# Comparative Study of Radiative Effects on Double Diffusive Convection in Nongray Air-CO2 Mixtures in Cooperating and Opposing Flow

Authors: Siham Laouar-Meftah; Denis Lemonnier; Didier Saury; Abderrahmane Benbrik; Mohamed Cherifi
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This study analyses the effects of nongray gas radiation on double diffusive convection, in a square differentially heated cavity filled with air-CO2 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 CO2 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.


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## Body

## 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 NH3-N2 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 CO2. 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 CO2, H2O, and CO2-H2O 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/CO2/H2O mixture [16] and in turbulent regime for a 2D cavity filled with an air-H2O 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-CO2 (or air-H2O) gas mixtures and Ibrahim and Lemonnier [27] in transient laminar flow of N2-CO2 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 (T and C) influence the radiative transfer through absorbing properties of radiating species, characterized by the absorption coefficient κ 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-H2O mixture only [26] or a cooperating flow in gas mixtures (air-CO2 or air-H2O) 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 Figure1. It consists of a square cavity of widthL, filled with air-CO2 at different average concentration in CO2. The vertical walls of the cavity are black and maintained at constant temperatures (TH>TC) and concentrations (CL<CH). 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% CO2), is, respectively, 0.80 kg/m3 and 0.61 kg/m3 in cooperating flow and 0.66 kg/m3 and 0.73 kg/m3 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/m3) calculated at the average temperature (T0=(TH+TC)/2) and concentration (C0=(CH+CL)/2). Therefore, we have considered that the Boussinesq approximation remains valid within the frame of our study.Figure 1
Physical model: (a) cooperating flow, (b) opposing flow.
(a)
(b)


### 2.2. Governing Equations

The flow is assumed to be two-dimensional, stationary, and laminar. The fluid in the cavity is a mixture of pure air (considered as perfectly transparent) and CO2 (an absorbing, emitting, and nonscattering species) acting as a pollutant. It is assumed Newtonian, incompressible, with constant thermophysical properties—except in the buoyancy term of the momentum equation—and satisfying the Boussinesq approximation. Under these assumptions, the fluid motion is described by the following set of equations, expressed in the vorticity (ω)-stream function (ψ) form:(1)∂ω∂t+u∂ω∂x+v∂ω∂y=V∂2ω∂x2+∂2ω∂y2+gβT∂T∂x+βC∂C∂x,(2)∂T∂t+u∂T∂x+v∂T∂y=α∂2T∂x2+∂2T∂y2+SR,(3)∂C∂t+u∂C∂x+v∂C∂y=D∂2C∂x2+∂2C∂y2,(4)-ω=∂2ψ∂x2+∂2ψ∂y2,(5)u=∂ψ∂y,v=-∂ψ∂x,where βT and βC are the thermal and mass expansion coefficients, respectively, defined by(6)βT=-1ρ∂ρ∂Tp,c=1T0,βC=-1ρ∂ρ∂Cp,T=Mair-MCO2ρ0.Mair and MCO2 are the molecular masses of air and CO2.In the energy equation (2), SR stands for the radiation source field. It is calculated as explained below.


### 2.3. Radiation Model

The general equation for radiative transfer in an absorbing emitting medium is, in the(x,y) Cartesian coordinate frame,(7)μ∂I(x,y,s)∂x+η∂Ix,y,s∂y+κx,yIx,y,s=κx,yIbx,y.I is the spectral radiation intensity in the direction s, κ represents the local spectral absorption coefficient, and Ib stands for the spectral blackbody intensity at local temperature. The absorption coefficient is directly linked to the concentration of pollutant since(8)κx,y=Cabs×Cx,y,where Cabs is the spectral absorption cross-section of the considered species, evaluated at the local thermodynamic conditions (pressure, temperature).In this paper, the discrete ordinate method (DOM), together with the SLW model of Denison and Webb [28], is used to solve the radiative transfer equation (RTE) and get the radiative fluxes and sources within the medium. The DOM is based on the selection of Nd discrete directions sm to account for the directional dependence of the radiation intensity. The SLW model aims at representing the spectral radiative properties of the medium in a global yet accurate manner. It consists in replacing the real gas by a sum of Ng gray gases, each of them being associated with a given value of the absorption cross-section, Cabsk, and a blackbody factor ak which transforms the gas emission term into Ib,k=akσT4/π. Index k varies from 0 to Ng with, conventionally, Cabs0=0. Then the RTE is solved, for each gray gas k and each discrete direction m, as(9)μm∂Ik,mx,y∂x+ηm∂Ik,mx,y∂y+κkx,yIk,mx,y=κkx,yakσT4x,yπ,where Ik,m(x,y) is a shorthand for Ikx,y,sm and(10)κkx,y=Cabsk×Cx,y.The values of ak and Cabsk vary in the medium with the local temperature and concentration (we assume atmospheric pressure everywhere). The ak’s are deduced from the closed-form functions of the reordered wave number developed by Denison and Webb [28, 29].When using both the DOM and the SLW model, the incident fluxes at the walls are given by(11a)qx,kinc0,y=∑μm<0μmwmIk,m0,y,qx,kincL,y=∑μm>0μmwmIk,mL,y,(11b)qy,kincx,0=∑ηm<0ηmwmIk,mx,0,qy,kincx,L=∑ηm>0ηmwmIk,mx,L,where wm is the weighting coefficient of direction m in the quadrature set. Equations (11a) apply to the vertical (active) walls while (11b) are for the horizontal (passive) ones. More generally, the x- and y-components of the total radiative flux at any point in the medium are (12)qR,xx,y=∑k=1Ng∑m=1NdμmwmIk,m(x,y),qR,yx,y=∑k=1Ng∑m=1NdηmwmIk,mx,y,and the radiation source field (to be input in the energy equation) is obtained as(13)SRx,y=∑k=1Ngκkx,y∑m=1NdwmIk,mx,y-4akσT4x,y.


### 2.4. Initial and Boundary Conditions

Initially, the fluid is motionless, isothermal, and homogeneous at average temperatureT0 and average concentration C0. The thermal, mass, and dynamic boundary conditions of the problem are summarized in Table 1. It should be noted that, to create a cooperating double diffusive flow (i.e., thermal and mass buoyancy forces augment each other), we impose at the hot wall the lowest concentration (CL) of CO2 (pollutant heavier than air) and the highest concentration (CH) at the cold wall. For an opposing double diffusive flow, we must reverse (compared to cooperating case) these mass boundary conditions (CH at hot wall and CL at cold wall).Table 1
Boundary conditions of the problem.
Wall
Temperature
Concentration
Stream function
Vorticity
Cooperating case
Opposing case
1  (x=0)
T
H
C
L
C
H
ψ
=
0
ω
=
-
∂
2
ψ
/
∂
x
2
2  (x=L)
T
C
C
H
C
L
ψ
=
0
ω
=
-
∂
2
ψ
/
∂
x
2
3  (y=0)
q
t
=
0
∂
C
/
∂
y
=
0
ψ
=
0
ω
=
-
∂
2
ψ
/
∂
y
2
4  (y=L)
q
t
=
0
∂
C
/
∂
y
=
0
ψ
=
0
ω
=
-
∂
2
ψ
/
∂
y
2Regarding the radiative part of the problem, intensities leaving the vertical (black) walls correspond to the blackbody emission at the wall temperature:(14a)Ik,m0,y=akσTH4πforμm>0,Ik,mL,y=akσTC4πforμm<0,while the horizontal walls are assumed to be perfectly diffusively reflecting:(14b)Ik,mx,0=qy,kincx,0πforηm>0,Ik,mx,L=qy,kinc(x,L)πforηm<0.


