# Investigation of heat distribution using non-integer order time derivative

• Tomasz Raszkowski Department of Microelectronics and Computer Sciences, Łódź University of Technology
• Agnieszka Samson Department of Microelectronics and Computer Sciences, Łódź University of Technology
• Mariusz Zubert Department of Microelectronics and Computer Sciences, Łódź University of Technology
Keywords: Dual-Phase-Lag model, Gr¨unwald Letnikov, heat transfer approximation, Fourier-Kirchhoff modification, non-linear order time derivative

### Abstract

This paper presents the analyses of heat distribution based on non-linear order time derivative. The described problem has been demonstrated on a simple rectangular structure made of the silicon. Moreover, the thermal model called Dual-Phase-Lag has been employed to obtain the solution. Furthermore, the new approximation of Dual-Phase-Lag model has been proposed. This modification has been based on Grünwald-Letnikov definition of fractional derivative. The time derivative order, which appears in Fourier-Kirchhoff model, has been modified to non-integer order. Next, received normalized rises of the temperature have been compared with results obtained using Dual-Phase-Lag equation. Then, the orders of the fractional time derivative have been matched to different values of the heat flux and temperature time lags. Eventually, the final formula, which takes into consideration the order of time derivative and both model parameters of Dual-Phase-Lag

### References

J. B. J. Fourier, Th´eorie analytique de la chaleur, Firmin Didot, Paris, 1822.

M. Zubert, M. Janicki, T. Raszkowski, A. Samson, P. S. Nowak, K. Pomorski, The Heat Transport in Nanoelectronic Devices and PDEs Translation into Hardware Description Languages, Bulletin de la Societe des Sciences et des Lettres de Lodz, Serie: Recherches sur les D´eformations vol. LXIV, 69–80, Lodz, 2014.

A. Nabovati, et al, On the lattice boltzmann method for phonon transport, Journal of Computational Physics, 5864–5876, 2011.

T. Raszkowski, A. Samson, The Numerical Approaches to Heat Transfer Problem in Modern Electronic Structures Computer Science, vol. 18 (1) , 71–93, 2017.

D. Y. Tzou, A Unified Field Approach for Heat Conduction From Macro- to Micro-Scales, Transactions of ASME J. Heat Transfer, 8–16, 1995.

T. Raszkowski, A. Samson, M. Zubert, Dual-Phase-Lag Model Order Reduction Using Krylov Subspace Method for 2-Dimensional Structures, Bulletin de la Societe des Sciences et des Lettres de Lodz, Serie: Recherches sur les Deformations, In press

B. A. Jacobs, A new Grunwald Letnikov derivative from second order scheme, Abstract and Applied Analysis, vol. 2015, 2015.

T. Raszkowski, M. Zubert, A. Samson, Analysis of Implementation of Differential Equations of Non-Integer Orders to Dual-Phase-Lag Model Approximation, 24th Proc. of Thermal Investigations of ICs and Systems, September 26–28, 2018.

Published
2019-04-26
Issue
Section
Articles