Combined Heat and Mass Transfer in a Horizontal Channel with an Open Cavity

Combined heat and mass transfer from a horizontal channel with an open cavity heated from below is numerically examined in this paper. The fluid considered is air (Pr = 0.7). The main focus of the study is on mass-transfer driven flows (│N│> 1). The governing parameters considered are the buoyancy ratio N, Lewis number Le, Reynolds number Re, and Grashoff number Gr. First, the flow, temperature, and concentration fields were examined to understand the effects of these governing parameters on the heat and mass transfer processes. Based on the scale analysis, correlations for the entire convection regime, from natural, mixed, to forced convection, were then proposed. Nomenclature c concentration, [kg/m] C dimensionless concentration of species, C=(ccc)/(ch-cc) cp specific heat, [J/kg⋅K] D mass diffusivity of species in fluid, [m/s] G gravitational acceteration, [m/s] h average heat transfer coefficient H height of channel, [m] k effective thermal conductivity of porous medium, [W/m⋅K] Nu average Nusselt number, hH/k p pressure, {Pa] Pe Peclet number, Pe = Re⋅Pr = uH/α Ra Rayleigh number, Ra = Gr⋅Pr Re Reynolds number, Re = uH/ν T temperature, [K] u horizontal velocity, [m/s] u0 uniform velocity at the channel inlet, [m/s] U dimensionless horizontal velocity, U = u/u0 v vertical velocity, [m/s] V dimensionless vertical velocity, V = v/u0 1 Graduate Research Assistant, Student Member 2 Associate Professor, Senior Member AIAA, ASME x, y Cartesian coordinate, [m] α thermal diffusivity of porous medium, k/(ρcp)f, [m/s] βc coefficient of volumetric expansion due to concentration change,(-1/ρ)(∂ρ/∂c)p βT coefficient of volumetric expansion due to temperature change,(-1/ρ)(∂ρ/∂T)p μ dynamic viscosity of fluid, [kg/m⋅s] ν kinematic viscosity of fluid, [m/s] θ dimensionless temperature, (T-Tm)/(Ti-To) ρ density of fluid, [kg/m] Ψ dimensionless stream function Introduction Combined heat and mass transfer is an important transport process in which the flow is driven by a combination of thermal and solute diffusion effects. The problems of this kind are encountered in a wide range of fields, for example, in geology, oceanography, meteorology, biology, and astrophysics. Gebhart and Pera were among the first to numerically study the double-diffusion natural convection from vertical surfaces and plumes. This was followed by Nilson and Baer who studied the double-diffusive counter-buoyant boundary-layer flows. Kamotani et al. conducted an experimental study on combined heat and mass transfer by natural convection in a configuration which is commonly used for crystal growth. Bejan presented a scale analysis for heat and mass transfer processes in a cavity subjected to horizontal temperature and concentration gradients. This was followed by Trevisan and Bejan who performed an analytical and numerical study on combined heat and mass transfer by natural convection in a vertical enclosure. Beghein et al. presented a numerical study on double-diffusive natural convection in a square cavity. Their correlation of the Nusselt number and Sherwood number were in good agreement with those derived by Trevisan and Bejan for a cavity subject to heat and mass fluxes along the vertical wall. 8th AIAA/ASME Joint Thermophysics and Heat Transfer Conference 24-26 June 2002, St. Louis, Missouri AIAA 2002-3086 Copyright © 2002 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Kamakura and Ozoe examined the transient flow structure resulting from a solution with an initial linear concentration gradient and differentially heated from vertical walls. Costa presented the results of heat and mass transfer through walls in a single cavity and a vertical stack of communicating cavities. His results have more practical significance than those obtained for an impermeable and adiabatic wall when dealing with building elements. For mass transfer alone, the time-dependent natural and forced convection in a two-dimensional open cavity was reported by Shin and Economou. The effect of cavity aspect ratio and Rayleigh number on the local and spatial-average Sherwood number was examined. There are several studies on mixed convection heat transfer that are related to the present work. These include the numerical study by Humphrey and Jacobs on heat transfer to flows passing over an open cavity. A similar study was reported by Moallemi and Jang for the Prandtl number effects on laminar mixed convection heat transfer from a lid-driven cavity heated from below. Mixed convection from a bottom heated open cavity in a horizontal channel was reported by Leong and Lai. The effects of Reynolds number, Grashof number and the cavity aspect ratio on the flow stability were investigated. It was found that the heat transfer rate from the cavity base approaches that of natural convection at a low Reynolds number and approaches that of forced convection at a high Reynolds number. In the transitional regime, the heat transfer rate is reduced and the flow becomes unstable. In addition, the heat transfer rate per unit cavity width has been found to depend on the aspect ratio. As is clear from the above literature review, most of the previous studies on combined heat and mass transfer have focused on natural or forced convection. Mixed convection, on the other hand although encountered frequently in applications, has received rather little attention. The present study extends the numerical work of Leong and Lai to include the effects of mass transfer. The numerical calculations are performed over a wide range of buoyancy ratio (-10 ≤ N ≤ 10), Lewis number (0.1 ≤ Le ≤ 10), Reynolds number (1 ≤ Re ≤ 10), and Grashof number (0 ≤ Gr ≤ 10). It is hoped that the wide range of the parameters considered will help to bring a deeper insight to the problem and to complement the existing literature. Formulation and Numerical Method The geometry considered is a two-dimensional channel with an open cavity located at its bottom wall (Fig. 1). The cavity base is heated at a constant temperature Th while the rest of the bottom wall is well insulated. The top wall of the channel is maintained at a constant temperature Tc (< Th). A chemical species which has a maximum concentration ch is released from the cavity base while it has a minimum concentration cc at the top wall of the channel. Fresh air is introduced to the channel at a uniform velocity u0, temperature Tc, and concentration cc. Assuming that the flow to be two-dimenional, steady and laminar, the governing equation for the problem are, , y v x u 0 = ∂ ∂ + ∂ ∂ (1) , y u x u x p y u v x u u     ∂ ∂ + ∂ ∂ ν + ∂ ∂ ρ − = ∂ ∂ + ∂ ∂ 2 2