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

Alassane Diene¹, Mamadou Lamine Lo¹, Abdoulaye Sene¹, Ablaye Fame¹, Youssou Traore², Seydou Faye², Issa Diagne², Mamadou Babacar Ndiaye³, Gregoire Sissoko^2,4

¹Polytechnic School of Thiès, Thiès, Senegal.
²Laboratory of Semiconductors and Solar Energy, Physics Department, Faculty of Science and Technology, University Cheikh Anta Diop, Dakar, Senegal.
³University Institute of Technology, University Iba Der THIAM of Thiès, Thiès, Sénégal.
⁴Groupe International de Recherche en Energie Renouvelable, Dakar, Senegal.

DOI: 10.4236/epe.2021.1312027                

Abstract

The study is carried out in imperfect contact with a concrete slab wall attached to a panel based on rice straw compressed in a dynamic frequency regime. We will propose the characterization of thermal insulation for thermal resistance of contact (x = 0.05 m). The impact of heat exchange coefficients on the front face (x = 0 m) and the rear face (x = 0.1 m) on these resistors is shown.

Keywords

Concrete Slab, Rice Straw Board, Thermal Resistance of Contact, Frequency Dynamic

1. Introduction

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].

2. Presentation and Mathematical Modelling of the Insulation System: The Wall

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 $\omega$. 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:

$\frac{{\partial }^2{T}_i\left(x,t\right)}{\partial x^2}-\frac{1}{{\alpha }_i}\frac{\partial T_i\left(x,t\right)}{\partial t}=0$ (1)

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

${\alpha }_i=\frac{{\lambda }_i}{{\rho }_i\ast c_i}$

${\alpha }_i\left({\text{m}}^2\cdot {\text{s}}^{-1}\right)$ 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.

Figure 1. Study model diagrams. T₀₁ = 45˚C; T₀₂ = 20˚C; T_i = 23˚C; T_i is the initial wall temperature. h₁ and h₂ are the thermal exchange coefficients at the interface of the external and internal media, respectively.

$\{\begin{array}{l}{{\lambda }_1\frac{\partial T_1\left(x,t\right)}{\partial x}|}_{x=0}=h_1\left[T_1\left(0,t\right)-T_{01}\cdot {\text{e}}^{j\omega t}\right](2)\\ {-{\lambda }_1\frac{\partial T_1\left(x,t\right)}{\partial x}|}_{x=l}={-{\lambda }_2\frac{\partial T_2\left(x,t\right)}{\partial x}|}_{x=l}(3)\\ {-{\lambda }_1\frac{\partial T_1\left(x,t\right)}{\partial x}|}_{x=l}=\frac{T_1\left(l,t\right)-T_2\left(l,t\right)}{R_c}(4)\\ {-{\lambda }_2\frac{\partial T_2\left(x,t\right)}{\partial x}|}_{x=L}=h_2\left[T_2\left(L,t\right)-T_{02}\cdot {\text{e}}^{j\omega t}\right](\; 5\; )\end{array}$

Considering that the wall is at an initial temperature $T_i$

$\bar{T}\left(x,t\right)$ the addition temperature is therefore:

$T_i\left(x,t\right)={\bar{T}}_i\left(x,t\right)+T_{0i}$ Avec $i=1,2$ (6)

The expression of Equation (1) of heat becomes:

$\frac{{\partial }^2\left(\bar{T}+T_i\right)}{\partial x^2}-\frac{1}{\alpha }\frac{\partial \left(\bar{T}+T_i\right)}{\partial t}=0$ (7)

New boundary conditions become:

$\{\begin{array}{l}{{\lambda }_1\frac{\partial {\bar{T}}_1\left(x,t\right)}{\partial x}|}_{x=0}=h_1\left[{\bar{T}}_1\left(0,t\right)+T_{0i}-T_{01}\cdot {\text{e}}^{j\omega t}\right](8)\\ {-{\lambda }_1\frac{\partial {\bar{T}}_1\left(x,t\right)}{\partial x}|}_{x=l}={-{\lambda }_2\frac{\partial {\bar{T}}_2\left(x,t\right)}{\partial x}|}_{x=l}(9)\\ {-{\lambda }_1\frac{\partial {\bar{T}}_1\left(x,t\right)}{\partial x}|}_{x=l}=\frac{{\bar{T}}_1\left(l,t\right)-{\bar{T}}_2\left(l,t\right)}{R_c}(10)\\ {-{\lambda }_2\frac{\partial {\bar{T}}_2\left(x,t\right)}{\partial x}|}_{x=L}=h_2\left[{\bar{T}}_2\left(L,t\right)+T_{0i}-T_{02}\cdot {\text{e}}^{j\omega t}\right](\; 11\; )\end{array}$

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

${\bar{T}}_1\left(h_1,h_2,\alpha ,\omega ,x,t\right)=\left[A_1\sinh \left({\beta }_1\cdot x\right)+A_2\cosh \left({\beta }_1\cdot x\right)\right]{\text{e}}^{j\omega t}$ (12)

${\bar{T}}_2\left(h_1,h_2,\alpha ,\omega ,x,t\right)=\left[A_3\sinh \left({\beta }_2\cdot x\right)+A_4\cosh \left({\beta }_2\cdot x\right)\right]{\text{e}}^{j\omega t}$ (13)

${\beta }_1=\sqrt{\frac{\omega }{2\cdot {\alpha }_1}}\left(1+j\right)$ (14)

${\beta }_2=\sqrt{\frac{\omega }{2\cdot {\alpha }_2}}\left(1+j\right)$ (15)

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

3. Results and Discussion

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).

Table 1. Thermal resistance of critical and optimal contact under the influence of the heat exchange coefficient on the front face.

Table 2. Thermal resistance of critical and optimal contact under the influence of the heat exchange coefficient on the rear face.

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).

(a)(b)

Figure 2. (a) Temperature variation as a function of the contact heat resistance. Influence of the exchange coefficient on the front face. x = 0.05 m, h₂ = 0.01 W∙m⁻²∙K⁻¹. ω = 2 × 10⁻⁴ rad/s. (b) Temperature variation as a function of the contact heat resistance. Influence of the exchange coefficient on the rear side. x = 0.05 m, h₂ = 0.01 W∙m⁻²∙K⁻¹. ω = 2 × 10⁻⁴ rad/s.

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

(a)(b)

Figure 3. Evolution of heat flux density as a function of contact heat resistance. Influence of exchange coefficients on the front (x = 0 m) and rear side (x = 0.05 m), ω = 2 × 10⁻⁴ rad/s.

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.

(a)(b)

Figure 4. Temperature evolution through the wall as a function of its depth; Influences of the exchange coefficient on the front face w = 2 × 10⁻⁴ rad/s, h₂ = 0.01 W∙m⁻²∙K⁻¹. (a) optimal contact resistance Rcop = 10^1.3 W∙m⁻²∙K⁻¹. (b) critical contact resistance Rcc = 10^−1.3 W∙m⁻²∙K⁻¹.

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.

Table 3. Development of the temperature gap at the contact zone for different values of the exchange coefficient on the front face.

Table 4. Evolution of the temperature gap at the contact zone for different values of the exchange coefficient on the rear face.

(a)(b)

Figure 5. Evolution of the density of heat flow through the wall as a function of its depth; Influences of h₁. h₂ = 0.01 W∙m⁻²∙K⁻¹. (a) optimal contact resistance RTcop = 10^1.3 W∙m⁻²∙K⁻¹; (b) critical contact resistance RTcc = 10^−1.3 W∙m⁻²∙K⁻¹.

4. Conclusion

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.

Conflicts of Interest

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

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