Entropy generation analysis of mixed convective flow in an inclined channel with cavity with Al2O3-water nanofluid in porous medium

International Communications in Heat and Mass Transfer 89 (2017) 198–210

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Entropy generation analysis of mixed convective flow in an inclined channel with cavity with Al2O3-water nanofluid in porous medium

MARK

S. Hussaina,b,*, K. Mehmooda, M. Sagheera, A. Farooqa a b

Department of Mathematics, Capital University of Science and Technology, Islamabad, Pakistan Institut für Angewandte Mathematik (LS III), Technische Universität, Dortmund, Germany

A R T I C L E I N F O

A B S T R A C T

Keywords: Mixed convection Channel with cavity Entropy generation Porous medium Newton-multigrid Finite element analysis

A numerical study has been carried out in the analysis of two dimensional, incompressible and steady mixed convective flow in an inclined channel with cavity. The cavity is filled with Al2O3-water nanofluid saturated with porous medium using the Darcy-Brinkman-Forchheimer model. The temperature at the left wall of the cavity is considered as TH and the inlet temperature of the channel is TC while the rest of the walls are thermally insulated. The governing equations are discretized in space using finite element pair Q2/P1disc which leads to the third and second order accuracy in the L2-norm for velocity/temperature and pressure, respectively. The discrete system of nonlinear equations is treated by using Newton's method and the associated linear systems are computed using monolithic geometric multigrid solver with Vanka-type smoother. The effects of some physical parameters in the specific ranges such as Richardson number (0.01–20), Reynolds number (10–200), Darcy number (10 −6–10 −3), inclination angle (0°–360°), porosity (0.2–0.8) and solid volume fraction (0–0.04) on the flow are presented. The obtained results are shown in the form of isotherms, streamlines and some other useful plots. It is found that an increase in the inclination angle up to γ = 135°, maximum temperature gradient occurs and the temperature distribution is enhanced in the cavity that results an increase in the heat transfer. For an inclination angle greater than or less than this value, less heat transfer is observed.

1. Introduction Mixed convection is a kind of convection which occurs in the process of heat transfer in fluids where both forced and natural convective mechanisms are discussed simultaneously. Mixed convection flow and heat transfer in open cavities have received great attention in recent years due to wide range of engineering applications such as nuclear reactors, crystal growth, heat exchangers, and solar collectors. Some considerable studies on mixed convection in cavity can be found in [1–5]. Nanofluid is a combination of the base fluid and solid nanoparticles. It was introduced by Choi [6]. Moreover, study of the induction of nanoparticles in the base fluid inside an inclined cavity, shows the enhancement of heat transfer rate with increment in the solid volume fraction ϕ. Mixed convective study in case of nanofluid can be consulted in many recent articles, e.g., [7–10]. Convective heat transfer in porous media has vast applications in science and engineering including civil, chemical and mechanical engineering. Heat exchangers [11], boilers [12], oil and gas flowing in reservoirs [13], water filtration [14], ground-water flows [15], transfer

*

of drugs in tissues [16], fuel cells [17], packed-bed energy storage systems [18], thermal insulation [19,20] and fluidized beds [21] are some examples out of many applications. Khanafer and Chamkha [22] numerically studied the unsteady mixed convection in a lid driven enclosure within porous media. Applying the Brinkman-extended Darcy model, the average Nusselt number is significantly affected. It was found that by increasing Da, heat transfer rate increases whereas by increasing Da and decreasing Ri, Nu increases. A similar observation is presented, with the unsteady flow and carried the same conclusions, in Rahman et al. [23] where Brinkman Darcy model is used with the introduction of semi-circular heaters at the bottom wall of lid driven cavity. Hassan and Ismael [24] introduced the porous medium with the Maxwell Brinkman model inside a lid driven square cavity. The results clearly show that by reducing the Darcy number, enhancement is seen in the heat transfer due to the existence of porous layer. A survey on the literature has also revealed an advanced model confining the porous medium, known as the BrinkmanForchheimer extended Darcy model which is the combination of BDM (Brinkman Darcy model) and BFDM (Brinkman Forcheimer Darcy model). Hadim and Chen [25] numerically concluded the mixed convection flow in a

Corresponding author at: Department of Mathematics, Capital University of Science and Technology, Islamabad, Pakistan. E-mail address: [email protected] (S. Hussain).

https://doi.org/10.1016/j.icheatmasstransfer.2017.10.009

0735-1933/ © 2017 Elsevier Ltd. All rights reserved.

