Effect of Magnetic Field on Entropy Generation Due to Micropolar Fluid Flow in a Rectangular Duct

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Procedia Engineering 127 ( 2015 ) 1150 – 1157

Peer-review under responsibility of the organizing committee of ICCHMT – 2015 doi: 10.1016/j.proeng.2015.11.443

ScienceDirect

International Conference on Computational Heat and Mass Transfer-2015

E

ﬀect of Magnetic ﬁeld on Entropy Generation due to Micropolar

Fluid Flow in a Rectangular duct

D. Srinivasacharya

∗

, K. Hima Bindu

Department of Mathematics, National Institute of Technology, Warangal-506004, Telangana, India.

Abstract

The entropy generation due to heat transfer, fluid friction and magnetic field has been calculated for the steady, incompressible micropolar fluid flow in a rectangular duct with constant wall temperatures. An external uniform magnetic field is applied which

is directed arbitrarily in a plane perpendicular to the ﬂow direction. The governing partial diﬀerential equations of momentum,

angular momentum and energy are solved numerically using Finite diﬀerence method. The obtained velocity, microrotation and temperature distributions are then used to evaluate the entropy generation and Bejan number. Finally the graphs are presented to show the inﬂuence of various physical parameters on entropy generation rate and Bejan number.

c

2015 The Authors. Published by Elsevier Ltd.

Peer-review under responsibility of the organizing committee of ICCHMT – 2015.

Keywords: Rectangular duct; Magnetic parameter; Micropolar ﬂuid; Entropy generation; Bejan number.

1. Introduction

The present trend in the field of heat transfer and thermal design is to conduct second law (of thermodynamics) analysis including, design related concept of entropy generation and its minimization. Entropy generation is a measure of irreversibility associated to the real process. Entropy generation is present in all heat transfer processes. The Magnetic effect, viscous effect, heat transfer down temperature gradient etc., are responsible for the generation of entropy. The entropy generation is associated in many energy related applications such as geothermal energy systems, cooling of modern electronic systems and solar power collectors.

Initially Bejan ([1],[2],[3]) analysed the entropy generation in heat transfer and fluid flow processes. Since then several researches have been conducted theoretically on the entropy generation under various flow configurations. The fluid dynamics and heat transfer behaviour of laminar flow through non-circular ducts is an area of special interest as it has got wide applications for such geometries in compact heat exchangers. Among these heat transfer enhancement in a rectangular duct is of great interest and importance in many industrial applications like heat exchangers, cooling devices and gas turbines because of higher heat transfer rates increase the efficiency of a system and reduce thermal

∗ _{Corresponding author. Tel.:}_{+91-9849187249; fax: +91-870-2459547.}

E-mail address: dsc@nitw.ac.in

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load. Narusawa [4] investigated the mixed convection and entropy generation numerically in three dimensional rect-angular duct with heating at the bottom. Oztop [5] studied the entropy generation in a semicircular duct with constant heat flux. Ko and Ting [6] analysed the entropy generation in a curved rectangular duct caused by forced convection with external heating. Haji-Sheikh [7] considered the fully developed forced convection in a duct of rectangular cross section. Hooman et al. [8] analysed heat transfer and optimization of entropy generation in porous saturated ducts of rectangular cross section. Jarungthammachote [9] studied the entropy generation for laminar fluid flow through a hexagonal duct. Yang et al. [10] investigated heat transfer and entropy generation in the entrance region of a three di-mensional vertical rectangular duct. They noticed that increase in Reynolds number increases the entropy generation rate due to heat transfer and fluid friction in the channel. Leong and Ong [11] discussed the characteristics of entropy generation in various shapes of cross section ducts with constant heat flux.

However, it is quite common to encounter non-Newtonian fluids in many industries such as chemical, pharma-ceutical and food industries. Thus, it is important to understand the heat transfer behaviour of such non-Newtonian fluids flowing through non-circular ducts. Among these the micropolar fluid theory has received more attention than other non-Newtonian fluids because of its distinct features. The micromotion of fluid elements and its local structure are responsible for microscopic effects. This aspect is presented in the micropolar fluid model introduced by Eringen [12]. The rigid fluid particles which are randomly oriented, suspended in a viscous medium and whose deformation is neglected are represented in the micropolar fluid model. The flow characteristics of haematological and colloidal suspensions, polymeric additives, liquid crystals etc. accurately resemble the micropolar fluids. The study of microp-olar fluid flows with heat transfer has important engineering applications such as in electric transformers, transmission lines, power generators, heating elements, and refrigeration coils.

