THERMAL IMPEDANCE ESTIMATIONS BY SEMI-DERIVATIVES AND SEMI-INTEGRALS: 1-D semi-infinite cases
Abstract
Simple 1-D semi-infinite heat conduction problems enable to demonstrate the potential of the fractional calculus in determination of transient thermal impedances under various boundary conditions imposed at the interface (x=0). The approach is purely analytic and very effective because it uses only simple semi-derivatives (half-time) and semi-integrals and avoids development of entire domain solutions.
Dates
- Submission Date2012-05-22
- Revision Date2012-11-13
- Acceptance Date2012-11-20
References
- . Carslaw, H.S. , Jaeger, J.C., Conduction of Heat in Solids. Oxford University Press, London, 1959.
- Breaux , H.J. , Schlegel, P.T. , Transient heating of thin plates, Int. J Heat Mass Transfer, 13(1970) , 1, pp.218-211.
- Muzychka, Y.S., Yovanovich M. M., Culham, J.R., Thermal Spreading Resistance in Compound and Orthotropic Systems, J. Thermophysics and Heat Transfer, 18 (2004) ,1, pp. 45-51.
- Schneider, G. E. , Strong, A. B. , Yovanovich , M. M. , Transient thermal response of two bodies , communicating through a small circular contact area , Int. J. Heat Mass Transfer 20 (1977), 4, pp.301-308.
- Belghazi, M. H. , Analytical modelling of transient heat transfer due to moving heat sources in bi-layer materials with imperfect contact, PhD Thesis, University of Limoges, Limoges, France, 2008. (in French).
- Laraqi , N. , Thermal impedance and transient temperature due to a spot of heat on a half- space , Int. J. Therm Sci., 49 (2010) ,3, pp.529-533,
- Laraqi , N. , Contact temperature and flux partition coefficient of heat generated by dry friction between two solids. New approach to flux generation, Int. J. Heat Mass Transfer 35 (11) (1992) ,11, pp.3131-3139.
- Laraqi , N. , Bairi, A., Ségui,L., Tempertaure and thermal resistance in frictional devices, Appl. Therm. Eng. 24 (2004) ,17-18, pp. 2567-2581.
- Aderghal, N., Loulou, T., Bouchoucha, A. , Rogeon , Ph.Analytical and numerical calculation of surface temperature and thermal constriction resistance in transient dynamic strip contact , Appl. Therm. Eng., 31 (2011) ,8-9, pp.1527-1535 .
- Gabano, J.-D. Poinot , T. Fractional modelling and identification of thermal systems , Signal Processing, 91 (2011) ,3, pp.531-541.
- Zubair, S.M., Chaudhry, M.A., Some analytical solutions of time-dependent, continuously operating heat sources. Heat Mass Transfer, 28(1993) ,4, pp.217-223.
- Hou, Z.B. , Komanduri, R. , General solutions for stationary/moving plane heat source problems in manufacturing and tribology, Int. J. Heat Mass Transfer. 43 (2000) ,10, pp.1679-1698.
- Agrawal , O. P. , Application of Fractional Derivatives in Thermal Analysis of Disk Brakes , Nonlinear Dynamics , 38( 2004) ,1-4, pp.191-206.
- Ganaoui, M. El., Laraqi, N. Analytical computation of transient heat transfer macro-constriction resistance: application to thermal spraying processes, Comptes Rendus Mécanique, 30 (2012) ,7, PPP. 536-542.
- Turyk, P.J. , Yovanovich, M.M. , Transient constriction resistance for elemental flux channels heated by uniform flux sources 22nd Heat Transfer Conference,ASME (1984) ,Paper No 84-HT-52, August, Niagara Falls, N.Y.
- Oldham , K.B. , Spanier ,J. , The Fractional calculus , Academic Press, New York, 1974.
- Kulish , V.V. , Lage, J.L. , Fractional-diffusion solutions for transient local temperature and heat flux, J. Heat Transfer, 122 (2000) ,2, pp. 372-376.
- Siddique, I. , Vieru, D. , Stokes flows of a Newtonian fluid with fractional derivatives and slip at the wall, Int. Rev. Chem. Eng., 3 (2011) ,6, pp.822- 826.
- Qi , H. , Xu, M. , Some unsteady unidirectional flows of a generalized Oldroyd-B fluid with fractional derivative, Appl. Math. Model., 33 (2009) ,11, pp.4184-4191.
- dos Santos, M.C, Lenzi, E. , Gomes, E.M., Lenzi,, M.K. , Lenzi, E.K. , Development of Heavy Metal sorption Isotherm Using Fractional Calculus, Int. Rev. Chem. Eng., 3 (2011) ,6, pp.814-817.
- Hristov J., Starting radial subdiffusion from a central point through a diverging medium (a sphere): Heat-balance Integral Method, Thermal Science, 15 (2011), supl. 1 ,pp.S5-S20 .
- Pfaffenzeller, R.A., Lenzi, M.K., Lenzi, E.K. , Modeling of Granular Material Mixing Using Fractional Calculus, Int. Rev. Chem. Eng., 3 (2011) ,6, pp. 818-821.
- Meilanov, R.P., Shabanova, M.R. , Akhmedov, E.N. , A Research Note on a Solution of Stefan Problem with Fractional Time and Space Derivatives, Int. Rev. Chem. Eng., 3 (2011) ,6, pp. 810-813.
- Voller, V.R. , An exact solution of a limit case Stefan problem governed by a fractional diffusion equation, Int. J. Heat Mass Transfer, 53 (2010) ,23-24, pp.5622-5625.
- Liu, J. , Xu, M. , Some exact solutions to Stefan problems with fractional differential equations, J. Math. Anal. Appl., 351(2010) ,2, pp. 536-542
- Debnath, L., Nonlinear Partial Differential Equations for Scientist and Engineers, Birkhouser, Boston, 1997.
- Babenko, Yu. I., Heat-Mass Transfer. Methods for calculation of thermal and diffusional fluxes, Khimia Publ., Moscow, 1984 (in Russian).
- Abramovitz, M., Stegun, I.A., Handbook of mathematical functions. Dover, New York, 1964.
Volume
17,
Issue
2,
Pages581 -589