Minimization of entropy generation due to MHD double diffusive mixed convection in a lid driven trapezoidal cavity with various aspect ratios

Entropy generation minimization has significant importance in fluid flow, heat and mass transfer in an enclosure to get the maximum efficiency of a system and to reduce the loss of energy. In the present study, the analysis of mixed convection fluid flow, heat and mass transfer with heat line and mass line concept and entropy generation due to the effects of fluid flow, heat flow, mass flow and magnetic field in a trapezoidal enclosure with linearly heated and diffusive left wall, uniformly heated and diffusive lower wall, cold and nondiffusive right wall, adiabatic and zero diffusion gradient top wall have been reported. Parametric studies for the wide range of Prandtl number (Pr = 0.7 for air cooling system and Pr = 1000 for the engines filled with olive or engine oils), Rayleigh number (Ra = 103–105), aspect ratio (A = 0.5–1.5) and inclination angle of the cavity (ϕ = 45°–90°) have been performed, which help to construct the perfect shape of cavity in many engineering and physical applications so that the entropy is minimum to get the maximum efficiency of any system. The finite-difference approximation has been used to find out the numerical solutions. Biconjugate Gradient Stabilized (BiCGStab) method is used to solve the discretized nonhomogeneous system of linear equations.


Introduction
Mixed convection is a combination of forced convection and natural convection, which exists when a flow is determined simultaneously by both an outer forcing system and inner volumetric forces. In many engineering applications, engineers often encounter forced convection to design or analyze heat exchangers, pipe flow, cavity flow, flow over a flat plate etc. If the natural convection is not much less than the forced convection, then the flow is considered as mixed convection flow. In any mixed convection, flow stream lines, heat lines and mass lines are routinely considered as very efficient method for the adequate visualization tools of fluid flow, heat transfer and mass transfer, respectively. The heat function and heat line concept were first introduced by Kimura and Bejan [19] to visualize the heat transfer phenomena and in the field of mass transfer, mass function and mass line concept were first discussed by Trevisan and Bejan [35]. Heat and mass transfer phenomena have received considerable attention due to its various engineering and natural applications such as air cooling [10,14], heat exchanger [33], solar plate collector [36], geothermal system [7] etc. Air cooling is mostly preferable method for the cooling of computer system and other electronic equipments because of its simplicity and low cost. Lid-driven cavity flow is also very important due to its applications in crystal growth, flow and heat transfer in solar ponds [9], dynamics of lakes [18], thermal hydraulics of nuclear reactors [16], food processing and float glass production [25] etc. One useful control parameter for convective heat and mass transfer in cavities is magnetic field. Effect of magnetic field on double diffusive natural convection heat transfer in trapezoidal enclosure has been discussed by Teamah and Shehata [34]. Sensitivity analysis on MHD heat transfer and entropy generation in inclined trapezoidal cavity filled with nanofluid has been discussed by Shirvan et al. [2]. Together with these analyses, it is also very important to save energy. The second law of thermodynamics analyzes and estimates the efficiency of a thermal system, which states that when energy is transformed from one state to another, there is a loss in the amount of available energy, which reduces the efficiency of the system. In order to minimize the irreversibilities associated with the system, a new methodology is improved, known as entropy generation minimization. Therefore, it is important to analyze the entropy generation minimization to increase the efficiency of the system. Many researches have been done on this topic. Entropy generation minimization concept was first introduced by Bejan [4][5][6]. The heat transfer rate and entropy generation due to heat transfer and fluid friction during natural convection in trapezoidal enclosures have been discussed by Basak et al. [3]. Ramakrishna et al. [26] analysed the entropy generation during free convection within trapezoidal cavity. Singh et al. [32] investigated the role of entropy generation on thermal management during natural convection in tilted porous square cavities. A numerical analysis of double-diffusive MHD natural convection and entropy generation inside a tilted sinusoidal corrugated porous enclosure has been discussed by Hussain [15].
Recently, entropy generation in natural convection of nanofluid inside a square cavity having hot solid block has been analyzed by Sheremet et al. [30]. Sheremet et al. [31] also discussed natural convection of nanofluid inside a wavy cavity with a nonuniform heating. More recently, using Buongiorno's mathematical model, Sheremet et al. [29] discussed natural convection and entropy generation in a square cavity with side walls having variable temperature filled with a nanofluid. The natural convection flow with hot horizontal wall and an adjacent cold vertical wall has an important application in the heat transfer analysis of electronic equipment, mainly in slim rack described by Carvalho et al. [8]. In this case, the power dissipation from the horizontal circuit board is limited by its temperature. MHD mixed convection in a flexible walled and nanofluids filled lid-driven cavity with volumetric heat generation has been analyzed by Selimefendigil and Öztop http://www.journals.vu.lt/nonlinear-analysis [28]. Hasanuzzaman et al. [13] discussed magnetohydrodynamic natural convection in trapezoidal cavities. Effects of moving lid direction on MHD mixed convection in a linearly heated cavity have been studied by Al-Salem et al. [1]. Raza et al. [27] examined the combined effects of thermal radiation and magnetic field on molybdenum disulfide nanofluid in a channel with changing walls. A numerical model of MHD stability of natural convection in a cylindrical configuration filled with fluid is studied by Oudina and Bessaïh [23]. Numerical simulation of the effect of Prandtl number on oscillatory MHD natural convection in cylindrical annulus is discussed by Oudina and Makinde [24]. Oudina [22] modeled numerically the hydrodynamic stability in vertical annulus having heat source of different lengths. Recently, Alkasassbeh et al. [2] studied the heat transfer of convective fin with temperature dependent internal heat generation by hybrid block method.
In the present study, mixed convection fluid flow, heat and mass transfer with heat line and mass line concept and entropy generation minimization have been analyzed. The main objective of the present study is the analysis of entropy generation due to heat transfer, mass transfer and magnetic field and the total entropy generation in a trapezoidal enclosure with various aspect ratios in which the upper wall of the cavity is moving with a constant velocity. Our purpose is to analyze the thermal behavior of the fluid inside the trapezoidal enclosure under the effect of various parameters in terms of stream line, isotherm, isoconcentration, heat line, mass line, entropy generation due to heat transfer, fluid friction, mass transfer and magnetic field. The goal of our study is to achieve minimum entropy generation in order to enhance the overall efficiency of the system.