Nomenclature

a length of the duct in the x-direction

A, B micropolar parameters

b length of the duct in the y-direction

B0 Magnetic induction Br Brinkman number Ha Hartman number Kf thermal conductivity l non-dimensional parameter m micropolar parameter N coupling number

NF entropy generation due to viscous dissipation

NH entropy generation due to heat transfer

NM entropy generation due to magnetic eﬀect

Ns dimensionless entropy generation number

y0 =b/a

Greek Symbols

α material constant(viscosity coefficient) β material constant(viscosity coefficient) γ material constant(viscosity coefficient) κ material constant(viscosity coefficient) ρ density of the fluid

ν microrotation component σ electric conductivity of the ﬂuid

Most of the studies reported in the literature are on entropy generation in a duct without considering the micropolar fluid flow and the magnetic field. An electrically conducting fluid in a duct with the presence of a magnetic field has special technical significance because of its frequent occurrence in many industrial applications such as cooling

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of nuclear reactors, MHD marine propulsion, electronic packages, microelectronic devices etc. The main aim of this work is to investigate the effects of magnetic field on entropy generation with micropolar fluid flow in a rectangular duct. The governing equations are simplified and solved using Finite difference method. The effects of coupling number, Hartman number and Brinkman number on entropy generation and Bejan number are presented through graphs.

2. Mathematical formulation

Consider a steady flow of incompressible micropolar fluid through a rectangular duct with uniform cross-section. Choose the coordinate system such that z- axis along the axis of the duct, x, y-axis along the sides and origin at the centre of the duct. Let a and b be the lengths of the duct in x and y-directions respectively. An external uniform magnetic field is applied in a plane normal to the z -axis, which has a constant magnetic flux density B0 that is

assumed constant by taking the magnetic Reynolds number much smaller than the fluid Reynolds number. This is a reasonable assumption for the flow of certain working fluids in novel nuclear-MHD propulsion systems and in nuclear engineering liquid metals, e.g., liquid sodium [13]. The four walls are kept at uniform temperature T1. The velocity

vector of the ﬂuid is given asq = w(x, y)k and the microrotation vector asν = ν1(x, y)i+ ν2(x, y)j. Under these

assumptions the governing equations for the MHD ﬂow of micropolar ﬂuid in the absence of both body force and body couple are

−∂p_∂z + κ ∂ν2 ∂x −∂ν 1 ∂y + (μ + κ) ∂2_w ∂x2 + ∂2_w ∂y2 − σB2 0w= 0 (1) −2κν1+ κ∂w_∂y − γ_∂y∂ ∂ν2 ∂x − ∂ν1 ∂y + (α + β + γ)_∂x∂ ∂ν1 ∂x + ∂ν2 ∂y = 0 (2) −2κν2− κ∂w_∂x+ γ_∂x∂ _∂ν 2 ∂x − ∂ν1 ∂y + (α + β + γ)_∂y∂ _∂ν 1 ∂x + ∂ν2 ∂y = 0 (3) Kf _∂₂ T ∂x2 + ∂2_T ∂y2 + (μ + κ) ⎡ ⎢⎢⎢⎢⎢ ⎣ _∂w ∂x 2 + _∂w ∂y 2⎤ ⎥⎥⎥⎥⎥ ⎦ + 2κ ν2 1+ ν22− ν1∂w_∂y + ν2∂w_∂x + α _∂ν 1 ∂x + ∂ν2 ∂y 2 + γ ⎡ ⎢⎢⎢⎢⎢ ⎣ _∂ν 1 ∂x 2 + _∂ν 2 ∂x 2 + _∂ν 1 ∂y 2 + _∂ν 2 ∂y 2⎤ ⎥⎥⎥⎥⎥ ⎦ + β ⎡ ⎢⎢⎢⎢⎢ ⎣ _∂ν 1 ∂x 2 + 2∂ν1 ∂y ∂ν2 ∂x + _∂ν 2 ∂y 2⎤ ⎥⎥⎥⎥⎥ ⎦ = 0 (4)