### 2.5. Heat and Mass Transfer

The average Nusselt numbers (convective, radiative, and total) and Sherwood numbers at vertical walls are defined as follows:(15)Nu¯c=1TH-TC∫0L∂T∂xx=0orLdy,Nu-R=1λTH-TC∫0LqRnetx=0orLdy,where qRnet represents the net radiative flux (emitted + absorbed) at vertical walls, given by(16)qRnetx=0orL=εpσTw4-∑k=1Ngqx,kincx=0orL,Nu-t=Nu-c+Nu-R,Sh-=1CH-CL∫0L∂C∂xx=0orLdy.


## 2.1. Physical Model

The studied physical system is represented in Figure1. It consists of a square cavity of widthL, filled with air-CO2 at different average concentration in CO2. The vertical walls of the cavity are black and maintained at constant temperatures (TH>TC) and concentrations (CL<CH). 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% CO2), is, respectively, 0.80 kg/m3 and 0.61 kg/m3 in cooperating flow and 0.66 kg/m3 and 0.73 kg/m3 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/m3) calculated at the average temperature (T0=(TH+TC)/2) and concentration (C0=(CH+CL)/2). Therefore, we have considered that the Boussinesq approximation remains valid within the frame of our study.Figure 1
Physical model: (a) cooperating flow, (b) opposing flow.
(a)
(b)


## 2.2. Governing Equations

The flow is assumed to be two-dimensional, stationary, and laminar. The fluid in the cavity is a mixture of pure air (considered as perfectly transparent) and CO2 (an absorbing, emitting, and nonscattering species) acting as a pollutant. It is assumed Newtonian, incompressible, with constant thermophysical properties—except in the buoyancy term of the momentum equation—and satisfying the Boussinesq approximation. Under these assumptions, the fluid motion is described by the following set of equations, expressed in the vorticity (ω)-stream function (ψ) form:(1)∂ω∂t+u∂ω∂x+v∂ω∂y=V∂2ω∂x2+∂2ω∂y2+gβT∂T∂x+βC∂C∂x,(2)∂T∂t+u∂T∂x+v∂T∂y=α∂2T∂x2+∂2T∂y2+SR,(3)∂C∂t+u∂C∂x+v∂C∂y=D∂2C∂x2+∂2C∂y2,(4)-ω=∂2ψ∂x2+∂2ψ∂y2,(5)u=∂ψ∂y,v=-∂ψ∂x,where βT and βC are the thermal and mass expansion coefficients, respectively, defined by(6)βT=-1ρ∂ρ∂Tp,c=1T0,βC=-1ρ∂ρ∂Cp,T=Mair-MCO2ρ0.Mair and MCO2 are the molecular masses of air and CO2.In the energy equation (2), SR stands for the radiation source field. It is calculated as explained below.


## 2.3. Radiation Model

The general equation for radiative transfer in an absorbing emitting medium is, in the(x,y) Cartesian coordinate frame,(7)μ∂I(x,y,s)∂x+η∂Ix,y,s∂y+κx,yIx,y,s=κx,yIbx,y.I is the spectral radiation intensity in the direction s, κ represents the local spectral absorption coefficient, and Ib stands for the spectral blackbody intensity at local temperature. The absorption coefficient is directly linked to the concentration of pollutant since(8)κx,y=Cabs×Cx,y,where Cabs is the spectral absorption cross-section of the considered species, evaluated at the local thermodynamic conditions (pressure, temperature).In this paper, the discrete ordinate method (DOM), together with the SLW model of Denison and Webb [28], is used to solve the radiative transfer equation (RTE) and get the radiative fluxes and sources within the medium. The DOM is based on the selection of Nd discrete directions sm to account for the directional dependence of the radiation intensity. The SLW model aims at representing the spectral radiative properties of the medium in a global yet accurate manner. It consists in replacing the real gas by a sum of Ng gray gases, each of them being associated with a given value of the absorption cross-section, Cabsk, and a blackbody factor ak which transforms the gas emission term into Ib,k=akσT4/π. Index k varies from 0 to Ng with, conventionally, Cabs0=0. Then the RTE is solved, for each gray gas k and each discrete direction m, as(9)μm∂Ik,mx,y∂x+ηm∂Ik,mx,y∂y+κkx,yIk,mx,y=κkx,yakσT4x,yπ,where Ik,m(x,y) is a shorthand for Ikx,y,sm and(10)κkx,y=Cabsk×Cx,y.The values of ak and Cabsk vary in the medium with the local temperature and concentration (we assume atmospheric pressure everywhere). The ak’s are deduced from the closed-form functions of the reordered wave number developed by Denison and Webb [28, 29].When using both the DOM and the SLW model, the incident fluxes at the walls are given by(11a)qx,kinc0,y=∑μm<0μmwmIk,m0,y,qx,kincL,y=∑μm>0μmwmIk,mL,y,(11b)qy,kincx,0=∑ηm<0ηmwmIk,mx,0,qy,kincx,L=∑ηm>0ηmwmIk,mx,L,where wm is the weighting coefficient of direction m in the quadrature set. Equations (11a) apply to the vertical (active) walls while (11b) are for the horizontal (passive) ones. More generally, the x- and y-components of the total radiative flux at any point in the medium are (12)qR,xx,y=∑k=1Ng∑m=1NdμmwmIk,m(x,y),qR,yx,y=∑k=1Ng∑m=1NdηmwmIk,mx,y,and the radiation source field (to be input in the energy equation) is obtained as(13)SRx,y=∑k=1Ngκkx,y∑m=1NdwmIk,mx,y-4akσT4x,y.


## 2.4. Initial and Boundary Conditions

Initially, the fluid is motionless, isothermal, and homogeneous at average temperatureT0 and average concentration C0. The thermal, mass, and dynamic boundary conditions of the problem are summarized in Table 1. It should be noted that, to create a cooperating double diffusive flow (i.e., thermal and mass buoyancy forces augment each other), we impose at the hot wall the lowest concentration (CL) of CO2 (pollutant heavier than air) and the highest concentration (CH) at the cold wall. For an opposing double diffusive flow, we must reverse (compared to cooperating case) these mass boundary conditions (CH at hot wall and CL at cold wall).Table 1
Boundary conditions of the problem.
Wall
Temperature
Concentration
Stream function
Vorticity
Cooperating case
Opposing case
1  (x=0)
T
H
C
L
C
H
ψ
=
0
ω
=
-
∂
2
ψ
/
∂
x
2
2  (x=L)
T
C
C
H
C
L
ψ
=
0
ω
=
-
∂
2
ψ
/
∂
x
2
3  (y=0)
q
t
=
0
∂
C
/
∂
y
=
0
ψ
=
0
ω
=
-
∂
2
ψ
/
∂
y
2
4  (y=L)
q
t
=
0
∂
C
/
∂
y
=
0
ψ
=
0
ω
=
-
∂
2
ψ
/
∂
y
2Regarding the radiative part of the problem, intensities leaving the vertical (black) walls correspond to the blackbody emission at the wall temperature:(14a)Ik,m0,y=akσTH4πforμm>0,Ik,mL,y=akσTC4πforμm<0,while the horizontal walls are assumed to be perfectly diffusively reflecting:(14b)Ik,mx,0=qy,kincx,0πforηm>0,Ik,mx,L=qy,kinc(x,L)πforηm<0.


## 2.5. Heat and Mass Transfer

The average Nusselt numbers (convective, radiative, and total) and Sherwood numbers at vertical walls are defined as follows:(15)Nu¯c=1TH-TC∫0L∂T∂xx=0orLdy,Nu-R=1λTH-TC∫0LqRnetx=0orLdy,where qRnet represents the net radiative flux (emitted + absorbed) at vertical walls, given by(16)qRnetx=0orL=εpσTw4-∑k=1Ngqx,kincx=0orL,Nu-t=Nu-c+Nu-R,Sh-=1CH-CL∫0L∂C∂xx=0orLdy.


