Abstract

Convective heat transfer coefficients, materializing exchanges between solid wall (here typha) and its environment, influence its behavior under excitation pulse. Temperature of wall and its density of flow vary with these coefficients according to its thickness (in depth). This study therefore focuses on the evaluation of convective heat transfer coefficient on front face and the optimal insulation thickness.

Keywords

Typha, Heat Transfer Coefficients, Spectroscopy, Optimal Insulation Thickness

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1. Introduction

Use of local insulation materials from vegetable (biodegradable) or mineral origin [1] is alternative for environmental protection, on the one hand, and needs to ensure good energy efficiency [2] [3], on the other hand. Use of synthetic materials (polystyrene [4] [5], polyurethane [6] [7], glass wool [8] [9]) guarantees efficiency and profitability, but is nevertheless harmful to environment [10]. Option of substituting and/or combining synthetic materials with natural ones (local materials) has been subject of several research studies [11] - [16]. To ensure good insulation of buildings, material must have low thermal conductivity and be able to stand heat exchanges between it and its surrounding environment.

External parameters that can influence behavior of wall are: excitation pulse, solar radiation and convection coefficients [17]. Knowledge of thickness of insulation material is also an important parameter to consider for optimal insulation [18] [19] [20].

Heat transfer coefficient influences behavior of the material in response to excitations it undergoes. It materializes heat exchanges between walls of material and its surrounding environment (exterior and interior). Much research has focused on his determination [21] - [26].

In this paper, we will study spectroscopy of convection coefficient at front face of typha panel [27] [28] and also determine corresponding optimal thickness of insulator.

2. Theory

2.1. Study Model

Study model is shown in Figure 1, it is panel made of typha with thickness L. Transverse dimensions are large enough to consider that heat transfer is unidirectional. Heat exchanges between material and two sides (exterior and interior) are assumed to be convective. They are quantified by heat transfer coefficients on front and back sides.

· T₁ (˚C) and T₂ (˚C): temperature in frequency dynamic mode of external and indoor environment respectively;

· T₀₁ and T₀₂ (˚C): maximum amplitude of T₁ and T₂ respectively;

· T₀ (˚C): initial temperature of insulating material;

· L (m): length of material along x-axis;

· h₁ and h₂ (W∙m⁻²∙K⁻¹): heat transfer coefficient at front and back face panel respectively.

2.2. Mathematical Formulation

Conservation of energy at any point of material is governed by following heat equation:

$\rho C\frac{\partial T}{\partial t}=\frac{\partial }{\partial x}\left(\lambda \cdot \frac{\partial T}{\partial x}\right)+P$ (1)

where:

· ρ (kg∙m⁻³): density of material;

· C (J∙kg⁻¹∙K⁻¹): mass thermal capacity;

· λ (W∙m⁻¹∙K⁻¹): thermal conductivity of material;

· P (W∙m⁻³): internal heat supply (heat sink) of material;

· x (m): depth position.

Simplified form of this equation, in absence of internal heat sinks and for constant thermal conductivity (assumed isotropic material) is given by:

$\frac{\partial T}{\partial t}=\frac{\lambda }{\rho Cp}\cdot \Delta T$ (2)

Study is done in one dimension and equation becomes:

$\frac{{\partial }^{2}T\left(x,h_1,h_2,\omega ,t\right)}{\partial x^2}=\frac{1}{\alpha }\frac{\partial T\left(x,h_1,h_2,\omega ,t\right)}{\partial t}$ (3)

where:

· $T\left(x,h_1,h_2,\omega ,t\right)$ : Temperature in material.

$\alpha =\frac{\lambda }{\rho Cp}$ (4)

· h₁: heat exchange coefficient front face;

· h₂: heat exchange coefficient rear face;

· α: Thermal diffusivity coefficient of the material (m²∙s⁻¹).

Solving this equation requires establishment of boundary conditions:

$\lambda {\frac{\partial T}{\partial x}|}_{x=0}=h_1\left(T\left(0,h_1,h_2,\omega ,t\right)-T_1\right)$ (5)

$-\lambda {\frac{\partial T}{\partial x}|}_{x=L}=h_2\left(T\left(L,h_1,h_2,\omega ,t\right)-T_2\right)$ (6)

Form of the solution of Equation (3) in dynamic frequency regime is:

$T\left(x,h_1,h_2,\omega ,t\right)=A\sinh \left(\beta x\right)+B\cosh \left(\beta x\right){\text{e}}^{i\omega t}+T_0$ (7)

where:

$\beta =\sqrt{\frac{\omega }{2\alpha }}\left(1+i\right)$ (8)

$\frac{1}{\beta }$ : complex diffusion length.

