The heat transfer equation is used to determine the heat flow by conduction through a composite material along the real axis. An analytical dimensionless analysis is implemented in the framework of a separation of variables method (SVM). This approach leads to an Eigenvalues problem that is solved by the Newton’s method. Two types of dynamics are found: An unsteady condition (in the form of jumps or drops in temperatures depending on the considered case), and a permanent equilibrium (tending to the ambient temperature). The validity and effectiveness of the proposed approach for any number of adjacent layers is also discussed. It is shown that, as expected, the diffusion of the temperature is linked to the ratio of the thermo-physical properties of the considered layers and their number.

The heat equation, describing heat transfer by conduction, is a partial differential equation, established by J. Fourier in the end of the 19^{th} century after certain experiments. It has a great interest in mathematics and physics. This transfer can be studied either in a single body or through several systems with different thermo-physical properties. In the literature, there are several interesting works, based on certain original references [

In our paper, the transfer problem by conduction across bilayer and three-layer materials is investigated, adopting the same notations, as used in [

We consider a material composed of two different regions

where,

The one-dimensional heat conduction without thermal source for the two slabs, is given by [

with the set of boundary conditions (BCs) [

and on the contact surface, we have

The initial conditions (ICs), for each layer, are supposed constants

We have now, a full solvable system constructed by

For each layer, the solutions are obtained easily by SVM, where

Dimensionless calculus has been used for several reasons and purposes. We are talking about: The simplicity of the calculations, the reduction of the parameters, the permanent verification of the homogeneity of the equations, the writing of the formulas will be more compact and more significant as well as the ease of reading the curves because the intervals of variations of the solution temperatures are reduced to acute and positive intervals. Then, the dimensionless group is defined by [

and the Biot numbers

Then, after some manipulations (see [

called Eigen values, representing the non-vanishing dimensionless roots of the function

With

We reconstruct our solutions under the dimensionless form, as

where, we introduce the space-time dimensionless coordinates:

The determination of

1 | 1.174564997 |

2 | 2.670341640 |

3 | 4.100755655 |

4 | 5.902293511 |

5 | 7.463000332 |

6 | 9.311800164 |

7 | 10.99791545 |

8 | 12.76394020 |

9 | 14.59650467 |

10 | 16.25466773 |

11 | 18.18431142 |

12 | 19.80816753 |

13 | 21.72556781 |

14 | 23.42691725 |

15 | 25.23379476 |

16 | 27.07242465 |

17 | 28.75147230 |

18 | 30.68935659 |

19 | 32.32010618 |

20 | 34.24837236 |

An order higher than 20 roots can be reached, having a small influence on the result (see

where,

where, the dimensionless initial temperatures

The error is defined as the relative difference, at zero time, between calculated temperature, obtained by

As can be seen, ^{th} root, the reason why, we stopped in the sum (

The same reasoning can be used, for the case of three-layer material. We consider a material composed of three different regions

with

where,

Thus, the temporal and space solutions take, according to the Eigen values set

The final dimensionless solution is expanded as a linear combination of products of the temporal and space parts

We can regroup, similarly as

Note that the unsteady equilibrium temperature, between any two adjacent slabs

The final dimensionless solution is expanded as a linear combination of products of the temporal and space parts. Therefore, the dimensionless unsteady equilibrium temperature is

The error is defined by

By taking the instance, presented in

To reach that, in the first layer, the temperature increases, proportionally to the distance from the contact surface, with small values, called temperature jumps, while the taken time is inversely proportional to this distance. Hence, the final thermal equilibrium is achieved. For the second layer, the temperature converges to the equilibrium one, with a much slower pace, due to its thickness that is greater than that of the first layer.

We also emphasize that the unsteady equilibrium is clear, for

The heat transfer, in bilayer material, is well described according to thermo-physical properties (

The error with respect to the number of roots

Similarly, one can observe the same reaction of heat transfer

In this case, we consider the example where

_{1} and S_{2}), it takes place at _{2} and S_{3}), it takes

The continuity of

The divergence of the graphs in the boundaries (

This analytical study shows that there is a similarity in the evolution of heat transfer between two layers as well as between three layers. In both cases, we note that an unsteady thermal equilibrium occurs moments before the final thermal equilibrium. Mathematically, we notice that we get the same form for the solutions, except for the coupling function

Concerning the accuracy of the results, this is based on two points having almost significant equivalence: The first is the exact analytical calculation by SVM and using the Maple Program, which brings us back to the similar results detailed in the literature. The second point is the concordance with the experimental prediction, which explains the existence of two kinds of equilibrium: Unsteady, and permanent similar to the damped movement in the mechanical vibration of solids or fluids (Transitional and permanent diverge).

Layer’s index (

Layer’s temperature [K]

Ambient temperature [K]

Initial temperature in i^{th} layer [K]

Initial deference in temperatures [K]

Deference in temperatures [K]

Layer’s thickness [m]

Conductivity [W.m^{−1}.K^{−1}]

Diffusivity [m^{2}.s^{−1}]

Specific heat [J.Kg^{−1}.K^{−1}]

Density [Kg.m^{−3}]

Convection coefficient [W.m^{−2}.K^{−1}]

Dimensionless diffusivity

Dimensionless conductivity

Dimensionless thickness

Biot numbers

Dimensionless Eigen-values

Dimensionless time

Dimensionless position

Dimensionless initial, temporal and unsteady equilibrium temperatures respectively

T. Sahabi would like to thank Prof. F. de Monte for his assistance in this research in terms of providing references and answering questions, and Prof. Mohamed El Ganaoui for his encouragement and providing references.

_{2}O

_{3}and TiO

_{2}nanoparticles in order to reduce the energy demand in the conventional buildings by integrating the solar collectors and phase change materials