The present study deals with doublediffusive convection within a twodimensional inclined cavity filled with an airCO_{2} binary gas mixture. The left and the right vertical walls are differentially heated and subjected to different locations of (CO_{2}) contaminants to allow for the variation of the buoyancy strength (N). However, the horizontal walls are assumed adiabatic. The simulations are conducted using the finite volume method to solve the conservation equations of continuity, momentum, energy, and species transport. Good agreement with other numerical results in the literature is obtained. The effect of multiple parameters, namely, buoyancy ratio (N), thermal Rayleigh number (Ra), and inclination angle (α) on entropy generation rate is analyzed and discussed in the postprocessing stage, while considering both laminar and turbulent flow regimes. The computations reveal that these parameters considerably affect both the heat and mass transfer performances of the system.
Conjugate heat and mass transfer, widely referred to as doublediffusive convection or thermosolutal convection, is a buoyancydriven flow governed by two components with different molecular diffusivities and opposing contributions to the vertical density gradient. Owing to its engineering and industrial applications, including the design of chemical processing equipment, formation, and dispersion of fog, environmental pollution, oceanography, etc., doublediffusive convection received great interest among researchers. Beghein et al. [
In spite of the turbulence’s huge effect and according to the extensive bibliographic research examined so far, previous studies’ attention was mostly devoted to entropy production under the laminar regime. The turbulent flow was widely neglected. Accordingly, the present work aims to solve numerically doublediffusive convection and entropy generation under turbulent flow regime. Entropy production under opposing flow i.e.,
The governing partial differential equations of a mathematical model that describes heat and mass transfer are continuity, momentum, energy, and species diffusion equations. From a simplification point of view, the working fluid is assumed as a Newtonian and incompressible perfect binary gas mixture. Soret and Dufour effects are also neglected in addition to the radiative exchanges between surfaces. On the other hand, all the fluid physical properties were constant except density in the buoyancy force term. Therefore, the Boussinesq approximation is satisfied. In view of these considerations, the governing equations could be written as [
where,
The total entropy generation due to conjugate heat and mass transfer is the sum of three different irreversibilities, i.e., friction, heat, and species diffusion. For a Newtonian and incompressible perfect binary gas mixture over a twodimensional domain it takes the following form [
Using the following dimensionless variables:
The dimensionless form of entropy production is obtained as follows:
Therefore:
where,
The discretization of the governing equations described above was ensured by way of the finite volume method technique. To couple pressure and velocity, the SEMPLEC algorithm was employed in this work. The standard
Entropy generation due to laminar and turbulent doublediffusive natural convection process inside a square cavity for different inclination angles and multiple buoyancy ratios is analyzed in this section. The Prandtl number, as well as the Lewis number, were taken to be constant:
Multiple quadratic grids were tested in order to satisfy an optimum balance between computational time and results accuracy. The Mean, the minimum, and the maximum values of the Nusselt number at the hot wall beside the dimensionless components of the vertical and horizontal velocities at the cavity midsection were calculated and verified for
Mesh  

75 × 75  5.377  18.9  9.200  19.14  0.504  9.52  0.130  0.00  0.256  0.39 
125 × 125  4.522  0.02  7.722  0.03  0.557  23.5  0.130  0.00  0.257  0.00 
175 × 175  4.521  0.00  7.720  0.00  0.728  0.00  0.130  0.00  0.257  0.00 
225 × 225  4.521  –  7.720  –  0.728  –  0.130  –  0.257  – 
Once the optimum mesh was indicated, it is preceded to the numerical code validation. Every simulation must undergo this process to verify the accuracy of its numerical outputs. The numerical results were confronted quantitively and qualitatively with various data found in the literature.
First, the issue of doublediffusive convection under the laminar regime was simulated and evaluated against various authors for multiple buoyancy ratios (
Present work  [ 
[ 
[ 

–0.01  16.5  16.4  13.4  16.3 
–0.1  16.1  16.0  16.0  15.9 
–0.2  15.6  15.5  15.3  15.4 
–0.5  13.7  13.6  13.6  13.5 
–0.8  10.7  10.6  10.6  10.5 
–0.9  8.8  8.8  8.8  8.6 
–1.5  13.7  13.6  13.5  13.5 
–5.0  23.8  23.7  23.7  23.6 
Notes  

The flow is dominated by solutal buoyancy force. As a result, a counterclockwise motion is obtained.  
Convections of thermal and solutal buoyancy forces compete with equal forces. The only forces driving the flow are heat and mass diffusion. Thus, a circular motion is generated at the cavity center.  
The flow is purely thermoconvective. Hence, the flow rotates in the clockwise direction.  
–1.0 < 
The flow is dominated by thermal buoyancy force. Hence, a righthanded motion is observed. 
The solutal and thermal forces assist each other to rotate in the clockwise direction. Thus, identical comportment as pure natural convection was detected. 
The inclination effect was more significant on entropy generation for low and large buoyancy ratios, i.e.,
Therefore, the inclination angle should be taken into account in order to define the optimum design of any apparatus due to its important implication.
For the assessment of the dominant irreversibility of the entropy components, i.e., thermal, viscous, and diffusion, the Bejan number was investigated using the following formula:
where if
In the present study, doublediffusive convection under the laminar and turbulent regimes inside an inclined cavity was explored numerically. The effects of three different inclination angles and various buoyancy forces (
The buoyancy ratio (
Due to convection nonexistence when
The influence of inclination angle is more significant under the turbulent regime, especially at small and high buoyancy ratios (
Except
Aspect ratio
Bejan number
Concentration, kg m^{–3}
Dimensionless concentration
Mass diffusivity, m^{2} s^{–1}
Lewis number
Buoyancy ratio
Entropy number
Prandtl number
Gas constant, J kg^{–1} K^{–1}
Thermal, solutal Rayleigh numbers
Entropy generation, W m^{–3} K^{–1}
Temperature, K
Horizontal, vertical velocity component, m s^{–1}
Horizontal, vertical dimensionless velocity component
Dimensionless coordinates
Inclination angle
Thermal diffusivity, m^{2} s^{–1}
Solutal expansion coefficient, m^{3} kg^{–1}
Thermal expansion coefficient, K^{–1}
Dimensionless temperature
Thermal conductivity, W m^{–1} K^{–1}
Dynamic viscosity, kg m^{–1} s^{–1}
Error
Thermal entropy
Viscous entropy
Diffusive entropy