Problem formulation
A trapezoidal cavity filled with incompressible viscous fluid, with right and left walls inclined at an angle φ with the positive x-axis and negative x-axis, respectively, has been considered with uniformly heated and massive bottom wall, adiabatic and of zero concentration gradient top wall, hot and diffusive side walls. The nondimensional governing equations for two-dimensional, double-diffusive, steady, mixed convection flow in the presence of magnetic field can be written from the dimensional governing equations in [15] with the help of following nondimensional quantities: where X, Y are dimensional distances along the coordinate axes; U , V , T and C are dimensional velocities along x and y directions, temperature and concentration, respectively; x and y are dimensionless distances along x and y coordinates, respectively; u, v, θ and S are dimensionless velocities along x and y coordinates, temperature and concentration, respectively; L is the length of the enclosure, α is the thermal diffusivity, β T is the coefficient of thermal expansion, β S is the coefficient of solutal expansion, T h and T c are the temperature at the hot and cold wall, respectively; C c and C h are low and high concentrations at the wall of the cavity, respectively; ν is the kinematic viscosity, P is the dimensional pressure, p is the dimensionless pressure, ρ is the fluid density, g is the acceleration due to gravity, B is the magnetic induction, Pr is the Prandtl number, Ra is the Rayleigh number, Ha is the Hartmann number, N is buoyancy ratio, D is mass diffusivity, σ is electrical conductivity. The upper wall of the cavity is moving with a constant velocity U 0 = α/L towards positive x direction. Using stream function (ψ) and vorticity function (ω), momentum Eqs. (1) and (2) yield governing equation of vorticity and governing equation of stream function Eliminating ω from (3) and (4), we get biharmonic equation of ψ as Temperature equation: Concentration equation: Equations of heat function: which yield a single equation Equations of mass function: which yield a single equation where Le is the Lewis number, π is the heat function and τ is the mass function. The boundary conditions are as follows: (i) At the top wall: u = 1 (lid-driven upper wall), v = 0 (no slip condition), ∂θ/∂y = 0 (adiabatic condition), ∂S/∂y = 0 (zero concentration gradient), π = 0, τ = 0. (ii) At the bottom wall: u = v = 0 (no slip conditions), θ = 1 (uniformly heated), S = 1 (uniformly diffusive),n · ∇π = 0 (uniform heating),n · ∇τ = 0 (uniformly diffusive), where Nu l and Nu r , Sh l and Sh r are given in Sections 3 and 4. (iii) At the left wall: u = v = 0 (no slip conditions), θ = 1 − y (linearly heated), S = 1 − y (linearly diffusive),n · ∇π = sin φ,n · ∇τ = sin φ. (iv) At the right wall: u = v = 0 (no slip conditions), θ = 0 (cold), S = 0 (nondiffusive),n · ∇π = 0,n · ∇τ = 0, wheren denotes the unit normal vector to the wall.