International Communications in Heat and Mass Transfer 89 (2017) 198–210

S. Hussain et al.

Nomenclature Cp g Gr k Nuavg Nu p P Pr Re Ri Da H l L w TH − TC T u, v U, V u0 x, y X, Y

Be

Specific heat (J kg −1 K −1) Gravitational acceleration (m s −2) Grashof number, βg ΔTL3 / νf2 Heat conductivity (W m −1 K −1) Average Nusselt number Local Nusselt number Pressure (N m −2) Dimensionless pressure Prandtl number, νf/αf Reynolds number, U0L/νf Richardson number, Gr/Re2 Darcy number (κ/H2) Cavity height (m) Channel length (m) Cavity length (m) Channel height (m) Temperature gradient Temperature (K) Dimensional velocity components (m s −1) Dimensionless velocity components Velocity of the flow at the inlet (m s −1) Dimensional space coordinates (m) Dimensionless space coordinates

Bejan number

Greek symbols α ν β ρ θ μ κ ϵ ϕ

Thermal diffusivity (m2 s −1) Kinematic viscosity (m2 s −1) Thermal expansion coefficient (K −1) Density (kg m −3) Dimensionless temperature Dynamic viscosity (kg m −1 s −1) Permeability of porous medium Porosity Volume fraction of the nanoparticles

Subscripts nf C H avg s f

Nanofluid Cold Hot Average Solid Fluid

bisecting the moving plate kept at the middle. Slight change in the parameters does not affect the Nuavg, as an increment in the Darcy number increases the heat transfer rate. A significant comparison is found in Kumar et al. [40], between BDM and BFDM. The results depicted in this model are the same as above but effect of Darcy number, Grashof number and Richardson number makes different changes in both cases. In the absence of the inertial term shown in BFDM, BDM obtains increasing values of Nuavg. Jeng and Tzeng. [41] explored the study including aluminium foams in the fluid saturated porous media. Due to greater porosity effect on momentum and energy equation, higher heat transfer rate was observed for low Darcy number. Similarly, Kumar and Gupta [42,43] reported the heat transfer and flow in non-Darcy porous media considering the thermal fields and flow characteristics in wavy cavities. Furthermore, Chen et al. [44] also discussed the CuO-nanoparticles in the fluid flow conducted in small tube. Results have been deduced by the induction of pressure drop of the nanofluid greater than water. Pressure drop has a linear relationship with Reynolds number in the laminar region while it is increased sharply with an increment of Reynolds number in the turbulent region.

porous vertical channel with heat sources at the walls. With the effect of porosity, as Da decreases, Nuavg increases in the vertical flow. Recently, a similar model is considered for porous geometry with Cu-nanoparticles observed by Sureshkumar and Muthtamilselvan [26]. Conclusions are made to find the effect of heat transfer rate. Nuavg increases with high Darcy number whereas the porosity is fixed because higher value of Darcy number increases the flow conductance with the permeability in the fluid. Furthermore, decreasing the Richardson number, higher heat transfer rate is seen in an enclosure. Entropy generation suppresses the thermodynamic efficiency of a system. It indicates the location of a system in which more energy dissipation occurs. Bejan [27] has investigated the fundamental principles to mitigate the entropy generation. Since entropy is one reason out of many for the wastage of energy in heat transfer process, therefore sometimes it becomes necessary to measure entropy generation in a very accurate way. Further literature on entropy generation can be found in [28–38]. Nagarajan and Akbar [39] studied the numerical investigation of the mixed convection with Cu-nanoparticles in a square filled enclosure

Fig. 1. Schematic diagram of the physical model.