## 3. Numerical Procedure

The thermophysical properties of the mixture are calculated atT0 and C0 (reference state) according to the ideal gas laws (for molecular mass and density), Wilke and Wassiljewa formulas (for dynamic viscosity and thermal conductivity), and the empirical relation of Fuller et al. [30] (for mass diffusivities of water vapour into air). Equations (1)–(3) are solved by a finite difference method based on an ADI scheme. The first and second space derivatives are approximated by central differences and the time derivatives by a first-order forward difference. The Poisson equation (4) is solved by the successive over relaxation method. A nonuniform grid (80×80) with hyperbolic tangent distribution in x- and y-direction is used herein:(17)xi=0.51+thδi/80-0.5thδ/2,yi=0.51+thδj/80-0.5thδ/2,where δ is a stretching parameter, set to 4 in this study.All the calculations are performed using a false transient method. A steady state solution is assumed to be reached when the two following criteria are met:(18)fi,jn+1-fi,jnfi,jn+1≤10-4,(19)Nu-t,Sh-x=0-Nu-t,Sh-x=LNu-t,Sh-x=0+Nu-t,Sh-x=L/2≤10-4,where f stands for T,C and ψ; n is the iteration number. The second constraint equation (19) ensures that, at steady state, the total heat and mass flux input to the cavity through one vertical wall must be equal to those leaving the other.


### 3.1. Validation of the Computer Code

The accuracy of the numerical results was checked through several tests based on the effects of grid size (Table2) and number of gray gases Ng (Table 3) required to characterize the radiative properties of the fluid (SLW model). For this, we consider a square cavity filled with a mixture at 10% CO2 and submitted to the same boundary conditions as shown in Figure 1(a). It can be seen that an 80×80 grid and Ng=11 gray gases for the SLW model (one of them being transparent) are sufficient to ensure a relative accuracy of order 0.1%. This conclusion was found to remain valid with other concentrations of CO2.Table 2
Grid size effect on the average Nusselt (convective and total) and Sherwood numbers.
Grid size
Nu
-
c
Nu
-
t
Sh
-
4
0
×
40
15,819
198,486
21,431
80
×
80
15,970
198,564
21,444
160
×
160
15,968
198,562
21,445Table 3
Effect of the number of gray gases on the average Nusselt (convective and total) and Sherwood numbers.
Number of gray gases
Nu
-
c
Nu
-
t
Sh
-
Hot wall
Cold wall
5
15,71
18,95
198,02
21,49
11
15,97
18,67
198,56
21,45
17
15,98
18,66
198,59
21,45The predictions of our model were also successfully validated against previously published results in different situations. First, in double diffusive convection, the average Nusselt and Sherwood numbers (Table4) agree within a maximum relative error of 0,44% with that obtained by Béghein et al. [31]. Dealing with convection in gray fluid (Table 5), we note a maximum discrepancy of 0,3% on the radiative Nusselt number and of 0,54% on the total Nusselt number when comparing to [8] (the maximum error is reached when the medium optical thickness is about 5). In the case of nongray gas radiation, the average total Nusselt number obtained compared well with Fusegi’s data (laminar flow, L=7,62 cm, relative error 0,85%) [13]. Finally, The accuracy of the SLW spectral model (Figure 2) was more particularly assessed by comparison to [28, 29]. For this test we considered a homogenous and nonisothermal fluid, with a cosine temperature distribution Tx=T0+0.5TH-TCcos⁡⁡(πx/L), placed between infinite parallel black walls (εp=1) maintained at TH and TC.Table 4
Average convective Nusselt and Sherwood numbers: Ra =107; Pr = 0,71; Le = 1.
N
−0,01
−0,2
−0,5
−0,8
−1,5
−5
Nu
-
C (Sh-)
Present work(80×80)
16,40−16,40
15,49−15,49
13,65−13,65
10,64−10,64
13,66−13,66
23,77−23,77
Béghein et al. [31](45×45)
16,4−16,4
15,5−15,5
13,6−13,6
10,6−10,6
13,6−13,6
23,7−23,7Table 5
Average Nusselt (radiative and total) number for different opacity of a gray gas: Ra =5×106; Pr = 0,71;   Pl = 0,02; θ0 = 1,5.
τ
S
N
Present work  (80×80)
Yucel et al. [8] (50×50)
Nu
-
r
Nu
-
t
Nu
-
r
Nu
-
t
1
S
8
31,25
38,81
31,28
38,93
5
S
4
23,57
31,59
23,64
31,76
0,2
S
4
37,40
46,05
37,40
46,11Figure 2
Radiative source and net cold wall: (a) fluid at 30% CO2; TH-TC=1000 K; T0= 1250 K; (b) fluid at 10% H2O; TH-TC=500 K; T0=1000 K.
(a)
(b)


## 3.1. Validation of the Computer Code

The accuracy of the numerical results was checked through several tests based on the effects of grid size (Table2) and number of gray gases Ng (Table 3) required to characterize the radiative properties of the fluid (SLW model). For this, we consider a square cavity filled with a mixture at 10% CO2 and submitted to the same boundary conditions as shown in Figure 1(a). It can be seen that an 80×80 grid and Ng=11 gray gases for the SLW model (one of them being transparent) are sufficient to ensure a relative accuracy of order 0.1%. This conclusion was found to remain valid with other concentrations of CO2.Table 2
Grid size effect on the average Nusselt (convective and total) and Sherwood numbers.
Grid size
Nu
-
c
Nu
-
t
Sh
-
4
0
×
40
15,819
198,486
21,431
80
×
80
15,970
198,564
21,444
160
×
160
15,968
198,562
21,445Table 3
Effect of the number of gray gases on the average Nusselt (convective and total) and Sherwood numbers.
Number of gray gases
Nu
-
c
Nu
-
t
Sh
-
Hot wall
Cold wall
5
15,71
18,95
198,02
21,49
11
15,97
18,67
198,56
21,45
17
15,98
18,66
198,59
21,45The predictions of our model were also successfully validated against previously published results in different situations. First, in double diffusive convection, the average Nusselt and Sherwood numbers (Table4) agree within a maximum relative error of 0,44% with that obtained by Béghein et al. [31]. Dealing with convection in gray fluid (Table 5), we note a maximum discrepancy of 0,3% on the radiative Nusselt number and of 0,54% on the total Nusselt number when comparing to [8] (the maximum error is reached when the medium optical thickness is about 5). In the case of nongray gas radiation, the average total Nusselt number obtained compared well with Fusegi’s data (laminar flow, L=7,62 cm, relative error 0,85%) [13]. Finally, The accuracy of the SLW spectral model (Figure 2) was more particularly assessed by comparison to [28, 29]. For this test we considered a homogenous and nonisothermal fluid, with a cosine temperature distribution Tx=T0+0.5TH-TCcos⁡⁡(πx/L), placed between infinite parallel black walls (εp=1) maintained at TH and TC.Table 4
Average convective Nusselt and Sherwood numbers: Ra =107; Pr = 0,71; Le = 1.
N
−0,01
−0,2
−0,5
−0,8
−1,5
−5
Nu
-
C (Sh-)
Present work(80×80)
16,40−16,40
15,49−15,49
13,65−13,65
10,64−10,64
13,66−13,66
23,77−23,77
Béghein et al. [31](45×45)
16,4−16,4
15,5−15,5
13,6−13,6
10,6−10,6
13,6−13,6
23,7−23,7Table 5
Average Nusselt (radiative and total) number for different opacity of a gray gas: Ra =5×106; Pr = 0,71;   Pl = 0,02; θ0 = 1,5.
τ
S
N
Present work  (80×80)
Yucel et al. [8] (50×50)
Nu
-
r
Nu
-
t
Nu
-
r
Nu
-
t
1
S
8
31,25
38,81
31,28
38,93
5
S
4
23,57
31,59
23,64
31,76
0,2
S
4
37,40
46,05
37,40
46,11Figure 2
Radiative source and net cold wall: (a) fluid at 30% CO2; TH-TC=1000 K; T0= 1250 K; (b) fluid at 10% H2O; TH-TC=500 K; T0=1000 K.
(a)
(b)