In the present problem the flow variables are invariant in the flow direction (z-direction), except the pressure gradient ∂p_∂z, which is a constant. μ, κ, α, β and γ are the material constants(viscosity coefficients) which satisfy the following inequalities:

κ ≥ 0, 2μ + κ ≥ 0, 3α + β + γ ≥ 0, γ ≥| β | (5)

Introducing the following non-dimensional variables

x= a x, y = a y, w = U w, ν1= U aν 1, ν2= U aν 2, T− T1 T2− T1 = θ, ∂p ∂z = ρU2 a p0 (6)

into the equations (1) to (4) and dropping tildes we get −Rep0+ N 1− N ∂ν2 ∂x −∂ν 1 ∂y + 1 1− N ∂2_w ∂x2 + ∂2_w ∂y2 − Ha2_w_{= 0} ₍₇₎ −ν1+ 1 2 ∂w ∂y − 1 2 2− N m2 ∂ ∂y ∂ν2 ∂x −∂ν 1 ∂y + 1 l2 ∂ ∂x ∂ν1 ∂x +∂ν 2 ∂y = 0 (8) −ν2− 1 2 ∂w ∂x + 1 2 2− N m2 ∂ ∂x ∂ν2 ∂x − ∂ν1 ∂y + 1 l2 ∂ ∂y ∂ν1 ∂x + ∂ν2 ∂y = 0 (9)

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∂2_θ ∂x2+ ∂2_θ ∂y2 + Br ⎡ ⎢⎢⎢⎢⎢ ⎣₁_{− N}1 ⎛ ⎜⎜⎜⎜⎜ ⎝ ∂w ∂x 2 + ∂w ∂y 2⎞ ⎟⎟⎟⎟⎟ ⎠ +₁2N_{− N} ν2 1+ ν22− ν1∂w ∂y + ν2∂w ∂x + A ∂ν1 ∂x + ∂ν2 ∂y 2⎤ ⎥⎥⎥⎥⎥ ⎦ + Br ⎡ ⎢⎢⎢⎢⎢ ⎣_mN(22₍₁− N)_{− N)} ⎛ ⎜⎜⎜⎜⎜ ⎝ ∂ν1 ∂x 2 + ∂ν2 ∂x 2 + ∂ν1 ∂y 2 + ∂ν2 ∂y 2⎞ ⎟⎟⎟⎟⎟ ⎠ + B ⎛ ⎜⎜⎜⎜⎜ ⎝ ∂ν1 ∂x 2 + 2∂ν1 ∂y ∂ν2 ∂x + ∂ν2 ∂y 2⎞ ⎟⎟⎟⎟⎟ ⎠ ⎤ ⎥⎥⎥⎥⎥ ⎦ = 0 (10) where N = κ

μ + κ is the coupling number, m2 =

a2_{κ(2μ + κ)}

γ(μ + κ) is the micropolar parameter, l2 = 2a2_κ

α + β + γ is the non-dimensional parameter, Ha= B0a

_σ

μ is the Hartman number, Br=

μU2

Kf(T2− T1)

is the Brinkman number, and

A= α

μa2, B=

β

μa2 are micropolar parameters. The usual no-slip and hyper stick boundary conditions are given by

w= 0 at x = ±1 and y = ±y0 where y0=

b

a (11a)

ν1= ν2= 0 at x = ±1 and y = ±y0 (11b)

θ = 0 at x = ±1 and y = ±y0 (11c)

3. Entropy Generation

Non-equilibrium conditions arise due to the exchange of energy, magnetic field and momentum with in the fluid and at the walls of the duct. This causes a continuous entropy generation. The mechanisms of entropy generation are heat transfer, fluid friction and magnetic effects. The volumetric rate of entropy generation for incompressible micropolar fluid given as

SG= Kf T2 1 ⎡ ⎢⎢⎢⎢⎢ ⎣ _∂T ∂x 2 + _∂T ∂y 2⎤ ⎥⎥⎥⎥⎥ ⎦ +μ + κ_T 1 ⎡ ⎢⎢⎢⎢⎢ ⎣ _∂w ∂x 2 + _∂w ∂y 2⎤ ⎥⎥⎥⎥⎥ ⎦ +2κ_T 1 ν2 1+ ν22− ν1∂w_∂y + ν2∂w_∂x + α T1 _∂ν 1 ∂x + ∂ν2 ∂y 2 +γ T1 ⎡ ⎢⎢⎢⎢⎢ ⎣ ∂ν1 ∂x 2 + ∂ν2 ∂x 2 + ∂ν1 ∂y 2 + ∂ν2 ∂y 2⎤ ⎥⎥⎥⎥⎥ ⎦ +_Tβ 1 ⎡ ⎢⎢⎢⎢⎢ ⎣ ∂ν1 ∂x 2 + 2∂ν1 ∂y ∂ν2 ∂x + ∂ν2 ∂y 2⎤ ⎥⎥⎥⎥⎥ ⎦ +σB 2 0 T1 w2 (12) which can be expressed in its dimensionless form by the following expression:

