Film Condensation Process Controlled by a Darcy-Cooling Fluid Flow

Abstract

The film-condensation process of a saturated vapor in contact with the external surfaces of a porous channel, caused by a Darcy cooling fluid flow, is studied. The longitudinal heat conduction effects in the walls of the channel are included. The momentum and energy balance equations are reduced to a system of integro‐differential equations with five nondimensional parameters: the Prandtl number of the condensed fluid Prc, the Jacob number Ja ,a nondimensional plate thermal conductivity α, and the aspect ratio of the walls e and β defined by the ratio of the thermal resistance of the condensed layer to the thermal resistance of the Darcy cooling flow. The resulting governing equations are integrated in the asymptotic limit Ja → 0t oobtain the spatial evolution of the condensedlayer thickness and temperature of the walls as a function of the longitudinal coordinate. For practical values of the parameters α and β, the present analysis shows that both effects modify the well-known Nusselt solution for an isothermal wall. Nomenclature c = specific heat of the cooling fluid cc = specific heat of the condensed phase fc = nondimensional stream function introduced in Eq. (10) g = acceleration of gravity h = thickness of the plate h fg = latent heat of condensation Ja = Jacob number defined in Eq. (2) L = length of the wall m � = mass flow rate of condensed fluid Nu c = Nusselt number defined in Eq. (26) Pr = Prandtl number of the cooling flow Prc = Prandtl number of the condensed fluid Re =R eynolds number of the cooling flow Rec =R eynolds number of the condensed fluid T = temperature Ts = temperature of the saturated vapor T∞ = freestream temperature of the cooling fluid flow U = uniform Darcy velocity of the cooling fluid flow u, v = nondimensional longitudinal and transverse velocities uc = characteristic longitudinal velocity of the condensed fluid u, v = longitudinal and transverse velocities in physical units x, y = Cartesian coordinates α = heat conduction parameter defined in Eqs. (7) β = nondimensional parameter defined in Eqs. (7) � = normalized thickness of the condensed layer δ = thermal thickness of the cooling flow δc = thickness of the condensed layer δcL = thickness of the condensed layer at χ = 1 e = aspect ratio of the plate ζ = nondimensional inner coordinate defined in Eq. (9) ηc = nondimensional transversal coordinate for the condensed fluid flow