periodic distribution of multiple spheres is similar.

If this result is applied to the percentage studied (20%), we can find that the equivalent thermal conductivity of the composite material wood-polyurethane when it is modelled into a 3D periodic sample is λ=0.1412 W/m°C. This result is in good agreement with that found by numerical calcul ation λ_num=0.1422 W/m°C with a negligible relative error of 0.7%. By comparison with the models treating the composites with isotropic components and the random distribution of spherical inclusions proposed by Eshelby [26] (Eqs. 30–32), we have

where

and

which predict λ_Esh=0.1343 W/m°C and that of Nan [27]. In addition, λ_Nan=0.1369 W/m°C, which proposes the expression given by the equation (Eqs. 33 and 34), where the interfacial thermal resistance R is zero:

where

Compared with the random distribution, the periodic distribution with a simple cubic structure improves the conductivity of the composite by almost 4.9%. Moreover, Figure 4 and Table 1 show that the 3D configuration of the PCM spheres has a thermal conductivity whose intermediate value is between those of the parallel planes and those of the planes frontal to flux. Otherwise, it is noted that, in practice, the equivalent conductivity of a composite material can be assimilated to the arithmetic conductivity (Figure 4) with a relative error of 2.54%.