Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 586913, 17 pages

http://dx.doi.org/10.1155/2015/586913

## Comparative Study of Radiative Effects on Double Diffusive Convection in Nongray Air-CO_{2} Mixtures in Cooperating and Opposing Flow

^{1}Faculty of Hydrocarbon and Chemistry, M’Hamed Bougara University, 35000 Boumerdes, Algeria^{2}Institut Pprime, CNRS, ENSMA, University of Poitiers, Futuroscope, 86961 Chasseneuil, France

Received 10 September 2014; Revised 27 January 2015; Accepted 28 January 2015

Academic Editor: Pedro Jorge Martins Coelho

Copyright © 2015 Siham Laouar-Meftah et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This study analyses the effects of nongray gas radiation on double diffusive convection, in a square differentially heated cavity filled with air-CO_{2} mixtures, when the buoyancy forces (thermal and mass) are cooperating or opposing. The radiative source term in the energy equation is evaluated by the discrete ordinate method (solving the radiative transfer equation) and the SLW spectral model (accounting for real radiative properties of absorbing species). Here, gas absorption varies with the local temperature and concentration of pollutant, which induces a strong direct coupling between the concentration and thermal fields that would not exist with gray gas. Simulations are performed at different concentrations of CO_{2} corresponding to different flow regimes (thermal, transitional, and mass). Results show the following: (i) in cooperating flow, radiation modifies essentially the heat transfer and the characteristics of temperature and concentration fields; (ii) in opposing flow, radiation effects are more important and depend on the nature of the flow regime.

#### 1. Introduction

Radiation heat transfer occurs in many engineering applications (cooling electronic components, nuclear reactors, industrial furnaces, combustion chambers, and so on) where it is coupled to other modes of heat transfer, like conduction and natural or forced convection. It can be substantial even at temperatures as low as 273 K [1, 2] and its influence on natural convection is more important than on forced convection [3] (because of direct coupling between thermal and dynamic fields in natural convection). Many investigations dealing with coupling natural convection and radiation in cavities [4–7] have been conducted with a transparent medium (only surface-to-surface radiation interaction, acting indirectly through heat flux boundary conditions at passive walls). However, many real engineering problems involve truly absorbing-emitting gases. In this case, volumetric radiation can significantly affect the temperature field which, in turn, induces changes in the fluid dynamic. Among works discussing the natural convection-radiation interaction in a confined semitransparent space, many use the simple assumption of a gray fictitious gas [8–11]. This approach is very unlikely to depict real situations, because the radiative transfer in a semitransparent medium (particularly in gases) depends on radiative properties of fluid that vary with the wavelength, the temperature, and the concentration (or partial pressure) of radiating species. It is noted that the absorption spectra of gases have very strong dynamics, consisting of hundreds of thousands of lines, with variables intensity and multiple quasitransparent bands between them. For these reasons, no (real) gas can be properly represented by a gray model, wherein an average value of the absorption coefficient of the spectrum is considered.

Among the first works taking into account nongray radiative properties of the fluid, we cite the analytical and experimental study of Bratis and Novotny [12] in a differentially heated rectangular enclosure filled with a NH_{3}-N_{2} gas mixture. The same problem was treated by Fusegi and Farouk [13] in 2D and Fusegi et al. [14] in a 3D thermal driven cavity filled with CO_{2}. The authors used the simplest nongray model, named weighted sum of gray gas model (WSGG), to account for the spectral aspect of the radiative transfer. Colomer et al. [15] also studied this coupled transfer phenomenon (in CO_{2}, H_{2}O, and CO_{2}-H_{2}O mixtures) using the SLW spectral model (spectral line weighted sum of gray gases), which is considered as a refinement to the WSGG model. They concluded that the use of any of the nongray models is justified since neither the gray gas nor the transparent model captures well the real gas behavior. Recently, Soucasse et al. [16] and Ibrahim et al. [17] considered the same coupled phenomena in laminar regime for a 3D cavity filled with an air/CO_{2}/H_{2}O mixture [16] and in turbulent regime for a 2D cavity filled with an air-H_{2}O mixture [17]. The spectral dependency of gas radiative properties is handled by the global ADF model [16] and the SLW model [17] (the two models are similar in their principle). Regarding the coupling of radiation with the double diffusive natural convection, most of the available investigations use the simple assumption of fictitious gray medium (uniform absorption over space and wavelengths) [18–21]. In these works, the fluid was generally regarded as optically thick and the radiative fluxes were calculated by using the Rosseland approximation. Rafieivand [22], Mezrhab et al. [23], and Moufekkir et al. [24] have investigated the same coupling phenomena in a gas mixture. They considered a more realistic situation of an absorption coefficient of fluid proportional to the local concentration of the absorbing species. These studies are still limited to the gray gas assumption.

Recently, some attempts have been made to study double diffusive convection coupled to radiation in participating gases, accounting for the real (nongray) radiative properties of the mixture (absorption varies with temperature, concentration, and wavelength). In this context, we can mention the numerical studies performed by Meftah et al. [25] and Laouar-Meftah et al. [26] in a stationary laminar flow of air-CO_{2} (or air-H_{2}O) gas mixtures and Ibrahim and Lemonnier [27] in transient laminar flow of N_{2}-CO_{2} mixture. The authors used the SLW spectral model of Denison and Webb [28] along with the discrete ordinate method to account for the real radiative participation of the medium. Here, as gas absorption varies with the local temperature and concentration of pollutant, a strong direct coupling between the concentration and thermal fields occurs, which does not exist with gray gas. There is a direct influence on the thermal field (through a volumetric heat source in energy equation) and an indirect influence on the dynamic field (by modifying buoyancy forces) and the field of concentration (through the dynamic field). In turn, these fields ( and ) influence the radiative transfer through absorbing properties of radiating species, characterized by the absorption coefficient which depends, in particular, on the local concentration of pollutant. Therefore, the objective of our investigation is to highlight the effect of nongray gas radiation on natural double diffusive convection (flow structures, heat and mass transfer, and so on) in the three convective flow regimes: thermal, intermediate, and mass dominated, when buoyancy forces are cooperating or opposing. We also note that this work is an extension and a further exploration of our previous studies, where we have considered an opposing flow in air-H_{2}O mixture only [26] or a cooperating flow in gas mixtures (air-CO_{2} or air-H_{2}O) at two average concentrations of pollutant only (10% and 25%) [25].

#### 2. Analysis and Modeling

##### 2.1. Physical Model

The studied physical system is represented in Figure 1. It consists of a square cavity of width* L*, filled with air-CO_{2} at different average concentration in CO_{2}. The vertical walls of the cavity are black and maintained at constant temperatures () and concentrations (). These conditions are disposed so as to create an opposing or cooperating fluid flow. The horizontal walls are adiabatic, impermeable, and completely reflecting. The mixture density at the cold and hot walls, in the most severe conditions investigated here (20% CO_{2}), is, respectively, 0.80 kg/m^{3} and 0.61 kg/m^{3} in cooperating flow and 0.66 kg/m^{3} and 0.73 kg/m^{3} in opposing flow. So, from these values, we can note that the maximum density variations within the fluid are of order of 14% (cooperating flow-cold wall) if related to the reference value (0.70 kg/m^{3}) calculated at the average temperature () and concentration (). Therefore, we have considered that the Boussinesq approximation remains valid within the frame of our study.