# Homogenization of the heat equation for a domain with a network of pipes with a well-mixed fluid

- 92 Downloads

## Abstract

We consider the linear heat equation in a domain occupied by a solid material with a network of pipes in which a well-mixed fluid is circulating. The temperature of the fluid in the pipe is uniform and its time variation is determined by the thermal flux on the wall of the pipe, plus a given internal source; continuity of the temperature across the pipe is also assumed. We suppose that we deal with a periodic geometry, with cells of size ε with inclusions of size r_{g}; we study in detail in the case r_{ε}∼ε, referring to a previous paper for the case r_{ε}≪ε In the limit ε»0 we get a homogenized equation. The limit depends strongly on the ratio between the time variation of the temperature in the inclusions and the thermal flux through the interface. The homogenized equation has a new specific heat, which depends on the porosity and the constant of proportionality between the time variation of temperature and the flux on the boundary of the pipe. We also have a new thermal conductivity depending on the microstructure, and volume sources appear. The main tool is the energy method and we generalize the classical results for the more standard boundary conditions for parabolic equations. Finally, we consider the network of pipes forming a random ball structure. We prove convergence for this case. The homogenized equation is of the same form as in the periodic case but auxiliary problems are stochastic.

## Keywords

Time Variation Parabolic Equation Heat Equation Solid Material Main Tool## Preview

Unable to display preview. Download preview PDF.

## Literature

- [1]E. Acerbi -V. Chiadò Piat -G. Dal Maso -D. Percivale,
*An extension theorem from connected sets,and homogenization in general periodic domains*, Nonlinear Anal. TMA,**18**(1992), pp. 481–496.Google Scholar - [2]D. Cioranescu -J. Saint Jean Paulin,
*Homogénéisation de problèmes d'évolution dans des ouverts à cavités*, C. R. Acad. Sci. Paris,**286**(1978), pp. 899–902.Google Scholar - [3]D. Cioranescu -J. Saint Jean Paulin,
*Homogenization in open sets with holes*, J. Math. Anal. Appl.,**71**(1979), pp. 590–607.Google Scholar - [4]D. Cioranescu -F. Murat,
*Un terme étrange venu d'ailleurs*, in*Nonlinear Partial Differential Equations and their Applications*, Collège de France Seminar, vol. II: 60, 93–138; vol. III:**70**, 154–178; Reserach Notes in Mathematics, Pitman, London (1981).Google Scholar - [5]A. Damlamian -P. Donato,
*Homogenization with small perforations of increasingly complicated shape*, SIAM J. Math. Anal.,**22**(1991), pp. 639–652.Google Scholar - [6]V. Fitt -J. R. Ockendon -M. Shillor,
*Counter-current mass transfer*, Int. J. Heat Mass Transfer,**28**(1985), pp. 753–759.Google Scholar - [7]
- [8]S. Kaizu,
*Behavior of solutions of the Poisson equation under fragmentation of the boundary of the domain*, Japan J. Appl. Math.,**7**(1990), pp. 77–102.Google Scholar - [9]S. M. Kozlov,
*Averaging of random operators*, Math. USSR Sbornik,**37**(1980), pp. 167–180.Google Scholar - [10]O. A.Ladyzenskaja - V. ASolonnikov - N. N.Ural'ceva,
*Linear and quasilinear equations of parabolic type*, Transl. A.M.S.,**23**(1968).Google Scholar - [11]J. L. Lions -F. Magenes,
*Problème aux limites non homogènes et applications*, vol. 1, Dunod, Paris (1968).Google Scholar - [12]J. L. Lions,
*Some Methods in the Mathematical Analysis of Systems and their Control*, Gordon and Breach, New York (1981).Google Scholar - [13]A. Mikelić -M. Primicerio,
*Homogenization of heat conduction in materials with periodic inclusions of a perfect conductor*, in*Progress in Partial Differential Equations: Calculus of Variations, Applications*(C. Bandle et al., eds.) Pitman Research Notes in Mathematics vol.**267**, pp. 244–257, Longman, London (1992).Google Scholar - [14]J. Saint Jean Paulin,
*Etude de quelques problèmes de mecanique et d'electrotechnique liés aux methodes d'homogénéisation*, Thèse d'état, Université P. et M. Curie, Paris (1981).Google Scholar - [15]E.Sanchez-Palencia,
*Non homogeneous media and vibration theory*, Lecture Notes in Physics,**127**, Springer-Verlag (1980).Google Scholar - [16]V. V. Zhikov,
*Problems of function extension related to the theory of homogenization*, Diff. Eqns. (Trans. of Differ. Uravn.),**26**(1990), pp. 33–34.Google Scholar - [17]V. V. Zhikov,
*Asymptotic problems connected with the heat transfer equation in perforated domains*, Math. USSR-Sbornik,**67**(1990), pp. 1283–1305.Google Scholar