Heat transfer
Nusselt number (Nu) is the ratio of heat transferred due to convection and the heat transferred due to conduction, and so it tells us about how much the heat transfer is enhanced due to fluid motion. It is a nondimensional heat transfer coefficient, which is mathematically defined as Nu = −∂θ/∂n, where n denotes normal direction to a plane. The local Nusselt number at the top, bottom, left and right walls of the cavity are, respectively, defined by respectively, where dS 1 and dS 2 are the small elemental lengths along the left and right walls, respectively. Here, since top wall is considered to be adiabatic, so Nu t = 0.

Mass transfer
Sherwood Number (Sh) is the ratio of the convective mass transfer and the diffusive mass transport, and so it tells us about how much mass transfer is enhanced due to fluid motion. It is a nondimensional mass transfer coefficient, which is mathematically defined as Sh = −∂S/∂n, where n and S denote normal direction to a plane and nondimensional concentration, respectively. The local Sherwood number at the top, bottom, left and right walls of the cavity are, respectively, given by and the average Sherwood number at bottom, left, right walls and average Sherwood numbers at the corresponding walls of the cavity are given by respectively, where dS 1 and dS 2 are the small elemental lengths along the left and right walls, respectively. Here, since top wall is considered to be of zero diffusive gradient, so Sh t = 0.

Entropy generation
Entropy, generated by fluid friction, heat transfer and mass transfer are discussed by Mourad et al. [21], and the effect of magnetic field on entropy generation has been discussed by Hussain [4]. The dimensionless forms of local entropy generations due to fluid friction (S ψ ), thermal gradients (S θ ), diffusion (S τ ) and magnetic field (S M ) are, respectively, given by where λ 1 , λ 2 and λ 3 are called irreversibility distribution ratios. In the present study, the values of λ 1 , λ 2 and λ 3 are taken as 10 −4 , 0.5 and 0.01, respectively.