Expression of heat flow density

$\phi =-\lambda gradT$ (9)

φ (W∙m⁻²): heat density flow modulus, After resolution of Equation (9), we obtain the following expression:

$\phi \left(x,h_1,h_2,\omega ,t\right)=-\lambda \beta A\cosh \left(\beta x\right)+B\sinh \left(\beta x\right){\text{e}}^{i\omega t}$. (10)

2.3. Spectroscopic Expression of Heat Exchange Coefficient

To determine spectroscopic expression of heat exchange coefficient h₁ at front face, we first study evolution of heat density flow as function of heat exchange coefficient at rear face h₂ (Figure 2).

Heat density flow increases with h₂ and reaches maximum for h₂ > 50 W∙m⁻²∙K⁻¹; derivative function of heat density flow (11) allows to obtain the expression of h₁ (13).

$\frac{\partial \phi \left(x,h_1,h_2,\omega ,t\right)}{\partial h_2}=0$ (11)

$\frac{\partial \phi \left(x,h_1,h_2,\omega ,t\right)}{\partial h_2}=-\lambda \beta \left[\frac{\partial A\cosh \left(\beta x\right)}{\partial h_2}+\frac{\partial B\sinh \left(\beta x\right)}{\partial h_2}\right]{\text{e}}^{i\omega t}=0$ (12)

Resolution of Equation (12) allows us to obtain following expression:

$h_1\left(L,\omega ,t\right)=-\frac{{\left(\lambda \beta \right)}^2\sinh \left(\beta L\right)\left(T_{02}-T_0{\text{e}}^{-i\omega t}\right)}{\left(\lambda \beta \right)\cosh \left(\beta L\right)\left(T_{02}-T_0{\text{e}}^{-i\omega t}\right)+\left(T_0{\text{e}}^{-i\omega t}-T_{01}\right)}$. (13)

3. Results and Discussions

Figure 3 shows evolution of heat exchange coefficient h₁ as function of excitation pulse under thickness influence. Heat transfer coefficient increases with excitation for the different thickness values. Each maximum of h₁ corresponds to resonance frequency ω_r.

Cutoff frequencies ω_c are intersections of tangent lines of two consecutive parts of the concavity of curve.

Table 1 shows that resonant and cutoff frequencies increase as depth decreases. Maximum value of heat exchange coefficient for a resonant frequency decreases with thickness of insulating panel.

Figure 4 gives us the variation of h₁ as a function of thickness, taking into account the resonance frequencies. The heat transfer coefficient increases with the

thickness of the material and reaches a maximum value. This maximum of h₁ corresponds to a minimum thickness that allows good insulation.

In fact, higher heat transfer coefficient on front panel, thicker insulating panel. This thickness is called optimal thermal insulation thickness: X_op (Table 2).

Maximum of heat transfer coefficient is more important when pulsation is low, this corresponds to an increase of heat flow in material due to excitation. Indeed, period being inversely proportional to excitation frequency, latter lasts longer optimal insulation thickness decreases due to relaxation phenomena.

When material thickness exceeds optimal thickness, heat transfer coefficient on front face of panel decreases, which means that heat transfer coefficient no longer, has any influence on material’s behavior.

Figure 5 is obtained from Table 2, in fact, we were able to plot logarithm of maximum heat transfer coefficient h_1max as function of logarithm of optimal thickness X_op.

The resulting curve can be assimilated linear function characterize by equation:

$\log \left(h_{1\max }\right)=a\log \left(X_{op}\right)+b$ (14)

$h_{1\max }={\text{e}}^b{\left[X_{op}\right]}^a$ (15)

Coefficients a and b are determined from curve by using Equation (15).

$h_{1\max }=1.48\times {10}^8{\left[X_{op}\right]}^{11.32}$ (16)

Figure 6 and Figure 7 show respectively phase diagram of the heat transfer coefficient and its corresponding Nyquist representations for different values of the material thickness [29].

These graphs make it possible to highlight equivalent electrical phenomena of typha panel such as capacitive, inductive or resistive aspects [14] [30] [31] [32] [33].

For values of 10^−4.4 ≤ ω ≤ 10^−4.2, heat transfer coefficient phase changes slightly in an almost linear way. For ω ≥ 10^−4.2, phase decreases considerably and this decrease is even more important when thickness is significant.

The phase is negative or zero which corresponds to an equivalent electrical circuit in R, L, C where the capacitive phenomena prevail over the inductive phenomena [30] [31] [32] [33].

4. Conclusion

In this article, method of characterizing heat transfer to face of material is studied from heat exchange coefficient. It was then evaluated order of magnitude of this coefficient in relation to optimal insulation thickness of typha panel. Indeed, it has been shown that convection coefficient influences insulation thickness, heat transfer coefficient is an important factor to consider when choosing the insulation thickness.

Conflicts of Interest

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

References