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20 2 Modeling of Thermal Non-equilibrium On the symmetry planes: @u @v @k @e ¼ ¼ ¼ ¼ 0; @y @y @y @y ð2:49Þ . where  u and v are components of u On the periodic boundaries:  ujoutlet ; ujinlet ¼  vjinlet ¼ vjoutlet ; hjinlet ¼ hjoutlet kjinlet ¼ kjoutlet ; ejinlet ¼ ejoutlet ;   T À Tw  T À Tw  , ¼ ; TB ðxÞ À Tw inlet TB ð xÞ À Tw outlet The bulk mean temperature of the fluid is given by: R  uTdy  TB ð xÞ ¼ R udy ð2:50Þ ð2:51Þ ð2:52Þ Computations are based on the Darcy velocity, the length of structural unit H and the temperature difference ðTB ðxÞ À Tw Þ, as references scales.

16), the momentum equation reads (see [1, 5] for details), q rÁ  ! n  Ão D u D u ÀrÁ l þ lt/ r uD Þ T uD þ ðr / : rel j l/ cF /qju urel pffiffiffiffi rel þ ¼ À rð/hpii Þ À u K K ð3:19Þ where lt/ is the macroscopic eddy viscosity given by lt/ ¼ q cl fl hkii 2 heii ; ð3:20Þ cl is dimensionless constant and fl is a damping function, which differs from unit if a Low-Reynolds turbulence model is applied. More on damping functions and model constants will be shown below. Thus, to obtain the eddy viscosity, lt/ , we used here the Low and High Reynolds number k – e models, whose equations for the turbulent kinetic energy and its dissipation rate, incorporating now a relative rel j, are given next [5].

As in the flow case, volume integration is performed over a Representative Elementary Volume (REV), resulting in, 8 0 > > > B i < 6 À Á 6@ /h Tf i B i  i Tf ii þ þ r Á / Bh ui hTf ii þ hi u q cp f 6 hu0 ii 0f ii þ > 4 @t @ |fflfflffl{zfflfflffl} |fflfflfflffl{zfflfflfflffl} > > : thermal disperson turbulent heat flux 2 3 Z Z À Á 1 1 6 7 ¼ r Á 4 kf r / hTf ii þ ni kf Tf dA 5 þ ni Á kf rTf dA DV Ai Ai |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 2 conduction 1 93 > > >7 C= C 7 i 0 i 0 i h u Tf i C 7 5 A> |fflfflfflffl{zfflfflfflffl} > > ; turbulent thermal dispersion interfacial heat transfer ð2:28Þ where the expansion, hu0 Tf0 ii ¼ hðhu0 ii þ i u0 Þ ðhTf0 ii þ i T 0 Þii ¼ hu0 ii hTf0 ii þ hi u0i Tf0 ii ð2:29Þ has been used in light of the double decomposition concept given by Eqs.

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