Ns= SG SGC = _∂θ ∂x 2 + _∂θ ∂y 2 + Br Tp ⎡ ⎢⎢⎢⎢⎢ ⎣₁_{− N}1 ⎛ ⎜⎜⎜⎜⎜ ⎝ _∂w ∂x 2 + _∂w ∂y 2⎞ ⎟⎟⎟⎟⎟ ⎠ +₁2N_{− N} ν2 1+ ν22− ν1∂w_∂y + ν2∂w_∂x + A _∂ν 1 ∂x + ∂ν2 ∂y 2⎤ ⎥⎥⎥⎥⎥ ⎦ +Br Tp ⎡ ⎢⎢⎢⎢⎢ ⎣_mN(22₍₁− N)_{− N)} ⎛ ⎜⎜⎜⎜⎜ ⎝ ∂ν1 ∂x 2 + ∂ν2 ∂x 2 + ∂ν1 ∂y 2 + ∂ν2 ∂y 2⎞ ⎟⎟⎟⎟⎟ ⎠ + B ⎛ ⎜⎜⎜⎜⎜ ⎝ ∂ν1 ∂x 2 + 2∂ν1 ∂y ∂ν2 ∂x + ∂ν2 ∂y 2⎞ ⎟⎟⎟⎟⎟ ⎠ ⎤ ⎥⎥⎥⎥⎥ ⎦ +_TBr_pHa2_w2 ₍₁₃₎ where SGC= Kf(T2− T1)2 a2_T2 1

is the characteristic entropy generation rate, Tp=

T2− T1

T1

is the dimensionless tempera-ture diﬀerence. The equation (13) can be expressed alternatively as follows

Ns= NH+ NF+ NM (14) The first term on the right hand side of this equation denotes the entropy generation due to heat transfer irreversibil-ity, the second term represents the entropy generation due to fluid friction irreversibility and the third term represents the entropy generation due to magnetic field.

To evaluate the irreversibility distribution, the another parameter Be (Bejan number), which is the ratio of heat transfer entropy generation to the overall entropy generation is deﬁned as follows

Be= NH Ns

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The Bejan number varies from 0 to 1. Subsequently, Be= 0 reveals that the fluid friction irreversibility and irreversibil-ity due to magnetic field are dominate over heat transfer irreversibilirreversibil-ity, whereas Be= 1 indicates the domination of heat transfer irreversibility on entropy generation. It is evident that Be= 0.5 is the case in which the contribution of irreversibility due to heat transfer is same as the sum of fluid friction and magnetic field irreversibilities.

4. Numerical details

The governing equations given by Eqs. (7) to (10) along with the boundary conditions Eq. (11) are solved numer-ically using Finite Difference Method. The derivatives are replaced by the central difference approximations to get the algebraic system of equations and these equations are solved using Gauss-Seidel iteration method. A numerical experiment was conducted with various meshes in the rectangular region and axial step lengths in x and y directions to check the independence of the mesh resolution of the numerical results. There are three mesh distributions tested in the analysis. They are 21× 21, 41 × 41 and 81 × 81 respectively. It is found that the deviations in the velocity, microrotation and temperature components calculated with 41× 41 and 81 × 81 are always less than 10−3. Therefore the computations with mesh distribution of 41× 41 are considered to be sufficiently accurate to describe the flow in this study. The obtained velocity, microrotation and temperature distributions are then used to evaluate the entropy generation and Bejan number.

5. Results and Discussion

Magneto hydrodynamic flow and heat transfer in a micropolar fluid flow through a rectangular duct has been solved numerically using Finite Difference method. Numerical expressions of velocity, microrotation and temperature have been used to compute the entropy generation. These quantities are evaluated numerically by dividing the rectangu-lar region into a grid of mesh points (xi, yj). The effects of various parameters like coupling number(N), magnetic parameter(Ha) and Brinkman number(Br) on entropy generation and Bejan number are described graphically.