## 4. Results and Discussion

### 4.1. Conditions of Simulations

The results presented in this investigation were obtained by considering the following parameters:L = 0.3 m, TH = 580 K, and TC = 530 K. In cooperating flow (i.e., where thermal and mass buoyancy forces augment each other), a null concentration of CO2 is prescribed at the hot wall (CL=0 mol/m3 or xL=0%) and, at the cold wall, the concentration is set to CH = 2.20, 4.39, and 8.78 mol/m3. These correspond, respectively, in reference conditions, to molar fractions xCO2=10%, 20%, and 40%. Now, to create an opposing double diffusive flow (i.e., thermal and mass buoyancy forces are opposite), the mass boundary conditions must be reversed: CL at the cold wall and CH at the hot wall.In the SLW model, the reference state values of temperature and molar fraction wereT0=555 K and the average molar fractions xCO2=5%, 10%, and 20%. Based on these data, the mass-to-thermal buoyancy ratio (N), thermal Rayleigh number (Ra), Prandtl (Pr), Planck (Pl), and Lewis number (Le) relating to the problem are summarized in Table 6. Values of |N| below unity (0.57) characterize a flow mainly driven by thermal forces (thermal regime), whereas values above unity (2.10) refer to a flow dominated by concentration gradients (mass regime). In between (|N|=1.10) mass and thermal effects are of comparable magnitude (intermediate regime).Table 6
Dimensionless numbers corresponding to cooperating and opposing flow.
Pollutant
x (%)
N
Ra
Pr
Pl
Le
CO2
5
±0,57
8,85 × 106
0,72
3,58 × 10−3
1,30
10
±1,10
9,44 × 106
0,72
3,55 × 10−3
1,26
20
±2,10
1,09 × 107
0,72
3,50 × 10−3
1,18It is worth noting that the effects of gas radiation on field characteristics were determined by comparison to the nonparticipating gas case (i.e., a fluid having the same thermophysical properties as the considered mixtures, but totally transparent).


### 4.2. Flow Patterns

#### 4.2.1. Cooperating Flow

Figures3 and 4 display the streamline, temperature, and concentration contours for different average molar fractions of CO2 in the cavity (xCO2=5%,10%, and 20%), when the gas is transparent or participating to radiation. On the whole, we can see (Figure 4(a)) that, when the buoyancy forces (mass and thermal) are cooperating, gas radiation weakly alters the flow pattern (slight slope and distortion of the streamlines, small secondary cells in the core of cavity). These modifications are less important at 20% CO2 than at 5% (and 10%), because a higher rate of absorbing species strengthens the mass forces and increases buoyancy ratio (N) from 0,57 to 2,10 (see Table 6). This makes the flow mainly driven by concentration (not temperature) gradients and hence less sensitive to radiation. Under gas radiation effects, the vertical boundary layers are broadened and partially accelerated (Figures 5(c) and 5(d)), whereas the core of the cavity is unaffected and remains at rest. This induces an increase of horizontal jets flowing from one active wall to the other along the adiabatic surfaces (Figures 5(a) and 5(b)). The strengthening of the vertical boundary layers, associated with fixed concentrations at vertical walls, leads to an excess of pollutant (CO2) near the left wall (at CH) and a deficit near the right one (at CL) (Figure 6(a)). To equilibrate this trend, some part of CO2 is diffused from left to right, resulting in negative gradient of concentration in the x-direction. This breaks the centrosymmetry of the field and replaces the vertical stratification by a tilted iso-concentration pattern (Figure 4(b)) [25].Figure 3
Dimensionless streamline, iso-concentration, and isotherm patterns for air-CO2 mixture at 5%, 10%, and 20% CO2, in cooperating flow. (a) ψ*=ψgβTTH-TCL3, Δψ*=0.00065; (b) C*=C-C0/CH-CL, ΔC*=0.0625; (c) T*=T-T0/TH-TC, ΔT*=0.0625.
(a)
Dimensionless streamlineψ*
(b)
Dimensionless iso-concentrationC*
(c)
Dimensionless isothermT*Figure 4
Dimensionless streamline, iso-concentration, and isotherm patterns for air-CO2 mixture at 5%, 10%, and 20% CO2, in cooperating flow. (a) ψ*=ψgβTTH-TCL3, Δψ*=0.00065; (b) C*=C-C0/CH-CL, ΔC*=0.0625; (c) T*=T-T0/TH-TC, ΔT*=0.0625.
(a)
Dimensionless streamlineψ*
(b)
Dimensionless iso-concentrationC*
(c)
Dimensionless isothermT*Figure 5
Dimensionless vertical and horizontal velocity at different levels in the cavity, for cooperating flow: air-CO2 mixtures at 5%, 10%, and 20% CO2. v*=vgβTTH-TCL-0.5, u*=ugβTTH-TCL-0.5. (a) Upper wall, (b) lower wall, (c) hot wall, and (d) cold wall.
(a)
(b)
(c)
(d)Figure 6
Dimensionless concentration (a) and temperature (b) profiles at different heights in the cavity, for a cooperating flow.C*=C-C0/CH-CL, T*=T-T0/TH-TC.
(a)
(b)Regarding the thermal field, the gas flowing under the upper wall emits radiation. It is slightly colder than it would be without radiative effects (Figure6(b), air-CO2 mixture, y/L=0.875), and this weakens the temperature gradient at vertical cold wall when the fluid flows downward. A symmetrical picture is observed along the bottom wall (Figure 6(b), air-CO2 mixture, y/L=0.125) where the fluid is preheated by absorbing radiation, leading to a reduction of the temperature gradient along the vertical hot wall. In the core of the cavity (outside boundary layers) (Figure 6(b), air-CO2, y/L=0.5), while the reference transparent gas remains practically isothermal, the gas is heated by absorption of radiation from the hot wall in the left part of the cavity and is cooled by emission toward the cold wall in the right part. As a result (in the center of the cavity) negative temperature gradients in the x-direction are formed and will create a thermal field with tilted stratification (Figure 4(c)). This trend is more pronounced at 20% CO2 than at 10% and 5%.


