Transient analysis of incompressible ~ow through a packed bed

The present investigation aims at numerically simulating the temporal energy transport for incompressible ~ows through a packed bed[ The governing equations are formulated according to the volume averaging method[ The generalized momentum equation which accounts for the inertial and viscous forces was used to carry out this research\ whereas the two!energy equation model was used for simulating the energy transport[ The implications of the non! Darcian terms and the thermal dispersion e}ects on the transient energy transport were explored[ Error maps were introduced for a wide range of ~ow conditions and bed con_gurations[ In addition\ the investigation also tackles the applicability of the local thermal equilibrium condition and the one!dimensional approach for studying the thermal response in packed bed[ Þ 0887 Elsevier Science Ltd[ All rights reserved[ Nomenclature asf speci_c surface area of the packed bed ðmŁ c speci_c heat at constant pressure ðJ kg KŁ dp sphere particle diameter ðmŁ dP:dx pressure gradient ðNmŁ Dh hydraulic diameter ðmŁ Da Darcy number ðK :HŁ F geometric function de_ned in equation "5# hsf solid!to!~uid heat transfer coe.cient ðW m KŁ H height of the packed bed ðmŁ J unit vector oriented along the pore velocity vector ðvp:=vp=Ł k thermal conductivity ðW m KŁ K permeability ðmŁ L length of the packed bed ðmŁ Nu Nusselt number Pr Prandtl number "nf:af# Rep particle Reynolds number "uDdp:nf# t time ðsŁ T intrinsic average temperature ðKŁ u velocity vector in the x!direction ðm sŁ uD Darcian velocity "−K :mf "dP:dx## Corresponding author[ Tel[ ] 990 503 181 5459 ^ fax ] 990 503 181 2052 ^ e!mail ] vafai[0Ýosu[edu v velocity vector ðm sŁ x\ y cartesian coordinates ðmŁ[ Greek symbols a thermal di}usivity ðm sŁ o porosity h dimensionless normal length scale "yo :K # m dynamic viscosity ðkg m sŁ n kinematic viscosity ðm sŁ u dimensionless ~uid phase temperature "Tw−Tf#:"Tw−Tm# V dimensionless solid phase temperature "Tw−Ts#: "Tw−Tm# r density ðk gmŁ t dimensionless time "u t:L# j dimensionless axial length scale "x:L#[ Subscripts e inlet f ~uid fe} e}ective property for ~uid in initial m mean s solid se} e}ective property for solid w wall asymptotic free stream[ A[ Amiri\ K[ Vafai:Int[ J[ Heat Mass Transfer 30 "0887# 3148Ð3168 3159 Superscripts f ~uid s solid reference quantity[ Symbol ðŁ volume!averaged quantity[