Figure.1 shows the proﬁles of entropy generation and Bejan number for y0 = 1, Ha = 2, m = 1, l = 0.5, N =

0.25, Re = 1, p0 = 1, Tp = 1, Br = 1, A = 1 and B = 0.1. It is clear from Figure 1(a) that the maximum mag-nitude of the entropy generation is observed at the boundaries of the rectangular duct and decreases at the center of the duct due to their suppressing effect on the flow and thermal fields. Furthermore, a similar phenomenon also occurs in 2D. Fig. 1(b) shows the Bejan number profile in the rectangular duct. It is observed that the fluid friction irreversibility dominates at the center of the duct and heat transfer irreversibility dominates at the boundary of the duct. The effect of coupling number on entropy generation is shown in Figs. 2(a) and 2(b) in x and y-directions. The coupling of linear and rotational motion arising from the micromotion of the fluid molecules are characterised by coupling number. Hence, the coupling between the Newtonian and rotational viscosities is represented by N. The microstructure effect is significant as N → 1, and for a smaller value of N the substructure individuality is limited. The fluid is non polar as its micropolarity is lost atκ → 0 i.e. N → 0. Thus, for viscous fluid N → 0. It is observed that the entropy generation decreases with increase in the values of N in x and y-directions .The peak value of entropy generation is noticed at the boundaries of the rectangular duct. It is seen from Figs. 2(c) and 2(d) that the Bejan number initially increases and then decreases with increase in the values of N in both x and y directions. We notice that the fluid friction irreversibility dominates around center of the duct.

Figs. 3(a)-3(d) present the magneto hydrodynamic effect on entropy generation and Bejan number in x and y di-rections. It is observed from Figs. 3(a) and 3(b) that the entropy generation near the boundaries of the duct is higher when compared to the center region of the duct, this is due to maximum gradients in velocity and temperature. With increase of Hartman number the entropy generation rises at the centre of the duct, since in this region the velocities are maximum and hence the contribution of MHD flow is maximum on the entropy generation. Like-wise near the boundaries of the duct less entropy generation is observed when Ha is more, because increase in the Hartman number implies to decrease the fluid friction irreversibility. Figs. 3(c) to 3(d), shows that the Bejan number decreases when

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Hartman number increases.

Figure. 4 analyses the effect of Brinkman number on entropy generation and Bejan number in x and y directions. Figs. 4(a) and 4(b) shows that the entropy generation increases with increase of Br in the entire rectangular duct. Entropy generation number is high in magnitude at the boundaries of the duct due to the presence of high temperature and velocity gradients. N s profiles are similar in shape and almost parallel to one another for any parameter but they do vary in magnitude. The Bejan number indicates the domination of irreversibility heat transfer, fluid friction on entropy generation. Figs. 4(c) and 4(d) shows that the Bejan number increases as Brinkman number increases due to decrease in viscous dissipation irreversibility.

6. Conclusions

The current investigations helps us to understand the influence of governing parameters on the micropolar fluid flow through a rectangular duct. The variations of entropy generation and Bejan number for different parameters are described in x and y directions. The results obtained can be summarized as follows.

1. The entropy generation decreases with increase in coupling number (N).

2. Maximum entropy generation occurs at the boundaries of the rectangular duct in x and y-directions.

3. With an increase in the Hartman number(magnetic ﬁeld), entropy generation decreases in the vicinity of the boundaries of the duct, whereas in the central region of the duct, the entropy generation number is raised in x and

y-directions.

4. Bejan number increases with increasing N and Br and decreases with increasing Ha in x and y-directions. 5. All the Bejan number proﬁles show the maximum value at the boundaries of the duct in x and y-directions due to

domination of heat transfer irreversibility in this region. Zero Bejan number represents the domination of ﬂuid friction and magnetic ﬁeld irreversibility.

References

[1] A. Bejan, A study of entropy generation in fundamental convective heat transfer, J.Heat Transfer. 101 (1979) 718-725. [2] A. Bejan, Second law analysis in heat transfer and thermal design, Adv. Heat Transfer. 15 (1982) 1-58.

