Chapter 5

FROM THE FORMAL CONCEPT ANALYSIS

TO THE NUMERICAL SIMULATION OF THE THERMAL DIFFUSION PHENOMENA IN A FINITE MEDIUM

Riad Assaf1,2, , Roy Abi Zeid Daou2,†,

Xavier Moreau1,‡ and Fady Christophy1,2,#

1University of Bordeaux, IMS Laboratory, Talence, Bordeaux, France

2Lebanese German University, Faculty of Public Health, Biomedical Technologies department, Sahel Alma, Jounieh, Lebanon

Abstract

This chapter shows the use of the fractional order approach in the study of the thermal interface behavior in view to facilitate the identification of this phenomenon and the optimization of its process control. After a short introduction, the main parameters are defined then the semi-infinite plane case study is considered. A representative model is developed based on the system equations. The operational, frequency and time responses are studied. The case study of a finite plane follows using the same methodology, nevertheless exact analytical solutions in the time domain don’t exist; their approximations will be treated in other publications.

1. Context and Problematic

In general, the development of diffusion phenomena equations naturally leads to fractional order systems (Podlubny, 2002). There are many examples in the fields of physics such as electrochemistry (Kuhn, 2005), electromagnetism (Benchellal, 2005) (Canat, 2005), or thermal (Agrawal, 2004) (Battaglia, 2001) (Kusiak, 2005).

E-mail addresses: riad.assaf@u-bordeaux.fr; r.assaf@lgu.edu.lb.

† E-mail address: r.abizeiddaou@lgu.edu.lb.

‡ E-mail address: xavier.moreau@u-bordeaux.fr.

# E-mail addresses: fady.chrystophy@u-bordeaux.fr; f.christophy@lgu.edu.lb.

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Actually, the recognition of fractional order behavior when identifying physical systems flourished starting the second half of the twentieth century (Oldham, 1974). One of these fields where the fractional order integration can show up is the thermal diffusive interfaces. In fact, for some special conditions, an integration of order 0.5 appears when identifying such system (Abi Zeid Daou, 2012, Assaf, 2012). The objective of this chapter is to study the presence of the fractional order and its effects on the system behavior, and to determine the relation between the applied flux, the conduction and the point of measurement of the temperature in a plane medium.

This objective is a milestone towards the definition of valid approximations for reliable numerical simulations of the thermal diffusion phenomena in a finite medium where exact analytical time response solutions don’t exist. For instance, one can define conditions for which a finite medium can be considered as a semi-infinite medium, and can define other conditions for which analytical approximations generate accurate time response simulations.

The understanding of the thermal diffusion phenomena, hence the establishment of a validation model has two main goals:

-As part of the identification process: to facilitate the definition of the model structure, and to pass from a black box approach to a gray box approach.

-As part of a process control model: to deduct from the validation model not only a nominal model of synthesis, but also structural and parametric uncertainties.

This chapter contains six main paragraphs other than the Abstract and the Introduction at the beginning; the Conclusion and the References at the end. A reminder of the main physical characteristics needed to describe the thermal diffusion are found in Para 2, in Para 3 a model of the diffusion phenomena in the semi-infinite medium is developed, then the relative operational, frequency and time responses are studied in Para 4. In Para 5 a model of the diffusion phenomena in the finite medium is developed, then only the corresponding operational and frequency responses are studied in Para 6 since no exact analytical solutions exist to represent the time response for the finite medium. In Para 7 one can find some step responses using Simulink, it opens the door for the need for simple and accurate approximations.

2. Parameters and Definitions

The main thermal parameters and definitions characterizing any thermal interface are presented in this paragraph.

Homogeneous and Isotropic medium [Özişik, 1980] is a material in which thermal conductivity is independent of direction.

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3. Semi-Infinite Plane

We consider a semi-infinite, homogeneous, isotropic, one-dimensional plane medium of

Figure 1. Diffusion in a semi-infinite plane medium.

3.1. System Definition

The one-dimensional heat conduction equation (Özişik, 1985) is given by a partial differential equation:

Since the initial temperature is zero all over the medium:

The Fourier law states the boundary condition (Özişik, 1985) where the flux is applied, the second condition is trivial:

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3.2. System Resolution

The use of the fractional calculus to resolve the problems of the diffusive interfaces dates back to the beginning of the 20th century [Battaglia, 2001]. The system of (Eq. 1-4) represents

This is a differential equation of the 2nd order in x. Its general solution is:

leading to the characteristic equation:

which has the following 2 roots ( ) and ( ):

Hence the solution of (Eq. 5) is:

Taking into consideration the definitions of the characteristic parameters and the outcome of (Eq. 10) then resolving (Eq. 9), the solution of (Eq. 5) is found to be:

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Under the assumption of an initial condition of the temperature being zero, the transfer function ( ) between the input flux ̅( ) and the output temperature ̅( ) is:

semi-infinite medium.

This new approach allows understanding the link existing between the two kinds of study encountered in the dynamics of thermal systems, i.e.:

Dirichlet (first-type) boundary condition.

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