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Nasrin and Alim [45] studied the numerical analysis of forced flow in horizontal open channel with cavity showing the induction of porous medium having TiO2 nanoparticles. Increment in TiO2 nanoparticles tremendously increased the heat transfer rate. So, average Nusselt number Nuavg increases for low values of Darcy number. Our aim is to investigate the entropy generation of mixed convective flow in an inclined channel along with cavity saturated with aluminawater nanofluid in a porous medium. In view of very careful literature survey and authors' knowledge, this problem has not been considered and investigated yet.

2.3. The dimensionless governing equations Following variables are utilized to transform the system into dimensionless form

X=

2. Problem formulation

x , H

Da =

κ , H2

Ri =

Gr . Re 2

Y=

y , H

Re =

U= ρf uo H μf

u , uo

,

V=

Gr =

θ=

gβ ΔTL3 , νf2

T − TC , TH − TC

Pr =

P=

νf ρf (Cp)f Kf

p , ρnf u02

,

After converting into dimensionless form, Eqs. (1)–(4) are reduced as follows:

2.1. The problem configuration

∂U ∂V + = 0, ∂X ∂Y

The present work considers two dimensional and laminar flow of an incompressible, Newtonian fluid in an inclined open channel with cavity in two dimensional porous cavity with height H and length L, similarly the channel height is represented by w and the channel length is unity (see Fig. 1). The Boussinesq approximation is applied with the constant fluid properties [26,39,46,47] of alumina-water nanoparticles having variation of density in the buoyancy term as shown in Table 1. The porous medium is considered to be saturated with the nanofluid. Assuming that the pure fluid (water) and Al2O3-spherical particles are in thermal equilibrium. The left side of the wall has uniform heat distribution TH and flow enters through the left side of the channel at constant temperature TC. The remaining walls of cavity are kept adiabatic. The size and shape of solid particles are characterized to be uniform. The constant thermo-physical properties of the nanoparticles and the base fluid are given in Table 1.

(5)

2 2 1 ⎛ ∂U ∂U ⎞ ∂P 1 ρf 1 ⎛∂ U + ∂ U ⎞ +V = − + U 2 2.5 2 ϵ ⎝ ∂X ∂Y ⎠ ∂X ϵRe ρnf (1 − ϕ) ⎝ ∂X ∂Y 2 ⎠ ⎜

ρf ⎛ ρβϕ ⎞ 1 − ϕ + s s (sin γ ) ⎟ θ ρf βf ρnf ⎜ ⎝ ⎠ μnf 1 1.75 − U− ( U 2 + V 2 )U, 3 ρnf νf ReDa 150Da ϵ 2 (6)

2 2 1 ⎛ ∂V ∂V ⎞ ∂P 1 ρf 1 ⎛∂ V + ∂ V ⎞ +V = − + U 2 2.5 2 ϵ ⎝ ∂X ∂Y ⎠ ∂Y ϵRe ρnf (1 − ϕ) ⎝ ∂X ∂Y 2 ⎠ ⎜

ρf ⎛ ρβϕ ⎞ 1 − ϕ + s s (cos γ ) ⎟ θ ρf βf ρnf ⎜ ⎝ ⎠ μnf 1 1.75 − V− ( U 2 + V 2 )V , 3 ρnf νf ReDa 150Da ϵ 2 (7)

∂θ ∂θ 1 αnf km ⎛ ∂2θ ∂ 2θ ⎞ U +V = + . ∂X ∂Y RePr αf knf ⎝ ∂X 2 ∂Y 2 ⎠ ⎜

(1)

2 2 μ ⎛u ∂u + v ∂u ⎞= − 1 ∂p + nf ⎛ ∂ u + ∂ u ⎞+ ϵ 2 ⎝ ∂x ρnf ∂x ϵ ⎝ ∂x 2 ∂y 2 ⎠ ∂y ⎠ + (ρβ )nf g (T − Tc )(sin γ ) μnf 1.75ρnf − u− ( u2 + v 2 ) u 3 K 150K ϵ 2 ⎜

⎟

(8)

Associated with the problem, the boundary conditions are given by

ρnf

⎟

⎟

+ Ri

Taking into account the above assumptions and in consideration of the Boussinesq approximation, two dimensional equations consisting of mass, momentum and energy [26] can be written as follows:

∂u ∂v + = 0, ∂x ∂y

⎟

+ Ri

2.2. The governing equations

⎜

v , uo

• On the inlet side of the channel

⎟

U = 1,

• On the left wall of the cavity

(2)

U = 0, 2 2 μ ⎛u ∂v + v ∂v ⎞= − 1 ∂p + nf ⎛ ∂ v + ∂ v ⎞ ∂y 2 ⎠ ϵ 2 ⎝ ∂x ∂y ⎠ ρnf ∂y ϵ ⎝ ∂x 2 + (ρβ )nf g (T − Tc )(cos γ ) μnf 1.75ρnf − v− ( u2 + v 2 ) v, 3 K 150K ϵ 2

V=, θ=0

V = 0,

θ=1

ρnf

⎜

u

⎟

⎜

⎟

• On the rest of the adiabatic walls of the channel and cavity V = 0,

∂θ = 0 for horizontal wall ∂Y

U = 0,

V = 0,

∂θ = 0 for vertical wall ∂X

(3)

∂T ∂T ∂ 2T ∂ 2T ⎞ +v = αnf ⎛ 2 + . ∂x ∂y ∂y 2 ⎠ ⎝ ∂x ⎜

U = 0,

⎟

(4)

• On the outlet side of the channel

Table 1 Thermo-physical properties of water (H2O) and alumina (Al2O3). Physical properties

H2O

Al2O3

ρ (kg m −3) Cp (J kg −1 K −1) k (W m −1 K −1) β (K −1) σ (Ωm) −1 ds (nm)

997.1 4179 0.613 21 × 10 −5 0.05 –

3970 765 40 1.89 × 10 −5 1 × 10 −10 47

∂U = ∂X

∂V = ∂Y

∂θ =0 ∂X

2.4. Effective nanofluid properties The effective density, thermal diffusivity, electrical conductivity, specific heat and the coefficient of thermal expansion of the nanofluid 200

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S. Hussain et al.

We used the glass fibers to simulate the porous medium [56,57].

can be written in the following way:

ρnf = (1 − ϕ) ρf + ϕρs ,

αnf =

knf (ρCp)nf

(9)

,

2.5. Entropy generation The entropy generation due to various physical sources can be written as follows:

(10)

(ρCp)nf = (1 − ϕ)(ρCp)f + ϕ (ρCp)s ,

(11)

(ρβ )nf = (1 − ϕ)(ρβ )f + ϕ (ρβ )s ,

knf ⎡ ∂T 2 ⎛ ∂T ⎞2⎤ ⎛ ⎞ + ⎥ T02 ⎢ ⎝ ∂y ⎠ ⎦ ⎣ ⎝ ∂x ⎠ 2 2 μnf ⎡ ⎛ ∂u 2 ∂v ∂u ∂v ⎞ ⎤ 2 ⎜ ⎛ ⎞ + ⎛ ⎞ ⎟⎞ + ⎛ + , + ⎢ T0 ∂x ∂x ⎠ ⎥ ⎝ ∂y ⎠ ⎠ ⎝ ∂y ⎦ ⎣ ⎝⎝ ⎠

s=

(12)

⎜

The Brownian motion plays a significant role on nanofluid's thermal conductivity. The model suggested by Koo and Kleinstreuer [48] for effective thermal conductivity is given by

keff = kstatic + kBrownian

(13)

3(kp/ kf − 1) ϕ ⎤ kstatic = kf ⎡1 + ⎥ ⎢ ( k / k p f + 2) − (k p/ k f − 1) ϕ ⎦ ⎣

(14)

⎜

kBrownian = 5 ×

(Cp)f

κb T ′ g (T , ϕ, dp) ρs dp

ST =

χ=

=

(22)

(23)

2

⎛ uo ⎞ ⎝ TH − TC ⎠

⎜

kf

⎟

(24)