#### 4.2.2. Opposing Case

For an average concentration of 5% of CO2, the fluid flow in the cavity is mainly driven by thermal forces if gas absorption is neglected (Figure 7(a), 5% CO2). When taken into account (Figure 8(a), 5% CO2), radiation accelerates boundary layers (Figure 9(b), 5% CO2) by enhancing the thermal forces near the horizontals walls (as in cooperating case). The stagnant core of the cavity (for transparent gas) is also set in motion (thermal forces are slightly weakened). Such behavior tends to homogenize the concentration and temperature fields, whose stratifications are destroyed (Figure 8(b), 5% CO2). In this case (at 5% CO2), the magnitude of radiation effects on the structure of the different fields seems to be greater in opposing flow than in cooperating case, indicating that a lower flow dynamics when the thermal and mass buoyancy forces are opposing (equivalent Rayleigh number: N+1Ra lower) promotes the influence of radiation.Figure 7
Dimensionless streamline, iso-concentration, and isotherm patterns for air-CO2 mixture at 5%, 10%, and 20% CO2, in opposing flow. (a) ψ*=ψgβTTH-TCL3, Δψ*=0.00065; (b) C*=C-C0/CH-CL, ΔC*=0.0625; (c) T*=T-T0/TH-TC, ΔT*=0.0625.
(a)
Dimensionless streamlineψ*
(b)
Dimensionless iso-concentrationC*
(c)
Dimensionless isothermT*Figure 8
Dimensionless streamline, iso-concentration, and isotherm patterns for air-CO2 mixture at 5%, 10%, and 20% CO2, in opposing flow. (a) ψ*=ψgβTTH-TCL3, Δψ*=0.00065; (b) C*=C-C0/CH-CL, ΔC*=0.0625; (c) T*=T-T0/TH-TC, ΔT*=0.0625.
(a)
Dimensionless streamlineψ*
(b)
Dimensionless iso-concentrationC*
(c)
Dimensionless isothermT*Figure 9
Dimensionless kinetic energy at different concentration of CO2, in opposing flow: (a) transparent gas; (b) participating gas. eC*=0.5u2+v2gβTTH-TCL-1.
(a)
(b)Independently of gas radiation, the increase in concentration of pollutant (to 10% CO2) slightly strengthens the mass forces, which become comparable to the thermal ones. In this configuration, a multicellular flow structure is observed (transitional flow) (Figure 7(a), 10% CO2): two cells of thermal origin near the horizontal walls and one cell of mass origin in the center of cavity. Under radiation effect, the horizontal temperature gradients, at the vicinity of the horizontals walls (where thermal forces are dominant even with transparent gas), are enhanced. This accelerates the flow in thermal cells (upper and lower) and consequently in mass cell as well (Figure 9(b), 10% CO2). Radiation does not practically alter the structure of the concentration field (since the flow dynamics which controls the mass transfer itself is unaffected). However, we note a decrease in pollutant concentration at the top and center of the cavity and an increase at the bottom (Figure 10(a), y/L=0.875 and 0.125), due to acceleration of the fluid within cells. The structure of the thermal field is also little affected by radiation (Figure 8(c), 10% CO2). Also, a warming of the fluid can be noted in the upper part of the cavity and its cooling in the center and in the lower part (Figure 10(b), y/L=0.875 and 0.125). This causes (in association with the fluid flow direction) an increase of the temperature gradients along the vertical walls.Figure 10
Dimensionless concentration (a) and temperature (b) profiles at different heights in the cavity, for an opposing flow: mixture at 5%, 10%, and 20% CO2, C*=C-C0/CH-CL, T*=T-T0/TH-TC.
(a)
(b)For mixture at 20% CO2, when the gas is transparent, the fluid dynamics is dominated by mass buoyancy forces (mass flow regime) with a monocellular flow in anticlockwise direction (Figure 7(a), 20% CO2). Under gas radiation effect, thermal forces are strengthened near the horizontal walls. As these two types of forces are present in opposing flow, dominated by mass forces, their resultant is reduced, yielding a slowdown (and even stop) of horizontal boundary layers. The vertical boundary layers are also slowed (partially) (Figure 9(b), 20% CO2), which causes inclination of streamlines. But, owing to the strong dominance of mass forces (N=-2.10), the flow structure is globally maintained. Radiation causes the inclination of iso-concentrations and isotherms (i.e., formation of negative horizontal gradients of concentration and temperature) and a decrease of gradients of temperature at vertical walls: fluid warmer at hot wall and colder at the cold wall. These behaviors have also been observed in the cooperating case.


### 4.3. Heat and Mass Transfer

#### 4.3.1. Cooperating Flow

The variation of average convective Nusselt number (Nu-c), with pollutant (CO2) concentration, at vertical walls is illustrated in Figure 11(a). It can be seen that radiation reduces the average convective flux because of the weakening of the temperature gradient at vertical walls. The same trend is observed for the total Nusselt number (convective + radiative) (Figure 11(b)) due to the attenuation by the fluid of the radiative exchange between the active walls. This attenuation gets stronger when increasing concentration of absorbing species. It is worth noting that mass transfer (average Sherwood number Sh-) is very weakly altered by gas radiation because it is primarily controlled by the flow dynamics, itself less sensitive to radiation than the thermal field (Figure 11(c)).Figure 11
Average convective Nusselt number (a), average total Nusselt number (b), and average Sherwood number (c) as a function of molar fraction of CO2 in cooperating flow.
(a)
(b)
(c)


#### 4.3.2. Opposing Flow

Contrarily to the cooperating case, when mass and thermal buoyancy forces are opposed (Figure12(a)) different radiation effects can be observed on the average convective Nusselt number (Nu-c). This parameter is essentially reduced at high pollutant concentration corresponding to a mass flow regime and is less altered in thermal regime (reduction due to the weakening of temperature gradients at active walls) and transitional regime (augmentation due to acceleration of motion and increase of temperature gradients at active walls). The total Nusselt number (Figure 12(b)) is decreased whatever the concentration of pollutant is (so regardless the flow regime: thermal, transitional, or mass). Regarding mass transfer, outside the thermal regime, average mass flux (Sh-) becomes a little more sensitive to gas radiation (than in the cooperating case) (Figure 12(c)): it is increased in transitional regime and reduced in mass regime.Figure 12
Average convective Nusselt number (a), average total Nusselt number (b), and average Sherwood number (c) as a function of molar fraction of CO2 in opposing flow; A: thermal; B: intermediate regime; C: mass regime.
(a)
(b)
(c)


## 4.1. Conditions of Simulations

The results presented in this investigation were obtained by considering the following parameters:L = 0.3 m, TH = 580 K, and TC = 530 K. In cooperating flow (i.e., where thermal and mass buoyancy forces augment each other), a null concentration of CO2 is prescribed at the hot wall (CL=0 mol/m3 or xL=0%) and, at the cold wall, the concentration is set to CH = 2.20, 4.39, and 8.78 mol/m3. These correspond, respectively, in reference conditions, to molar fractions xCO2=10%, 20%, and 40%. Now, to create an opposing double diffusive flow (i.e., thermal and mass buoyancy forces are opposite), the mass boundary conditions must be reversed: CL at the cold wall and CH at the hot wall.In the SLW model, the reference state values of temperature and molar fraction wereT0=555 K and the average molar fractions xCO2=5%, 10%, and 20%. Based on these data, the mass-to-thermal buoyancy ratio (N), thermal Rayleigh number (Ra), Prandtl (Pr), Planck (Pl), and Lewis number (Le) relating to the problem are summarized in Table 6. Values of |N| below unity (0.57) characterize a flow mainly driven by thermal forces (thermal regime), whereas values above unity (2.10) refer to a flow dominated by concentration gradients (mass regime). In between (|N|=1.10) mass and thermal effects are of comparable magnitude (intermediate regime).Table 6
Dimensionless numbers corresponding to cooperating and opposing flow.
Pollutant
x (%)
N
Ra
Pr
Pl
Le
CO2
5
±0,57
8,85 × 106
0,72
3,58 × 10−3
1,30
10
±1,10
9,44 × 106
0,72
3,55 × 10−3
1,26
20
±2,10
1,09 × 107
0,72
3,50 × 10−3
1,18It is worth noting that the effects of gas radiation on field characteristics were determined by comparison to the nonparticipating gas case (i.e., a fluid having the same thermophysical properties as the considered mixtures, but totally transparent).


