Fractional modelling and identification of thermal systems

doi:10.1016/j.sigpro.2010.02.005

Signal Processing

Volume 91, Issue 3, March 2011, Pages 531-541

https://doi.org/10.1016/j.sigpro.2010.02.005 Get rights and content

Abstract

Heat transfer in homogeneous media obeys to diffusion phenomenon which can be modelled with the help of fractional systems. In this paper, we use a parsimonious black box model based on an original fractional integrator whose order $\frac{1}{2}$ acts only over a limited spectral band. We carried out simulations of front face thermal experimentations which consist in measuring the temperature at the surface of a material where a random heat flux is applied. We consider the characterization of the thermal behaviour of a wall or a sphere. These simulations show the ability of the fractional model, thanks to an output error identification technique, to obtain accurate estimation of diffusion interface temperature evolution as well as frequency response using time data series for the two considered geometries. Experimental results are given in the case of the sphere.

Introduction

Non-integer order systems, also known as fractional systems have been introduced long ago in various fields of science for the modelling of diffusion processes such as electrochemical phenomena encountered in batteries [1], [2], the modelling of frequency effects (skin effect) in the squired-cages of induction machine [3] and heat transfer [4], [5], [6]. Such systems are characterized by long range memory transients [7]. In this paper, we analyse heat unidirectional conduction in a homogeneous material. We first determine and analyse the non-linear transfer functions characterizing the diffusion interface in the cases of front face thermal characterization experiments which consist in measuring the temperature at the surface of a material where a random heat flux is applied [8], [9]. We developed simulators based on finite differences which lead to linear state-space representations for two kinds of geometry: the wall and the sphere. We propose then a fractional model which allows, thanks to only three parameters, to estimate the frequency behaviour, especially the phase plot which strongly depends on the geometry using identification in the time domain. The proposed methodology is finally tested on an experimental thermal system with a spherical geometry.

Section snippets

Wall heat transfer modelling

Let us consider the problem which consists in computing the heating temperature T_f(t) at the front face of a wall where a random heat flux P_in(t) is applied. Heating temperature T(x, t) is assumed to be uniform on any plane parallel to the front and back face (see Fig. 1). The dimensions of the wall are characterized by its section S_w and thickness L. Let P(x, t) be the flux passing through the wall at abscissa x. T(x, t) and P(x, t) satisfy the following heat diffusion equations: $\frac{\partial T (x, t)}{\partial t} = a \partial^{2} T ($

Wall simulator

In order to simulate the heat diffusion equation, we divide the wall into I elementary cells of the same thickness x=L/I characterized by their thermal resistance R_i and capacity C_i [12]: $\{\begin{matrix} R_{i} = R_{i + 1} = \frac{Δ x}{λ S_{w}} \\ C_{i} = ρ {cS}_{w} Δ x \end{matrix}$ Considering the simulation scheme of each elementary cell (see Fig 4), the temperature is governed by $\frac{{dT}_{i}}{dt} = \frac{1}{R_{i} C_{i}} (T_{i - 1} - 2 T_{i} + T_{i + 1})$ In the considered simulated experiment, we impose T(L, t)=0 assuming the back-face heating temperature is kept constant. The numerical simulation is performed

Output error identification

Considering a set of K observed input/output data u_k and $y_{k}^{*} = y_{k} + b_{k}$ , acquired with a sampling period T_s, b_k being an output noise, the quadratic criterion: $J ({\underset{̲}{\hat{θ}}}_{ω_{b}}) = \sum_{k = 1}^{K} ɛ_{k}^{2} ({\underset{̲}{\hat{θ}}}_{ω_{b}})$ of output error: $ɛ_{k} ({\underset{̲}{\hat{θ}}}_{ω_{b}}) = y_{k}^{*} - {\hat{y}}_{k} ({\underset{̲}{\hat{θ}}}_{ω_{b}})$ is now minimized. As model ${\hat{H}}_{ω_{b}} (s)$ is not linear in ${\underset{̲}{\hat{θ}}}_{ω_{b}}$ , the Levenberg–Marquardt algorithm [25], which achieves a harmonious compromise between the stability of the Gradient method and the fast convergence rate of the Gauss–Newton method, is used to estimate ${\underset{̲}{\hat{θ}}}_{ω_{b}}$ iteratively: $\underset{̲}{\hat{θ}}$

Description

The system is a brass ball with 3 cm radius (see Fig. 8). A power transistor is placed at the centre of the ball to generate a heat flux. A sensor is fixed on the interface of the heat source. The ball is immersed in a thermally controlled enclosure filled with water where the ambient temperature is kept constant at the value 31 °C. The thermal power P_in(t) is controlled by the base control voltage of the transistor. The values of inner temperature T_in(t) have been measured by a data acquisition

Conclusion

In this paper, a contribution to the modelling of diffusion systems and their identification by a parsimonious fractional model has been presented. In the framework of thermal front face experiments, we studied the diffusive heat transfer functions of systems with two different geometries, the wall and the sphere. Starting from the non-rational transfer function of each system, which behaves as fractional integrators of order $\frac{1}{2}$ at high frequencies, we proposed a simplified linear fractional

References (25)

J. Sabatier et al.
Fractional system identification for lead acid battery state charge estimation
Signal Processing
(2006)
J.-L. Battaglia et al.
Représentation non entière du transfert de chaleur par diffusion. Utilité pour la caractérisation et le contrôle non destructif thermique
Int. J. Therm. Sci.
(2004)
A. Degiovanni
Correction of pulse length for the flash thermal diffusivity measurement method
Int. J. Heat Mass Transfer
(1987)
H. Wang et al.
Periodic thermal contact: a quadrupole model and an experiment
Int. J. Therm. Sci.
(2002)
T. Poinot et al.
A method for modelling and simulation of fractional systems
Signal Processing
(2003)
A. Benchellal et al.
Approximation and identification of diffusive interfaces by fractional models
Signal Processing
(2006)
J. Lin, T. Poinot, J.-C. Trigeassou, R. Ouvrard, Parameter estimation of fractional systems: application to the...
J. Lin, T. Poinot, J.-C. Trigeassou, H. Kabbaj, J. Faucher, Modélisation et identification d’ordre non entier d’une...
J.-D. Gabano, T. Poinot, Fractional identification algorithms applied to thermal parameter estimation, in: 15th IFAC...
R. Malti, M. Aoun, J. Sabatier, A. Oustaloup, Tutorial on system identification using fractional differentiation...

J. Beran

Statistical methods for data with long-range dependance

Statist. Sci.

(1992)

S. Orain et al.

Thermal conductivity of ZrO₂ thin films

Int. J. Therm. Sci.

(1999)

Cited by (0)

View full text

Fractional modelling and identification of thermal systems

Abstract

Introduction

Section snippets

Wall heat transfer modelling

Wall simulator

Output error identification

Description

Conclusion

Signal Processing

Int. J. Therm. Sci.

Int. J. Heat Mass Transfer

Int. J. Therm. Sci.

Signal Processing

Signal Processing

Statistical methods for data with long-range dependance

Statist. Sci.

Thermal conductivity of ZrO2 thin films

Int. J. Therm. Sci.

Thermal conductivity of ZrO₂ thin films