Study at Two Dimensions of Thermal Transfer through a Fibers Panel Subjected to Climatic Constraints in Dynamic Frequency Regulations Established

From resolution of two-dimensional equation of heat in dynamic frequency regime, we have plotted evolution curves of temperature according to depth of material or in lateral direction. They will allow us to evaluate thermal behavior of towed material. Aim of study is to use fibers as a thermal insulating material by proposing a method for determining effective thermal insulation layer in dynamic frequency regime.


Introduction
Controlling energy consumption in homes during hot weather, or in cold rooms requires, among other things, good thermal insulation. Problems of cost of synthetic materials and end-of-life management lead us to propose use of biodegradable materials. Thus study of thermal behavior of material for use in thermal insulation [1] [2] is of major interest in search for energy control.
Several methods for calculating transient or established dynamic regimes [3] [4] are developed for simulation of thermal behavior within material.
We propose in this study resolution in dynamic regime established by impos-ing boundary conditions particular to panel of fibers. Temperature curves along material depth or in lateral direction will allow us to evaluate thermal behavior of fibers material. Influence of the thermal exchange coefficient on front face will be highlighted.

Theory
Equation of heat transfer through rectangular material to absence of internal source, is written in two dimensions ( Figure 1): α is thermal diffusivity coefficient of the material (m 2 ·s −1 ) and is expressed as where: ρ (kg·m −3 )is density of material, -C (J·kg −1 ·K −1 ) is mass thermal capacity, -λ (W·m −1 ·K −1 ) is thermal conductivity of material.
For method by separating variables, we propose: (3) in (1) leads to following expressions: Equations (5)-(7) are deduced from following relationships (8): Solutions of preceding equations give: ( ) which gives general expression of temperature through material in form: T is the temperature imposed on the front face.
We pose: we obtain: Graphical representations of Figure 2 and Figure 3 correspond to transcendental Equation (22) and Equation (23) for obtaining eigenvalues m β and n µ respectively and given in Table 1. Energy and Power Engineering      Coefficient k is relative to the thermal effisitivity, it defines transmission of heat to wall of material: 0 1 k ≤ ≤ . k = 0: we have perfect thermal insulation behavior on surface of material; k = 1: we have perfect conductor behavior on surface of material. We worked with k = 0.85.
Taking into account results obtained in previous studies [4] [5], we consider an average value of coefficient translating heat exchanges on surface of material:       excitation towards amplitude of initial temperature of material; material warms up slightly. Figure 5 shows evolution of temperature along lateral axis. Overheating phenomenon in vicinity of y = 0 is confirmed because there is no heat exchange with respect to outside. Beyond y = 0.02 m, material temperature fluctuates around 10˚C, which corresponds to low overheating of material. Figure 6 and Figure 7 show influence of heat exchange coefficient at front face on temperature. Exchange coefficient contributes significantly to heating of wall and propagation of heat along axis of depths. On other hand, along lateral axis, this influence is practically nil.

Validation
Methods of thermal characterization in numerical simulation [6] or analytical [7] [8] present results of evolutions of temperature and density of heat flux comparable through different materials.

Conclusion
Starting from resolution of equation of two-dimensional heat in frequency dynamic regime, we obtained curves which show thermal behavior of fibers material subjected to different climatic constraints. Thermal insulating nature of material is highlighted by an evanescent tendency of external excitation inside material.