[3] A. Bejan, Entropy Generation Minimization, CRC Press, New York. 1996.

[4] U. Narusawa, The second-law analysis of mixed convection in rectangular ducts, Heat and Mass Transfer. 37(2-3) (2001) 197-203.

[5] H.F. Oztop, Effective parameters on second law analysis for semicircular ducts in laminar flow and constant wall heat flux, International communications in heat and mass transfer. 32(1) (2005) 266-274.

[6] T.H. Ko, K. Ting, Entropy generation and optimal analysis for laminar forced convection in curved rectangular ducts: A numerical study, International Journal of Thermal Sciences. 45(2) (2006) 138-150.

[7] A. Haji-Sheikh, Fully developed heat transfer to fluid flow in rectangular passages filled with porous materials, Journal of heat transfer. 128(6) (2006) 550-556.

[8] K. Hooman, H. Gurgenci, A.A. Merrikh, Heat transfer and entropy generation optimization of forced convection in porous-saturated ducts of rectangular cross-section, International Journal of Heat and Mass Transfer, 50(11) (2007) 2051-2059.

[9] S. Jarungthammachote, Entropy generation analysis for fully developed laminar convection in hexagonal duct subjected to constant heat ﬂux, Energy. 35(12) (2010) 5374-5379.

[10] G. Yang, J.Y. Wu, L. Yan, Flow reversal and entropy generation due to buoyancy assisted mixed convection in the entrance region of a three dimensional vertical rectangular duct, International Journal of Heat and Mass Transfer. 67 (2013) 741-751.

[11] K.Y. Leong, H.C. Ong, Entropy generation analysis of nanoﬂuids ﬂow in various shapes of cross section ducts, International Communications in Heat and Mass Transfer. 57 (2014) 72-78.

[12] A.C. Eringen, The theory of micropolar ﬂuids, J.Math.Mech, 16 (1966) 1-18 .

[13] C.C. Lee, O.C. Jones, M. Becker, Thermoﬂuid-neutronic stability of the rotating, ﬂuidized bed, space-power reactor., Nuclear engineering and design. 139(1) (1993) 17-30.

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-0.95 -0.2 0.55 0 0.25 -0.95 _-0.7 -0.45 _-0.2 0.05 0.3 0.55 0.8 0.2 2 (a) -0.95 -0.2 0.55 0 0.018 -0.95 _-0.7 -0.45 -0.2 _0.05 0.3 0.55 0.8 Be 0.2 (b) Fig. 1. 3-Dimensional proﬁles of entropy generation and Bejan nuber

-1.0 -0.5 0.0 0.5 1.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Ns x N=0.1 N=0.4 N=0.6 N=0.9 (a) -1.0 -0.5 0.0 0.5 1.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Ns y N=0.1 N=0.4 N=0.6 N=0.9 (b) -1.0 -0.5 0.0 0.5 1.0 0.000 0.004 0.008 0.012 0.016 0.020 0.024 Be x N=0.1 N=0.4 N=0.6 N=0.9 (c) -1.0 -0.5 0.0 0.5 1.0 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 Be y N=0.1 N=0.4 N=0.6 N=0.9 (d)

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-1.0 -0.5 0.0 0.5 1.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Ns x Ha=1 Ha=2 Ha=3 Ha=4 (a) -1.0 -0.5 0.0 0.5 1.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Ns y Ha=1 Ha=2 Ha=3 Ha=4 (b) -1.0 -0.5 0.0 0.5 1.0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Be x Ha=1 Ha=2 Ha=3 Ha=4 (c) -1.0 -0.5 0.0 0.5 1.0 0.000 0.005 0.010 0.015 0.020 Be y Ha=1 Ha=2 Ha=3 Ha=4 (d)

Fig. 3. Eﬀect of Hartman number on entropy generation and Bejan number for N = 0.25, Br = 1

-1.0 -0.5 0.0 0.5 1.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Ns x Br=1 Br=2 Br=3 Br=4 (a) -1.0 -0.5 0.0 0.5 1.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Ns y Br=1 Br=2 Br=3 Br=4 (b) -1.0 -0.5 0.0 0.5 1.0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Be x Br=1 Br=2 Br=3 Br=4 (c) -1.0 -0.5 0.0 0.5 1.0 0.00 0.01 0.02 0.03 0.04 Be y Br=1 Br=2 Br=3 Br=4 (d)