Modelling heat transfer in heterogeneous media using fractional calculus

D. Sierociuk, A. Dzielinski, G. Sarwas, I. Petras, I. Podlubny, T. Skovranek
2013 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences  
This paper presents the results of modelling the heat transfer process in heterogeneous media with the assumption that part of the heat flux is dispersed in the air around the beam. The heat transfer process in a solid material (beam) can be described by an integer order partial differential equation. However, in heterogeneous media, it can be described by a sub-or hyperdiffusion equation which results in a fractional order partial differential equation. Taking into consideration that part of
more » ... e heat flux is dispersed into the neighbouring environment we additionally modify the main relation between heat flux and the temperature, and we obtain in this case the heat transfer equation in a new form. This leads to the transfer function that describes the dependency between the heat flux at the beginning of the beam and the temperature at a given distance. This article also presents the experimental results of modelling real plant in the frequency domain based on the obtained transfer function. on June 1, 2017 Downloaded from for n − 1 < α < n. The initial conditions for the fractional order differential equations with the Caputo derivatives are in the same form as for the integer order differential equations. For the Caputo partial fractional derivative of order α of a function f (t, λ) with respect to variable t, we will use notation of the form ∂ α f (t, λ)/∂t α , which is often used in the related literature.
doi:10.1098/rsta.2012.0146 pmid:23547224 fatcat:yxawy2x5vrhb7aguxbr7w4t37m