Coordinate transformation and numerical procedure
Application of boundary conditions at various boundaries of an enclosure with complex geometry is a difficult tusk. In order to reduce the complexity of such situation, we use transformations to get a simple computational region. In the present study, the physical domain (trapezoidal domain in x, y-plane) has been transformed into the computational domain (square in ξ, η-plane) by using the following transformations (see Fig. 1): where A is the aspect ratio, ξ and η are horizontal and vertical coordinates in a unit square. Now it becomes very easier to obtain the finite-difference representation of the governing equations in the computational domain to solve the equations. Using the transformation (14), Eqs. (5)-(9) are transformed as: F π ξξ + Eπ ξη + 1 F τ ξξ + Eτ ξη + 1 and the expressions of entropy generation due to fluid friction (10), heat transfer (11), mass transfer (12) and magnetic effect (13) can be expressed, respectively, in ξ, η-coordinate system as where expressions for E, F , G, H, T 1 , T 2 , T 3 , T 4 , T 5 , T 6 , T 7 , B K and B KC are given in Appendix. The total entropy generation is given by Since Eqs. (15)- (19) are highly nonlinear, no analytical method is available to solve them. So, Eqs. (15)- (19) are solved numerically using finite-difference approximation method. Using second-order approximations of different derivatives of stream function, temperature, concentration, heat function and mass function (see [20]), the discretizations of Eqs. (15)- (19) can be written in matrix form as (τ ξ ) ij and (τ η ) ij as discussed by Mahapatra et al. [20]. This constitutes one outer cycle. We utilize a relaxation parameter λ inside both the inner as well as outer cycles for ψ, θ, S, π and τ . Then u i,j and v i,j are obtained from the relation of stream function with velocity components. The process continues till the maximum values of ψ error , θ error , S error , π error and τ error between two iterations are less than 0.5 × 10 −6 . At last, the entropy generations (S ψ ), (S θ ), (S τ ) and (S M ) are obtained from the discretized forms of Eqs. (10)-(13).

Results and discussion
The detailed discussion of stream functions, isotherms, isoconcentrations, heat lines, mass lines and entropy generation minimization for mixed convection flow within a lid-driven trapezoidal enclosure are illustrated in the present study. Till now entropy generation due to fluid friction, heat transfer, mass transfer and magnetic field in a lid-driven trapezoidal enclosure with various aspect ratios have not been discussed in any earlier researches. In order to check the accuracy of present result, a comparison has been made with the bench mark numerical solutions given by Basak et al. [3] and Davis [11]. The comparison has been made by the values of |ψ| max and Nu (Table 1) within a square cavity with hot left wall, cold right wall, adiabatic top and bottom walls with Pr = 0.71 and Ra = 10 3 , 10 4 , 10 5 . The computation has been done using grid size 81 × 81 and gives an excellent agreement with the results of Basak et al. [3] and Davis [11]. Also two graphical comparisons of stream function (ψ) and isotherm (θ) contours with Basak et al. [3] (Fig. 2) and the entropy generation contours due to fluid friction (S ψ ) and heat transfer (S θ ) with Hussain [15] and Ilis et al. [17] (Fig. 3) have been carried out. The excellent agreement can be seen. To check the convergence of the solution grid independence tests of |ψ| max , |π| max and |τ | max with different grid sizes have been performed (Table 2) in which almost same results have been found for different grid sizes.      As aspect ratio increases, it is observed that entropy generation increases due to fluid friction and due to the effect of magnetic field, but decreases due to heat transfer and mass transfer. The effect of entropy generation due to heat and mass transfer are much higher than that due to fluid friction and magnetic field. Therefore, to increase the efficiency of any system, the aspect ratio of the enclosure should be higher as much as possible, which minimizes the total entropy generation. The contours of θ and S are smooth and converges to the right bottom corner, the junction of hot-massive and cold-nonmassive walls. So, the significant entropy generation due to heat and mass transfer is observed in the right bottom corner with higher values. The contours of S ψ are dense along the top wall because of high velocity gradient due to adiabatic top wall, whereas the contours of S M are dense in the right portion of the cavity with multiple circulation eddies for A = 0.5, 1.0, and for A = 1.5, they are distributed through out the cavity with small values of S M at left bottom portion of the cavity. Contours of S T are smoothly distributed for A 1 and become complex for high aspect ratio forming a strong zone of entropy generation at right bottom corner. Figure 5 represents ψ, θ, S, π, τ , S ψ , S θ , S τ , S M and S T contours for Ra    Fig. 6 are almost similar to the previous observation in Fig. 5 for Ra = 10 3 and 10 4 , but when Ra = 10 5 , the contours become asymmetric due to high thermal and mass diffusivity. Heat line and mass line contours also change for large values of Ra (10 4 , 10 5 ) and make two sided heat and mass circulation cells with |π| max = 0.2564 (Ra = 10 3 ); 0.7761 (Ra = 10 4 ); 3.9372 (Ra = 10 4 ), and |τ | max = 0.2577 (Ra = 10 3 ); 0.8516 (Ra = 10 4 ); 3.7714 (Ra = 10 4 ) due to the effect of high convection. As Pr is the ratio of momentum diffusivity to thermal diffusivity, so large Pr implies momentum diffusivity is large compared to thermal diffusivity.
Nusselt number determines the quantity of heat exchanged between the fluid and the boundary, i.e., the ratio of convective to conductive heat transfer across the boundary.  As Nusselt number, a heat transfer coefficient represents the quality of heat transfer rather than the quantity at the boundary, it is observed that at the uniformly heated bottom wall of the cavity the qualitative heat transfer Nu b is strictly increasing with respect to Ra and also increases as inclination angle φ increases. But along the linearly heated left wall, Nu l is decreasing slowly for φ = 45 • and 60 • and decreasing in a quantitatively higher rate for φ = 90 • . It is also observed that Nu l decreases with the increment of φ. The same observations as in the case of Nu b follow for Sh b but it is observed that Sh l remains almost uniform for the increment of Ra when Ra is large with approximate values Sh l = 0.2 (for φ = 45 • ), Sh l = 0.32 (for φ = 60 • ) and Sh l = 0.6 (for φ = 90 • and Ra 20000). Therefore, it is clearly depicted in Fig. 7 that heat and mass transfer in the bottom wall of the cavity are much greater than that along left wall due to uniform heat and mass transfer at the bottom wall and linear heat and mass transfer at left wall.  Total entropy generation throughout the cavity has huge importance for the efficiency of any system. In Fig. 8, the variation of total entropy generation is shown with respect to the change of Rayleigh number and Hartmann number for various values of inclination angle and aspect ratio of the cavity, respectively. From Fig. 8(a) it is observed that for a fixed value of φ, S T is increasing very quickly with the increment of Ra and when the value of φ increases, S T decreases.
In Fig. 8(b), the variation of S T with Ha for various values of A are shown. It is clear from the figure that the total entropy generation is decreasing with the increment of Ha. It is also observed that as A increases, the total entropy generation decreases. Therefore, when the intensity of magnetic field is very high, cavities of high aspect ratios are preferable to minimize the total entropy generation.

