Analytical characterization and conceptual assessment of solid and ~uid temperature di}erentials in porous media

An analytical characterization of forced convective ~ow through a channel _lled with a porous medium is presented in this work[ Based on a two!equation model\ including transverse conduction contributions\ exact solutions are obtained for both ~uid and solid phase temperature _elds[ The Nusselt number is also obtained in terms of the pertinent physical parameters\ namely the Biot number for the internal heat exchange and the ratio of e}ective conductivities between the ~uid and solid phases[ It is shown that the heat transfer characteristics can be classi_ed within three regimes\ each of which is dominated by one of three distinctive heat transfer mechanisms\ i[e[\ ~uid conduction\ solid conduction and internal heat exchange between solid and ~uid phases[ Based on these results\ a complete electrical thermal network representative of transport through porous media is established[ In addition\ an analytical characterization and con! ceptual assessment of solid and ~uid temperature di}erentials is presented\ the validity of the one!equation model is investigated and a practical criterion is suggested for channels with di}erent cross sections[ Þ 0887 Elsevier Science Ltd[ All rights reserved[ Nomenclature a interfacial area per a unit volume of porous media ðmŁ Ac cross!section area of the channel ðmŁ Ai interfacial area between solid and ~uid phases ðmŁ Aw outer surface area of the porous medium ðmŁ Bi Biot number de_ned in equation "03# c constants de_ned in equation "39# cP speci_c heat of the ~uid ðJ kg KŁ c0\ c1 constants de_ned in equations "38# and "49# D hydraulic diameter of the channel ðmŁ E error in the Nusselt number de_ned by equation "52# Ea allowable error in the Nusselt number hi interstitial heat transfer coe.cient ðW m KŁ hw wall heat transfer coe.cient de_ned by equation "12# ðW m KŁ H characteristic length of the channel ðmŁ kf\e} e}ective thermal conductivity of the ~uid ðW m KŁ Corresponding author[ ks\e} e}ective thermal conductivity of the solid ðW m KŁ Nuw Nusselt number de_ned by equation "13# Nuwl Nusselt number calculated from one!equation model P perimeter of the channel ðmŁ q heat ~ux ðW mŁ qw heat ~ux at the wall ðW mŁ R thermal resistance ðm K WŁ T temperature ðKŁ u ~uid velocity ðm sŁ u1 nondimensional velocity de_ned in equation "6# V volume of the porous media _lled in the channel ðmŁ x longitudinal coordinate ðmŁ y transverse coordinate ðmŁ[ Greek symbols Du nondimensional temperature di}erence\ Du us−uf o porosity at the wall h nondimensional transverse coordinate de_ned in equation "6# g geometric constant de_ned by equation "7# D[!Y[ Lee\ K[ Vafai:Int[ J[ Heat Mass Transfer 31 "0888# 312Ð324 313 u nondimensional temperature de_ned in equation "6# k ratio of the e}ective ~uid conductivity to that of the solid\ kf\e}:ks\e} l parameter de_ned by equation "10# r ~uid density ðkg mŁ[ Subscripts f ~uid or conduction in ~uid phase i internal heat exchange o overall s solid or conduction in solid phase ðŁ average over the channel cross section\ ðfŁ ÐAc f dAc:Ac[