Fractional Order Model of the Two Dimensional Heat Transfer Process

Krzysztof Oprzędkiewicz,Wojciech Mitkowski,Maciej Rosół

doi:10.3390/en14196371

Abstract

In this paper, a new, state space, fractional order model of a heat transfer in two dimensional plate is addressed. The proposed model derives directly from a two dimensional heat transfer equation. It employes the Caputo operator to express the fractional order differences along time. The spectrum decomposition and stability of the model are analysed. The formulae of impluse and step responses of the model are proved. Theoretical results are verified using experimental data from thermal camera. Comparison model vs experiment shows that the proposed fractional model is more accurate in the sense of MSE cost function than integer order model.

Highlights

It is known that the non integer order calculus can be applied in modeling of processes and phenomena hard to analyse with the use of other tools
Non integer order (NIO) or fractional order (FO) models of different physical phenomena have been presented by many authors
The convergence of the proposed model will be tested using approach close to presented in papers [32,33]. This can be performed by estimating the orders M and N assuring a predefined value of Rate Of Convergence (ROC)

Summary

Introduction

It is known that the non integer order calculus can be applied in modeling of processes and phenomena hard to analyse with the use of other tools. This paper is devoted to present a new, discrete, FO model of heat transfer process in two dimensional plate Such a process is described by a partial differential equation (PDE) of parabolic type. The partial derivative along time is expressed by Caputo operator, both partial, spatial derivatives are integer order Such a model of thermal process has not been previously published. An idea of Mittag–Leffler function needs to be given It is a non integer order generalization of exponential function eλt and it plays crucial role in the solution of fractional order (FO) state equation. The fractional order, integro-differential operator (1) is described by different definitions, given by Grünwald and Letnikov (GL definition), Riemann and Liouville (RL definition) and Caputo (C definition). A fractional linear state equation using Caputo definition should be recalled

The Experimental System and Its FO Model

The Decomposition of the Model

The Convergence

Experimental Validation of Results

Discussion of Results and Final Conclusions