## 4.2. Flow Patterns

### 4.2.1. Cooperating Flow

Figures3 and 4 display the streamline, temperature, and concentration contours for different average molar fractions of CO2 in the cavity (xCO2=5%,10%, and 20%), when the gas is transparent or participating to radiation. On the whole, we can see (Figure 4(a)) that, when the buoyancy forces (mass and thermal) are cooperating, gas radiation weakly alters the flow pattern (slight slope and distortion of the streamlines, small secondary cells in the core of cavity). These modifications are less important at 20% CO2 than at 5% (and 10%), because a higher rate of absorbing species strengthens the mass forces and increases buoyancy ratio (N) from 0,57 to 2,10 (see Table 6). This makes the flow mainly driven by concentration (not temperature) gradients and hence less sensitive to radiation. Under gas radiation effects, the vertical boundary layers are broadened and partially accelerated (Figures 5(c) and 5(d)), whereas the core of the cavity is unaffected and remains at rest. This induces an increase of horizontal jets flowing from one active wall to the other along the adiabatic surfaces (Figures 5(a) and 5(b)). The strengthening of the vertical boundary layers, associated with fixed concentrations at vertical walls, leads to an excess of pollutant (CO2) near the left wall (at CH) and a deficit near the right one (at CL) (Figure 6(a)). To equilibrate this trend, some part of CO2 is diffused from left to right, resulting in negative gradient of concentration in the x-direction. This breaks the centrosymmetry of the field and replaces the vertical stratification by a tilted iso-concentration pattern (Figure 4(b)) [25].Figure 3
Dimensionless streamline, iso-concentration, and isotherm patterns for air-CO2 mixture at 5%, 10%, and 20% CO2, in cooperating flow. (a) ψ*=ψgβTTH-TCL3, Δψ*=0.00065; (b) C*=C-C0/CH-CL, ΔC*=0.0625; (c) T*=T-T0/TH-TC, ΔT*=0.0625.
(a)
Dimensionless streamlineψ*
(b)
Dimensionless iso-concentrationC*
(c)
Dimensionless isothermT*Figure 4
Dimensionless streamline, iso-concentration, and isotherm patterns for air-CO2 mixture at 5%, 10%, and 20% CO2, in cooperating flow. (a) ψ*=ψgβTTH-TCL3, Δψ*=0.00065; (b) C*=C-C0/CH-CL, ΔC*=0.0625; (c) T*=T-T0/TH-TC, ΔT*=0.0625.
(a)
Dimensionless streamlineψ*
(b)
Dimensionless iso-concentrationC*
(c)
Dimensionless isothermT*Figure 5
Dimensionless vertical and horizontal velocity at different levels in the cavity, for cooperating flow: air-CO2 mixtures at 5%, 10%, and 20% CO2. v*=vgβTTH-TCL-0.5, u*=ugβTTH-TCL-0.5. (a) Upper wall, (b) lower wall, (c) hot wall, and (d) cold wall.
(a)
(b)
(c)
(d)Figure 6
Dimensionless concentration (a) and temperature (b) profiles at different heights in the cavity, for a cooperating flow.C*=C-C0/CH-CL, T*=T-T0/TH-TC.
(a)
(b)Regarding the thermal field, the gas flowing under the upper wall emits radiation. It is slightly colder than it would be without radiative effects (Figure6(b), air-CO2 mixture, y/L=0.875), and this weakens the temperature gradient at vertical cold wall when the fluid flows downward. A symmetrical picture is observed along the bottom wall (Figure 6(b), air-CO2 mixture, y/L=0.125) where the fluid is preheated by absorbing radiation, leading to a reduction of the temperature gradient along the vertical hot wall. In the core of the cavity (outside boundary layers) (Figure 6(b), air-CO2, y/L=0.5), while the reference transparent gas remains practically isothermal, the gas is heated by absorption of radiation from the hot wall in the left part of the cavity and is cooled by emission toward the cold wall in the right part. As a result (in the center of the cavity) negative temperature gradients in the x-direction are formed and will create a thermal field with tilted stratification (Figure 4(c)). This trend is more pronounced at 20% CO2 than at 10% and 5%.


### 4.2.2. Opposing Case

For an average concentration of 5% of CO2, the fluid flow in the cavity is mainly driven by thermal forces if gas absorption is neglected (Figure 7(a), 5% CO2). When taken into account (Figure 8(a), 5% CO2), radiation accelerates boundary layers (Figure 9(b), 5% CO2) by enhancing the thermal forces near the horizontals walls (as in cooperating case). The stagnant core of the cavity (for transparent gas) is also set in motion (thermal forces are slightly weakened). Such behavior tends to homogenize the concentration and temperature fields, whose stratifications are destroyed (Figure 8(b), 5% CO2). In this case (at 5% CO2), the magnitude of radiation effects on the structure of the different fields seems to be greater in opposing flow than in cooperating case, indicating that a lower flow dynamics when the thermal and mass buoyancy forces are opposing (equivalent Rayleigh number: N+1Ra lower) promotes the influence of radiation.Figure 7
Dimensionless streamline, iso-concentration, and isotherm patterns for air-CO2 mixture at 5%, 10%, and 20% CO2, in opposing flow. (a) ψ*=ψgβTTH-TCL3, Δψ*=0.00065; (b) C*=C-C0/CH-CL, ΔC*=0.0625; (c) T*=T-T0/TH-TC, ΔT*=0.0625.
(a)
Dimensionless streamlineψ*
(b)
Dimensionless iso-concentrationC*
(c)
Dimensionless isothermT*Figure 8
Dimensionless streamline, iso-concentration, and isotherm patterns for air-CO2 mixture at 5%, 10%, and 20% CO2, in opposing flow. (a) ψ*=ψgβTTH-TCL3, Δψ*=0.00065; (b) C*=C-C0/CH-CL, ΔC*=0.0625; (c) T*=T-T0/TH-TC, ΔT*=0.0625.
(a)
Dimensionless streamlineψ*
(b)
Dimensionless iso-concentrationC*
(c)
Dimensionless isothermT*Figure 9
Dimensionless kinetic energy at different concentration of CO2, in opposing flow: (a) transparent gas; (b) participating gas. eC*=0.5u2+v2gβTTH-TCL-1.
(a)
(b)Independently of gas radiation, the increase in concentration of pollutant (to 10% CO2) slightly strengthens the mass forces, which become comparable to the thermal ones. In this configuration, a multicellular flow structure is observed (transitional flow) (Figure 7(a), 10% CO2): two cells of thermal origin near the horizontal walls and one cell of mass origin in the center of cavity. Under radiation effect, the horizontal temperature gradients, at the vicinity of the horizontals walls (where thermal forces are dominant even with transparent gas), are enhanced. This accelerates the flow in thermal cells (upper and lower) and consequently in mass cell as well (Figure 9(b), 10% CO2). Radiation does not practically alter the structure of the concentration field (since the flow dynamics which controls the mass transfer itself is unaffected). However, we note a decrease in pollutant concentration at the top and center of the cavity and an increase at the bottom (Figure 10(a), y/L=0.875 and 0.125), due to acceleration of the fluid within cells. The structure of the thermal field is also little affected by radiation (Figure 8(c), 10% CO2). Also, a warming of the fluid can be noted in the upper part of the cavity and its cooling in the center and in the lower part (Figure 10(b), y/L=0.875 and 0.125). This causes (in association with the fluid flow direction) an increase of the temperature gradients along the vertical walls.Figure 10
Dimensionless concentration (a) and temperature (b) profiles at different heights in the cavity, for an opposing flow: mixture at 5%, 10%, and 20% CO2, C*=C-C0/CH-CL, T*=T-T0/TH-TC.
(a)
(b)For mixture at 20% CO2, when the gas is transparent, the fluid dynamics is dominated by mass buoyancy forces (mass flow regime) with a monocellular flow in anticlockwise direction (Figure 7(a), 20% CO2). Under gas radiation effect, thermal forces are strengthened near the horizontal walls. As these two types of forces are present in opposing flow, dominated by mass forces, their resultant is reduced, yielding a slowdown (and even stop) of horizontal boundary layers. The vertical boundary layers are also slowed (partially) (Figure 9(b), 20% CO2), which causes inclination of streamlines. But, owing to the strong dominance of mass forces (N=-2.10), the flow structure is globally maintained. Radiation causes the inclination of iso-concentrations and isotherms (i.e., formation of negative horizontal gradients of concentration and temperature) and a decrease of gradients of temperature at vertical walls: fluid warmer at hot wall and colder at the cold wall. These behaviors have also been observed in the cooperating case.