In Eq. (23), involved terms can be written as

SHT =

2 knf ⎡ ∂θ 2 ⎛ ⎞ + ⎛ ∂θ ⎞ ⎤, ⎢ kf ⎣ ⎝ ∂X ⎠ ⎝ ∂Y ⎠ ⎥ ⎦

(25)

and (17)

where μstatic = μf/(1 − ϕ)2.5 is nanofluid's viscosity presented by Brinkman [52]. Interfacial thermal resistance, i.e., Rf = 4 × 10 −8Km2/ W and replacement of original kp in Eq. (14) by kp,eff is given by the relation

kp

μf To

(16)

with the constants ci (i = 1,2,…,10) are tabulated in Table 2. Furthermore, effective viscosity model for nanofluid proposed by Koo and Kleinstreuer [51] is given by

dp

(21)

In Eq. (22), χ is known as an irreversibility factor [47]. It can be expressed as

+ c5 ln(dp)2)ln(T ) + (c6 + c7 ln(dp) + c8 ln(ϕ)

Rf +

2 knf ⎡ ∂θ 2 ⎛ ⎞ + ⎛ ∂θ ⎞ ⎤ ⎢ kf ⎣ ⎝ ∂X ⎠ ⎝ ∂Y ⎠ ⎥ ⎦ 2 2 2 μnf ⎡ ∂ U ⎛ ⎞ + ⎛ ∂V ⎞ ⎟⎞ + ⎛ ∂U + ∂V ⎞ ⎤, +χ 2⎜⎛ ⎢ ∂X ⎠ ⎥ μf ⎣ ⎝ ⎝ ∂X ⎠ ⎝ ∂Y ⎠ ⎠ ⎝ ∂Y ⎦

ST = SHT + SFF .

g ′ (T , ϕ, dp)= (c1 + c2 ln(dp) + c3 ln(ϕ) + c4 ln(ϕ)ln(dp)

μeff = μstatic + μBrownian = μstatic

⎟

Total entropy generation is

where empirical g′ for Al2O3-water nanofluid can be calculated from the following relation:

μf k + Brownian × kf Prf

⎜

The first term shown on the right side of Eq. (21) where T0 = indicates entropy generation due to heat transfer and second term represents the entropy generation due to viscous dissipation. The dimensionless form of entropy generation [47] is given by

(15)

+ c9 ln(dp)ln(ϕ) + c10 ln(dp)2)

⎟

Th + Tc . 2

where kp, kf are the thermal conductivities of solid particles and pure fluid, respectively and kstatic is thermal conductivity presented by Maxwell [49]. Thermal conductivity due to Brownian motion proposed by Ko-Kleinstreuer-Li [50] given by

10 4ϕρf

⎟

SFF = χ

(18)

The effective heat capacity and effective thermal conductivity of the porous medium are calculated from the relations [54,55] given by

(ρCp)m = (1 − ϵ)(ρCp)s + ϵ(ρCp)nf ,

(19)

km = (1 − ϵ) ks + ϵknf .

(20)

ST ,avg =

Al2O3-water

c1 c2 c3 c4 c5 c6 c7 c8 c9 c10

52.813488759 6.115637295 0.6955745084 0.041745555278 0.176919300241 −298.19819084 −34.532716906 −3.9225289283 −0.2354329626 −0.999063481

1 ϑ

∫ϑ ST

d ϑ = SHT ,avg + SFF ,avg.

(27)

where ϑ indicates the whole volume of the entire domain. Moreover, SHT,avg and SFF,avg are the dimensionless entropy generation caused by the heat transfer and the viscous dissipation, respectively. The Bejan number Be [47,58] is an important dimensionless number that is given by the following ratio:

Table 2 The coefficients values of Al2O3-water nanofluid [53]. Coefficient values

(26)

Eq. (25) shows dimensionless entropy generation due to heat transfer while Eq. (26) represents dimensionless entropy generation due to fluid friction. The non-dimensional average total entropy generation ST,avg is induced by integrating Eq. (22) on the whole computational domain that can be expressed as

dp kp, eff

μnf ⎡ ∂U ⎞2 ⎛ ∂V ⎞2⎞ ⎛ ∂U ∂V ⎞2⎤ + + . 2 ⎜⎛ ⎛ ⎟ + ∂ ∂ ∂ ∂X ⎠ ⎥ μf ⎢ X Y Y ⎠ ⎝ ⎠⎠ ⎝ ⎣ ⎝⎝ ⎦

Be =

SHT ,avg SHT ,avg + SFF ,avg

.

