Entropy Generation due to MHD Mixed Convection of Nanofluid in a Vertical Channel with Joule Heating and Radiation effects

Abstract—In this article, the effects of Joule heating and radiation on entropy generation due to laminar mixed convective flow of an incompressible electrically conducting nanofluid in vertical channel is studied. The governing equations are non-dimensionalized and then solved by using homotopy analysis method. The effects of magnetic, Joule heating, Brinkman number and radiation on the velocity in flow direction, temperature, nanoparticle concentration, Bejan number and entropy generation are analyzed and presented geometrically. Keyword Entropy generation, Mixed convection, Radiation, Joule heating, MHD, Nanofluid, Vertical channel

The study of entropy generation, the amount of irreversibility associated with the real processes, has been growing interest as it is a powerful and useful optimization tool for a high range of thermal applications. The entropy generation destroys the system energy. Hence, the performance of the system can be improved by decreasing the entropy generation. Bejan [15,16] introduced the entropy generation minimization method and developed its applications in engineering sciences. Since then several researchers have been studying the entropy generation analysis for different types of geometries with diverse fluids. Omid et al. [17] analyzed the significance of radiation parameter effect on entropy generation within nanofluids. Dehsara et al. [18] numerically analyzed the entropy generation in nanofluid flow in presence of variable magnetic field, viscous dissipation and solar radiation. Govindaraju et al. [19] investigated the entropy generation in MHD flow of a nanofluid. Rashidi et al. [20] presented the influence of thermal radiation on entropy generation in MHD blood flow of a nanofluid.
The survey of the literature reveals that the interaction of Joule heating and radiation effects with nanoparticles in a mixed convectional heat transfer flow in a vertical channel occupied by nanofluid has not been considered. Hence, the main aim of this article is to investigate the mixed convective flow of a nanofluid passing through a vertical channel with Joule heating and thermal radiation effect. The homotopy analysis method is used to solve the nonlinear differential equations. HAM [21] was first introduced by Liao, which is one of the most powerful technique to find the solution of strongly nonlinear equations. The effect of radiation, Joule heating and magnetic parameter on the velocity along the fluid direction, temperature, nanoparticle concentration, Bejan number and entropy generation is investigated..

II. PROBLEM FORMULATION
Consider a steady, laminar incompressible electrically conducting nanofluid flow through a vertical channel of width 2d. The x-axis is taken to be in vertically upward direction of the flow through the central line of the channel and y-axis in the orthogonal direction to the flow as shown in figure 1. The plate y=-d is maintained at temperature T 1 and nanoparticle volume fraction  1 , whereas the plate y=d is maintained at T 2 and  2 respectively. Rate of flow through the plates is assumed to be constant v 0 and uniformly equal for both the plates. A uniform magnetic field B 0 is taken in y-direction. A uniform pressure gradient in x-direction and buoyancy forces caused the mixed convective flow. The induced magnetic field is ignored as the magnetic Reynolds number is very low and all the fluid properties are considered as constant apart from the density in the buoyancy term. The fluid is considered to be a gray, absorbing/emitting radiation, but non-scattering medium. The Rosseland approximation [22] is used to describe the radiative heat flux in the energy equation. u is the velocity along x-direction, T denotes the temperature and  denotes the nanoparticle concentration. Under these assumptions, the governing equations of the flow by considering the nanofluid model proposed by Buongiorno [23] are as follows: 2   2  3  2  2  2 2  0  2  2   16  2  1  3   m  T  B  p  f  p  m  p   T  D  T  T  u  T  T  T  v  D  B u  y  c  y  K  c  y y T  y  c  y where the density is ρ, the pressure is p, the electrical conductivity is σ, Brownian diffusion coefficient is D B , the specific heat capacity is Cp, the viscosity coefficient is μ, the coefficients of thermal expansion is β T , the effective thermal diffusivity is α , the acceleration due to gravity is g, the thermophoretic diffusion coefficient is D T , Stefan-Boltzman constant is σ 1 and coefficient of mean absorption is χ.
The conditions on the boundary are: Introducing the following non-dimensional variables in equations (1) -(5), we get the nonlinear differential equations as where the kinematic viscosity coefficient is  , III. HOMOTOPY SOLUTION To obtain the HAM solution (For more details on homotopy analysis method see the works of Liao [21,[24][25][26] ), first we guess the initial values as and the auxiliary linear operators asL i =  2 /y 2 , for i = 1,2,3 such that L 1 (c 1 + c 2 y) = 0, L 2 (c 3 + c 4 y) = 0 and L 3 (c 5 + c 6 y) = 0, where c i ,( i = 1,2, . . .6) are constants. The zero th order deformation, which is given by . .
Thus, as p varying from initial value 0 to final value 1, f , θ and S changes from f 0 , θ 0 and S 0 to the final solution f(η), θ(η) and S(η). Using Taylor's series one can write and choose the values of the auxiliary parameters for which the series (15) are convergent at p=1 i.e., is convergent. The equivalent boundary conditions are The m th -order deformation is given by where   2 1 1 1 for 1 m   IV. CONVERGENCE In HAM, it is essential to see that the series solution converges. Also, the rate of convergence of approximation for the HAM solution mainly depend on the values of h. To find the admissible space of the auxiliary parameters, h-curves are drown for 20 th -level of approximation and shown in figure 2. It is oserved from this figure that the permissible range for h 1 , h 2 and h 3 is -0.6 < h 1 < -0, -0.7< h 2 < -0 and -1.0 < h 3 < -0.2, respectively. In order to obtain the optimal values of the auxiliary parameters, the following average residual errors [25] are computed and found that these errors are least at h 1 = -0. 3, h 2 = -0.64 and h 3 = -0.73. 2 2 where 1/ t k   and k = 5. Also, for different values of m the series solutions are calculated and noticed that the series (16) converges in the total area of η. Further, the graphs of the ratio in contrast to the number of terms m in the homotopy series are calculated and observed that the series (16) converges to the exact solution.
V. ENTROPY GENERATION The volumetric rate of local entropy generation of a nanofluid in vertical channel can be expressed as where R is the universal gas constant and D is the species diffusivity through the fluid. The entropy generation number Ns is the ratio of the volumetric entropy generation rate to the characteristic entropy generation rate according to Bejan [16]. Therefore Ns is given by The dimensionless coefficients are 2  and 3  , called irreversibility distribution ratios which are related to diffusive irreversibility, given by The entropy generation due to heat transfer irreversibility is denoted by the first term on the right hand side of the eq.(25) and the entropy generation due to viscous dissipation is represented by second term of eq.(25). The ratio of the entropy generation (Nh) and the total entropy generation (Nh+Nv) is called Bejan number (Be). To understand the entropy generation mechanisms the Bejan number Be is specified. The Bejan number for this problem can be expressed as In general the limits of Bejan number is 0 to 1. Finally, the irreversibility due to viscous dissipation dominant represent by Be = 0, whereas Be = 1 represents the domination of heat transferirreversibility on Ns. It is clear that the heat transfer irreversibility is equal to viscous dissipation at Be=0.5.