## 4.2.1. Cooperating Flow

Figures3 and 4 display the streamline, temperature, and concentration contours for different average molar fractions of CO2 in the cavity (xCO2=5%,10%, and 20%), when the gas is transparent or participating to radiation. On the whole, we can see (Figure 4(a)) that, when the buoyancy forces (mass and thermal) are cooperating, gas radiation weakly alters the flow pattern (slight slope and distortion of the streamlines, small secondary cells in the core of cavity). These modifications are less important at 20% CO2 than at 5% (and 10%), because a higher rate of absorbing species strengthens the mass forces and increases buoyancy ratio (N) from 0,57 to 2,10 (see Table 6). This makes the flow mainly driven by concentration (not temperature) gradients and hence less sensitive to radiation. Under gas radiation effects, the vertical boundary layers are broadened and partially accelerated (Figures 5(c) and 5(d)), whereas the core of the cavity is unaffected and remains at rest. This induces an increase of horizontal jets flowing from one active wall to the other along the adiabatic surfaces (Figures 5(a) and 5(b)). The strengthening of the vertical boundary layers, associated with fixed concentrations at vertical walls, leads to an excess of pollutant (CO2) near the left wall (at CH) and a deficit near the right one (at CL) (Figure 6(a)). To equilibrate this trend, some part of CO2 is diffused from left to right, resulting in negative gradient of concentration in the x-direction. This breaks the centrosymmetry of the field and replaces the vertical stratification by a tilted iso-concentration pattern (Figure 4(b)) [25].Figure 3
Dimensionless streamline, iso-concentration, and isotherm patterns for air-CO2 mixture at 5%, 10%, and 20% CO2, in cooperating flow. (a) ψ*=ψgβTTH-TCL3, Δψ*=0.00065; (b) C*=C-C0/CH-CL, ΔC*=0.0625; (c) T*=T-T0/TH-TC, ΔT*=0.0625.
(a)
Dimensionless streamlineψ*
(b)
Dimensionless iso-concentrationC*
(c)
Dimensionless isothermT*Figure 4
Dimensionless streamline, iso-concentration, and isotherm patterns for air-CO2 mixture at 5%, 10%, and 20% CO2, in cooperating flow. (a) ψ*=ψgβTTH-TCL3, Δψ*=0.00065; (b) C*=C-C0/CH-CL, ΔC*=0.0625; (c) T*=T-T0/TH-TC, ΔT*=0.0625.
(a)
Dimensionless streamlineψ*
(b)
Dimensionless iso-concentrationC*
(c)
Dimensionless isothermT*Figure 5
Dimensionless vertical and horizontal velocity at different levels in the cavity, for cooperating flow: air-CO2 mixtures at 5%, 10%, and 20% CO2. v*=vgβTTH-TCL-0.5, u*=ugβTTH-TCL-0.5. (a) Upper wall, (b) lower wall, (c) hot wall, and (d) cold wall.
(a)
(b)
(c)
(d)Figure 6
Dimensionless concentration (a) and temperature (b) profiles at different heights in the cavity, for a cooperating flow.C*=C-C0/CH-CL, T*=T-T0/TH-TC.
(a)
(b)Regarding the thermal field, the gas flowing under the upper wall emits radiation. It is slightly colder than it would be without radiative effects (Figure6(b), air-CO2 mixture, y/L=0.875), and this weakens the temperature gradient at vertical cold wall when the fluid flows downward. A symmetrical picture is observed along the bottom wall (Figure 6(b), air-CO2 mixture, y/L=0.125) where the fluid is preheated by absorbing radiation, leading to a reduction of the temperature gradient along the vertical hot wall. In the core of the cavity (outside boundary layers) (Figure 6(b), air-CO2, y/L=0.5), while the reference transparent gas remains practically isothermal, the gas is heated by absorption of radiation from the hot wall in the left part of the cavity and is cooled by emission toward the cold wall in the right part. As a result (in the center of the cavity) negative temperature gradients in the x-direction are formed and will create a thermal field with tilted stratification (Figure 4(c)). This trend is more pronounced at 20% CO2 than at 10% and 5%.


## 4.2.2. Opposing Case

For an average concentration of 5% of CO2, the fluid flow in the cavity is mainly driven by thermal forces if gas absorption is neglected (Figure 7(a), 5% CO2). When taken into account (Figure 8(a), 5% CO2), radiation accelerates boundary layers (Figure 9(b), 5% CO2) by enhancing the thermal forces near the horizontals walls (as in cooperating case). The stagnant core of the cavity (for transparent gas) is also set in motion (thermal forces are slightly weakened). Such behavior tends to homogenize the concentration and temperature fields, whose stratifications are destroyed (Figure 8(b), 5% CO2). In this case (at 5% CO2), the magnitude of radiation effects on the structure of the different fields seems to be greater in opposing flow than in cooperating case, indicating that a lower flow dynamics when the thermal and mass buoyancy forces are opposing (equivalent Rayleigh number: N+1Ra lower) promotes the influence of radiation.Figure 7
Dimensionless streamline, iso-concentration, and isotherm patterns for air-CO2 mixture at 5%, 10%, and 20% CO2, in opposing flow. (a) ψ*=ψgβTTH-TCL3, Δψ*=0.00065; (b) C*=C-C0/CH-CL, ΔC*=0.0625; (c) T*=T-T0/TH-TC, ΔT*=0.0625.
(a)
Dimensionless streamlineψ*
(b)
Dimensionless iso-concentrationC*
(c)
Dimensionless isothermT*Figure 8
Dimensionless streamline, iso-concentration, and isotherm patterns for air-CO2 mixture at 5%, 10%, and 20% CO2, in opposing flow. (a) ψ*=ψgβTTH-TCL3, Δψ*=0.00065; (b) C*=C-C0/CH-CL, ΔC*=0.0625; (c) T*=T-T0/TH-TC, ΔT*=0.0625.
(a)
Dimensionless streamlineψ*
(b)
Dimensionless iso-concentrationC*
(c)
Dimensionless isothermT*Figure 9
Dimensionless kinetic energy at different concentration of CO2, in opposing flow: (a) transparent gas; (b) participating gas. eC*=0.5u2+v2gβTTH-TCL-1.
(a)
(b)Independently of gas radiation, the increase in concentration of pollutant (to 10% CO2) slightly strengthens the mass forces, which become comparable to the thermal ones. In this configuration, a multicellular flow structure is observed (transitional flow) (Figure 7(a), 10% CO2): two cells of thermal origin near the horizontal walls and one cell of mass origin in the center of cavity. Under radiation effect, the horizontal temperature gradients, at the vicinity of the horizontals walls (where thermal forces are dominant even with transparent gas), are enhanced. This accelerates the flow in thermal cells (upper and lower) and consequently in mass cell as well (Figure 9(b), 10% CO2). Radiation does not practically alter the structure of the concentration field (since the flow dynamics which controls the mass transfer itself is unaffected). However, we note a decrease in pollutant concentration at the top and center of the cavity and an increase at the bottom (Figure 10(a), y/L=0.875 and 0.125), due to acceleration of the fluid within cells. The structure of the thermal field is also little affected by radiation (Figure 8(c), 10% CO2). Also, a warming of the fluid can be noted in the upper part of the cavity and its cooling in the center and in the lower part (Figure 10(b), y/L=0.875 and 0.125). This causes (in association with the fluid flow direction) an increase of the temperature gradients along the vertical walls.Figure 10
Dimensionless concentration (a) and temperature (b) profiles at different heights in the cavity, for an opposing flow: mixture at 5%, 10%, and 20% CO2, C*=C-C0/CH-CL, T*=T-T0/TH-TC.
(a)
(b)For mixture at 20% CO2, when the gas is transparent, the fluid dynamics is dominated by mass buoyancy forces (mass flow regime) with a monocellular flow in anticlockwise direction (Figure 7(a), 20% CO2). Under gas radiation effect, thermal forces are strengthened near the horizontal walls. As these two types of forces are present in opposing flow, dominated by mass forces, their resultant is reduced, yielding a slowdown (and even stop) of horizontal boundary layers. The vertical boundary layers are also slowed (partially) (Figure 9(b), 20% CO2), which causes inclination of streamlines. But, owing to the strong dominance of mass forces (N=-2.10), the flow structure is globally maintained. Radiation causes the inclination of iso-concentrations and isotherms (i.e., formation of negative horizontal gradients of concentration and temperature) and a decrease of gradients of temperature at vertical walls: fluid warmer at hot wall and colder at the cold wall. These behaviors have also been observed in the cooperating case.