(28)

2.6. Calculation of average Nusselt number The local Nusselt number Nu shows the heat transfer in saturated fluid domain which is used to characterize heat flux between fluid in the cavity and assumed heated wall. The local Nusselt number and average Nusselt number are given by

Nu = −

201

knf ⎛ ∂θ ⎞ kf ⎝ ∂X ⎠

,

⎜

X =0

(29)

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S. Hussain et al.

Nuavg =

∫0

1

Nu

dX.

Table 3 Comparison of present study to Refs. [47,58,60,61].

(30)

Average Nusselt number is obtained by integrating local Nusselt number with the heated wall. So it is also known as the overall heat transfer in the cavity and expressed in Eq. (30).

Assisting forced flow

3. The numerical algorithm 3.1. Computational formulation The system of coupled non-linear partial differential equation together with given boundary conditions have been discretized numerically by the finite element formulation together with the Galerkin weighted residual technique. The numerical procedure used to solve the governing equations for the present work that is based on the Galerkin weighted residual method of finite-element formulation where U, V and θ are discretized by Q2 element of 3rd order accuracy and P is discretized by P1disc element of 2nd order accuracy (see [59] for details). By the Galerkin approximation Eqs. (5)–(8) are transformed into an algeN braic system of equations. Using the FEM approximation Uh = ∑ j = 1 Uj ξ j , N

N

K

N ∑i = 1 wi ξi

K ∑i = 1 qi ηi

Vh = ∑ j = 1 Vj ξ j , θh = ∑ j = 1 θj ξ j and Ph = ∑ j = 1 Pj ηj are the trial functions,

Ri Present study 0.01 0.57623 0.1 0.54379 1 0.42042 10 0.30294 100 0.20985 Opposing forced flow

[60] 0.576 0.544 0.42 0.303 0.209

Ri Present study 0.01 0.62331 0.1 0.62633 1 0.61660 10 0.23757 100 0.13155 Horizontal forced flow Ri Present study 0.01 1.09456 0.1 1.06117 1 0.85609 10 0.61370 100 0.43698

[60] – 0.627 0.617 0.237 0.132 [60] – 1.06 0.856 0.613 0.437

[47] 0.575 0.549 0.426 0.306 –

[58] 0.577 0.545 0.422 0.305 0.211

[61] – 1.07 0.871 0.620 –

and qh = are the test functions. The similarly wh = Galerkin finite element model for a typical element Ωe is given by solved using monolithic geometric multigrid solver with a smoother based on blocking of all cell unknowns (see [59] for details). The following adopted criteria is implemented in this study to ensure the relative convergence by specifying some tolerance value of dependent variables.

11 12 13 14 1 ⎡ [K ] [K ] [K ] [K ] ⎤ ⎡ { U } ⎤ ⎡F ⎤ 21 22 23 ⎢[K ] [K ] [K ] [K 24]⎥ ⎢ { V } ⎥ ⎢ F 2⎥ ⎢ 31 ⎥⎢ ⎥ = ⎢ 3⎥ , 32 33 34 ⎢[K ] [K ] [K ] [K ]⎥ ⎢ { P } ⎥ ⎢F ⎥ 4 ⎢[K 41] [K 42] [K 43] [K 44]⎥ ⎢ { θ} ⎥ ⎢ ⎦ ⎣F ⎥ ⎣  ⎦ ⎣ ⎦  

where

⏟

F 3,

F 2,

F4

and

(31)

F

U

A

F1,

Λn + 1 − Λn ≤ 10−6 Λn + 1

are the right hand side vectors, A is known

as block matrix, U is called the block solution vector and F is said to be a block load vector. In block matrix,

K11=

1 ρf 1 ϵRe ρnf (1 − ϕ)2.5

+ +

μnf

1 νnf ρnf ReDa 1 . 75 3

150Da ϵ2

K 22=

+ K 44=

3 150Da ϵ2

αnf

1 αf RePr

∂ξ j ∂ξi ∂Y ∂Y

where Λ is the dependent variable showing U, V, P and θ where n indicates the iteration index.

