The two necessary criteria of the thermal design [1] [2] [3] of the buildings are the protection of the occupants in an automatic and passive way, climatic factors: rain, wind, radiation, hot or cold walls and optimization of energy consumption [4] [5]. The designer must ensure that this consumption (production of hot or cold) remains within the limits set by the regulations and the financial possibilities of the occupants while ensuring a level of comfort defined by the client. The thermal insulation introduced by the different elements of the building envelope constitutes an important criterion of energy performance. It reduces heat loss, saves heating, limits greenhouse gas emissions and provides better living comfort [6].

For years, many researchers have been studying the characterization [7] and optimization of plant and agricultural materials [8] [9] for the energy performance of buildings. It is in these perspectives that we have turned to research by proposing the study of a concrete slab adjoining a compressed board of rice straw in imperfect contact. So in this study, we will try to determine the value of the optimal and critical thermal resistance [10] - [15].

The diagram of the wall [12] [14] consisting of concrete and rice straw is shown in Figure 1. Temperatures T₁ and T₂ of the outdoor and indoor environments respectively are defined in a frequency dynamic regime with an excitatory pulsation ω . The time is noted t.

The phenomenon of heat diffusion in the wall is governed by the equation of heat. In the absence of a heat source and sink, it is given by Equation (1) below:

∂ 2 T i ( x , t ) ∂ x 2 − 1 α i ∂ T i ( x , t ) ∂ t = 0 (1)

T ( x , t ) is the material temperature at a depth x and time t.

α i = λ i ρ i ∗ c i

α i ( m 2 ⋅ s − 1 ) is the material’s thermal diffusivity coefficient i.

i = 1 or 2 for concrete slab or rice straw board respectively.

The boundary conditions [11] [12] [13] reflecting the different thermal exchanges at the interfaces and the initial condition are given by the equations below.

{ λ 1 ∂ T 1 ( x , t ) ∂ x | x = 0 = h 1 [ T 1 ( 0 , t ) − T 01 ⋅ e j ω t ]                                               ( 2 ) − λ 1 ∂ T 1 ( x , t ) ∂ x | x = l = − λ 2 ∂ T 2 ( x , t ) ∂ x | x = l                                                           ( 3 ) − λ 1 ∂ T 1 ( x , t ) ∂ x | x = l = T 1 ( l , t ) − T 2 ( l , t ) R c                                                         ( 4 ) − λ 2 ∂ T 2 ( x , t ) ∂ x | x = L = h 2 [ T 2 ( L , t ) − T 02 ⋅ e j ω t ]                                       ( 5 )

Considering that the wall is at an initial temperature T i

T ¯ ( x , t ) the addition temperature is therefore:

T i ( x , t ) = T ¯ i ( x , t ) + T 0 i Avec i = 1 , 2 (6)

The expression of Equation (1) of heat becomes:

∂ 2 ( T ¯ + T i ) ∂ x 2 − 1 α ∂ ( T ¯ + T i ) ∂ t = 0 (7)

New boundary conditions become:

{ λ 1 ∂ T ¯ 1 ( x , t ) ∂ x | x = 0 = h 1 [ T ¯ 1 ( 0 , t ) + T 0 i − T 01 ⋅ e j ω t ]                                       ( 8 ) − λ 1 ∂ T ¯ 1 ( x , t ) ∂ x | x = l = − λ 2 ∂ T ¯ 2 ( x , t ) ∂ x | x = l                                                               ( 9 ) − λ 1 ∂ T ¯ 1 ( x , t ) ∂ x | x = l = T ¯ 1 ( l , t ) − T ¯ 2 ( l , t ) R c                                                               ( 10 ) − λ 2 ∂ T ¯ 2 ( x , t ) ∂ x | x = L = h 2 [ T ¯ 2 ( L , t ) + T 0 i − T 02 ⋅ e j ω t ]                             ( 11 )

The resolution of Equation (7) leads to the following solution:

T ¯ 1 ( h 1 , h 2 , α , ω , x , t ) = [ A 1 sinh ( β 1 ⋅ x ) + A 2 cosh ( β 1 ⋅ x ) ] e j ω t (12)

T ¯ 2 ( h 1 , h 2 , α , ω , x , t ) = [ A 3 sinh ( β 2 ⋅ x ) + A 4 cosh ( β 2 ⋅ x ) ] e j ω t (13)

β 1 = ω 2 ⋅ α 1 ( 1 + j ) (14)

β 2 = ω 2 ⋅ α 2 ( 1 + j ) (15)

coefficients A₁, A₂, A₃ et A₄ are determined from boundary conditions.

The changes in temperature and heat flux density as a function of the thermal resistance of contact are described below under the influence of the heat exchange coefficients on the front and rear faces (yielding respectively Table 1 and Table 2).

Figure 2 shows that the temperature module at the contact zone evolves along a strip of contact thermal resistances. For values of RTC ≤ 10² W∙m⁻²∙K⁻¹ and RTC ≥ 10² W∙m⁻²∙K⁻¹, the temperature module hardly evolves corresponding to a quasi-static regime. For values of contact thermal resistance between 10⁻² W∙m⁻²∙K⁻¹ ≤ RTC ≤ 10² W∙m⁻²∙K⁻¹, the temperature module varies considerably with the increase of the contact thermal resistance describing a dynamic velocity at the contact area.

For this purpose, the thermal flux density module (Figure 3) decreases with the increase in the thermal resistance of contact. The density of heat flux remains important for the large values of heat exchange coefficient on the front face due to the strong heat exchange between the external medium and the surface of the wall of the material leading to heat propagation by deep conduction. So according to the evolution of the thermal flux density at the contact area, the critical and optimal contact thermal resistance values describing the maximum and minimum flux density limit value at the contact area are defined.

We give below the value table of the critical and optimal thermal contact resistance under the influence of the heat exchange coefficient on the front face (Table 1).

Below we will study the evolution of the temperature as a function of the depth by highlighting the effect of the exchange coefficient on the front and rear face in imperfect contact with an optimal thermal contact resistance value and critical (Table 1 and Table 2).

Figure 4 and Figure 5 shows that the temperature module decreases with increasing depth. The temperature module is important at the wall surface for large values of heat exchange coefficients on the front face. The heat exchange between the exterior and the surface of the wall thanks to the manpower, leads to conduction of the heat received on the front face of the first layer in the depth of the materials. At the level of the contact zone, a discontinuity of the temperature

module is observed, thus generating a temperature gap that varies according to the increase of the exchange coefficient on the front face but remains invariant for values of exchange coefficients on the rear face.

Note that at the contact area level (Table 3 and Table 4) the gap is much more for a value of optimal contact thermal resistance thus promoting the insulating character of the contact zone with a value of the interstitial exchange coefficient h₀.

The study of the contact area between the concrete slab and the compressed panel shows the important role in the installation of two-layer materials. The ideal thermal resistance of contact characterizing a low heat exchange in this zone favors a significant drop in temperature leading to the existence of a conducto-convective flux. These heat losses in the contact area are essential to mitigate heat diffusion to the interior environment.

The authors declare no conflicts of interest regarding the publication of this paper.

Diene, A., Lo, M.L., Sene, A., Fame, A., Traore, Y., Faye, S., Diagne, I., Ndiaye, M.B. and Sissoko, G. (2021) Influence of Heat Exchange Coefficients on Both Optimized Thermal Contact (OTCR) and Critical (CTCR) Resistances at the Contact Interface of a Flat Concrete Slab and a Rice Straw Board. Energy and Power Engineering, 13, 392-402. https://doi.org/10.4236/epe.2021.1312027