VI. RESULTS AND DISCUSSION
The effects of radiation, Joule heating, magnetic and Brinkman number on non-dimensional velocity, temperature and nanoparticle volume fraction, Bejan number Be and entropy generation Ns are presented graphically in figures 3 -7. To study these effect of the parameters taking computations asNr = 1, Nb =0.5, Gr = 10, Re = 2, R = 1, Pr = 1, A = 1, Le = 1, Tp = 0.1. Figure (3) displays the effect of the radiation parameter Rd on velocity in flow direction, temperature, nanoparticle concentration, entropy generation and Bejan number. Figure (3a) reveals that the velocity increased as enhancement in the radiation Rd. This indicates that Rd have a retarding impact on the mixed convective flow. From (3b), it is noticed that θ(η) increased with an increment in the radiation Rd. A raise in the radiation Rd parameter leads to release of heat energy in the flow direction, therefore the fluid temperature increased. Figure (3c) depicts that the nanoparticle concentration S(η) decays with an enhancement in the radiation Rd. Figure (3d) shows that entropy generation reduces with an enhancement in the radiation parameter Rd. It is noticed from figure (3e) that Be (Bejan number) is increasing near the lower plate of the channel, meanwhile far away from the plate the trend is reversed due to more contribution of the heat transfer irreversibility on Ns and Be decreasing near the upper plate of the channel with enhance in Rd.
The variation of velocity in flow direction, temperature, nanoparticle concentration, Ns and Be with magnetic parameter M is presented in figure (4). It is noticed from figure (4a) that, the dimensionless velocity decreases with an increase in the magnetic parameter M. Figure ( Figure (6d) shows that the increment in Joule heating parameter J raises the entropy generation. As Joule heating parameter J increases, the Bejan number is decreasing near the lower plate of the channel, meanwhile far away from the plate the trend is reversed due to more contribution of heat transfer irreversibility on Ns and then Be increasing near the upper plate of the channel as represented in figure (6e). Figure (7) represents the effect of thermoporesis parameter Nt on velocity in flow direction f(η), nanoparticle concentration S(η), temperature θ(η), Bejan number Be and entropy generation Ns. The velocity f(η) decreased with enhancement in the thermoporesis parameter Nt as shown in figure (7a). Figure (7b) reveals that, the temperature θ(η) increased with enhancement in thermoporesis parameter Nt. Increase in the thermoporesis parameter Nt leads to increase in the effective-conductivity, hence the nanoparticle concentration S(η) is increases as depicted in figure (7c). It is noticed from figure (7d) that as increase in the thermoporesis parameter Nt the entropy generation is also increased. It is seen from figure (7e) that the Be is decreasing near the lower plate of the channel, meanwhile far away from the plate the trend is reversed due to more contribution of heat transfer irreversibility on Ns and then Be is increasing near the upper plate of the channel as Nt increases.

VII.
CONCLUSION In this article the laminar entropy generation in mixed convective nanofluid flow in a vertical channel has been investigated by including magnetic, Joule heating, Brinkman number and chemical reaction parameter effects. The non-dimensional non-linear equations are solved by the HAM procedure. The main observations are summarized below: • The dimensionless velocity, temperature increased whereas the nanoparticle concentration and entropy generation decreased with raise in the thermal-radiation Rd. • An increase in the Brinkman number leads to increase the velocity, temperature, entropy generation and leads to decrease the nanoparticle concentration.
• As the Joule heating parameter increases, the dimensionless temperature, velocity and entropy generation increases but the nanoparticle concentration decreases. • The maximum values of Bejan number are observed at upper plate of the channel due to more heat transfer irreversibility on Ns and minimum value near the lower plate of channel due to more contribution of the fluid friction irreversibility on Ns with the increase in Rd, M, Br, J and Nt.