Conclusions
In the present study, the analysis of stream lines, heat lines, mass lines and entropy generation due to fluid friction, heat transfer, mass transfer and magnetic field within a trapezoidal enclosure with various inclination angles and various aspect ratios with liddriven upper wall has been carried out. From the present study the following conclusions can be drawn: • When the aspect ratio is low, higher values of φ leads to an intensification of total entropy generation. Therefore, in case of air cooling system (specially for the cooling of electronic components) or any system with low Pr the enclosure of small φ together with small aspect ratio (A) can be used to reduce the loss of energy and to get maximum efficiency of the system. • In the case of large Pr (e.g., ogive or engine oil with Pr = 1000), it is noticed that the absolute values of stream function, heat function, mass function, entropy due to fluid friction and magnetic field are highly increasing, whereas entropy generation due to thermal gradient and mass gradient are uniform with the increment of Rayleigh number and ultimately the total entropy generation is increasing. • The effect of magnetic field is observed as follows. As the value of Ha is increasing total entropy generation is decreasing and the increased value of aspect ratio leads to decrease of an intensification of total entropy generation. Therefore, the cavity with high aspect ratio should be used when the intensity of the magnetic field is high. • As Ra increases both heat transfer coefficient in terms of average Nusselt number at bottom wall (Nu b ) and mass transfer coefficient in terms of average Sherwood number at the bottom wall (Sh b ) increase for increased values of φ, which leads to high heat and mass transfer at the bottom wall but in the left wall Nu l and Sh l are almost uniform.