## 4.3. Heat and Mass Transfer

### 4.3.1. Cooperating Flow

The variation of average convective Nusselt number (Nu-c), with pollutant (CO2) concentration, at vertical walls is illustrated in Figure 11(a). It can be seen that radiation reduces the average convective flux because of the weakening of the temperature gradient at vertical walls. The same trend is observed for the total Nusselt number (convective + radiative) (Figure 11(b)) due to the attenuation by the fluid of the radiative exchange between the active walls. This attenuation gets stronger when increasing concentration of absorbing species. It is worth noting that mass transfer (average Sherwood number Sh-) is very weakly altered by gas radiation because it is primarily controlled by the flow dynamics, itself less sensitive to radiation than the thermal field (Figure 11(c)).Figure 11
Average convective Nusselt number (a), average total Nusselt number (b), and average Sherwood number (c) as a function of molar fraction of CO2 in cooperating flow.
(a)
(b)
(c)


### 4.3.2. Opposing Flow

Contrarily to the cooperating case, when mass and thermal buoyancy forces are opposed (Figure12(a)) different radiation effects can be observed on the average convective Nusselt number (Nu-c). This parameter is essentially reduced at high pollutant concentration corresponding to a mass flow regime and is less altered in thermal regime (reduction due to the weakening of temperature gradients at active walls) and transitional regime (augmentation due to acceleration of motion and increase of temperature gradients at active walls). The total Nusselt number (Figure 12(b)) is decreased whatever the concentration of pollutant is (so regardless the flow regime: thermal, transitional, or mass). Regarding mass transfer, outside the thermal regime, average mass flux (Sh-) becomes a little more sensitive to gas radiation (than in the cooperating case) (Figure 12(c)): it is increased in transitional regime and reduced in mass regime.Figure 12
Average convective Nusselt number (a), average total Nusselt number (b), and average Sherwood number (c) as a function of molar fraction of CO2 in opposing flow; A: thermal; B: intermediate regime; C: mass regime.
(a)
(b)
(c)


## 4.3.1. Cooperating Flow

The variation of average convective Nusselt number (Nu-c), with pollutant (CO2) concentration, at vertical walls is illustrated in Figure 11(a). It can be seen that radiation reduces the average convective flux because of the weakening of the temperature gradient at vertical walls. The same trend is observed for the total Nusselt number (convective + radiative) (Figure 11(b)) due to the attenuation by the fluid of the radiative exchange between the active walls. This attenuation gets stronger when increasing concentration of absorbing species. It is worth noting that mass transfer (average Sherwood number Sh-) is very weakly altered by gas radiation because it is primarily controlled by the flow dynamics, itself less sensitive to radiation than the thermal field (Figure 11(c)).Figure 11
Average convective Nusselt number (a), average total Nusselt number (b), and average Sherwood number (c) as a function of molar fraction of CO2 in cooperating flow.
(a)
(b)
(c)


## 4.3.2. Opposing Flow

Contrarily to the cooperating case, when mass and thermal buoyancy forces are opposed (Figure12(a)) different radiation effects can be observed on the average convective Nusselt number (Nu-c). This parameter is essentially reduced at high pollutant concentration corresponding to a mass flow regime and is less altered in thermal regime (reduction due to the weakening of temperature gradients at active walls) and transitional regime (augmentation due to acceleration of motion and increase of temperature gradients at active walls). The total Nusselt number (Figure 12(b)) is decreased whatever the concentration of pollutant is (so regardless the flow regime: thermal, transitional, or mass). Regarding mass transfer, outside the thermal regime, average mass flux (Sh-) becomes a little more sensitive to gas radiation (than in the cooperating case) (Figure 12(c)): it is increased in transitional regime and reduced in mass regime.Figure 12
Average convective Nusselt number (a), average total Nusselt number (b), and average Sherwood number (c) as a function of molar fraction of CO2 in opposing flow; A: thermal; B: intermediate regime; C: mass regime.
(a)
(b)
(c)


## 5. Conclusions

In this work, we analyzed the effects of radiation of gas radiation on laminar double-diffusive convection in a nongray air-CO2, when the thermal and mass buoyancy forces are cooperating or opposing. The real radiative participation of mixtures is accounted for by using the SLW spectral model of Denison and Webb. Three average concentrations of pollutants were considered so as to cover the different flow regimes: thermal (xCO2=5%), transitional (10%), and mass (20%).(i)
In cooperating case, the results show that gas radiation has little influence on flow structure (only an acceleration of boundary layers and the core of the cavity remains practically at rest) but disturbs the temperature and concentration fields, particularly by inclining their iso-values lines. The global heat transfer (total Nusselt number) is decreased, whereas the mass transfer (Sherwood numbers) is weakly affected (slightly decreased). These effects are observed for all considered concentrations (i.e., for all convective flow regimes).(ii)
In opposing case, radiation effects are more important than in cooperating flow and depend on the nature of the flow regime (in the absence of radiation):
(a)
in thermal regime (xCO2=5%): the fluid is accelerated across the cavity and this strongly alters the structure of the dynamic field and breaks the vertical stratification of temperature and concentration fields (essentially). Convective heat transfer is reduced by a decrease of the horizontal gradient of temperature at vertical walls;(b)
in transitional regime (xCO2=10% CO2): the fluid is also accelerated across the cavity without altering the structures of different fields. Convective heat transfer at vertical walls is increased (essentially at hot wall), by acceleration of the fluid flow and increase of the wall gradient of temperature;(c)
in mass regime (xCO2= 20%): the fluid is decelerated and gas radiation tends to replace the vertically stratified distribution of the temperature and concentration by a field with inclined iso-value lines (as in cooperating case). Average convective heat transfers are greatly reduced;(d)
more generally, gas radiation weakens the global heat transfer (conductive + radiative) whatever the flow regime is (as cooperating case) and also affects the mass transfer in transitional and mass regime.



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*Source: 586913-2015-03-24.xml*