⎞ dΩ ⎠

3.2. Code validation and grid independence study ∂ξ

1 ϵ2

∂ξ

∫Ω ⎛Ūξi ∂Xj + V ξi ∂Yj ⎞ dΩ ⎝ ⎠ ∫Ω ( Ū Ū + V V ) ξ j ξi dΩ,

μnf 1 νnf ρnf ReDa 1 . 75

⎝

∫Ω ξ j ξi dΩ +

1 ρf 1 ϵRe ρnf (1 − ϕ)2.5

+

∂ξ

∂ξ ∫Ω ⎛ ∂Xj ∂Xi +

∂ξ

∂ξ ∫Ω ⎛ ∂Xj ∂Xi +

⎝

∫Ω ξ j ξi dΩ +

∂ξ j ∂ξi ∂Y ∂Y 1 ϵ2

∂ξ

∂ξ j ∂ξi ⎞

⎝

∂Y ∂Y

∂ξ j

∂ξ j

⎠

The numerical code has been validated in terms of the forced, mixed and natural convective regimes and presented in Table 3. The results are in excellent agreement with those of [47,58,60,61]. Grid independence test has been performed for the value of average Nusselt number with Re = 100,Ri = 1.0,Da = 10 −3,ϕ = 0.04,γ = 45° and is given in Table 4. It shows that the values at mesh level ℓ = 7 and ℓ = 8 are very closed. So the computations have been performed utilizing grid level ℓ = 7 with #EL = 36,864 and #DOFs = 556,035.

⎞ dΩ ⎠ ∂ξ

∂ξ

∫Ω ⎛Ūξi ∂Xj + V ξi ∂Yj ⎞ dΩ

⎝ ∫Ω ( Ū Ū + V V ) ξ j ξi dΩ,

∂ξ ∫Ω ⎛ ∂Xj ∂Xi +

⎠

dΩ

4. Results and discussion

+ ∫Ω ⎛Ūξi ∂X + V ξi ∂Y ⎞ dΩ, ⎝ ⎠ ρ

f K14= − Ri ρ ⎛1−ϕ + nf ⎝

ρ

f K 24= − Ri ρ ⎛1−ϕ + nf ⎝

ρs βs ϕ ρf βf ρs βs ϕ ρf βf

In this paper, the mixed convection and the entropy generation for alumina-water nanofluid flow in a channel with cavity is numerically studied. In the whole work, the standard values of physical parameters are Re = 100, Ri = 1, Da = 0.001, γ = 0° and Pr = 6.2 unless these are

⎞ sin γ ∫ ξ j ξi dΩ, Ω ⎠ ⎞ cos γ ∫ ξ j ξi dΩ, Ω ⎠

Table 4 Results of grid independence test.

∂ξ

K13= − ∫Ω ηj ∂Xi dΩ, ∂ξ

K 23= − ∫Ω ηj ∂Yi dΩ, ∂ξ j η dΩ, ∂X i ∂ξ j η dΩ, Ω ∂Y i K 21 = K 33 =

K 31= ∫Ω K 32= ∫ K12=

K 34 = K 41 = K 42 = K 43 = 0,

N ∑ j = 1 Ūj ξ j

(32)

N

whereŪ = and V = ∑ j = 1 Vj ξ j . The governing system of nonlinear equations is linearized with the help of Newton method and the associated linear subproblems are 202

ℓ

#EL

#DOFs

Nuavg (Ri = 1)

1 2 3 4 5 6 7 8

9 36 144 576 2304 9216 36,864 147,456

186 639 2355 9027 35,331 139,779 556,035 2,217,987

1.54828060 2.58925194 4.88866766 8.01078458 9.96863011 10.61045025 10.82043432 10.98363453

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S. Hussain et al.

(a)

(a)

(a)

(b)

(b)

(b)

203

Fig. 2. Influence of Re on (a) streamlines and (b) isotherms with Ri = 1, γ = 0, Da = 10 −3, ϵ = 1 and ϕ = 0.04.

Fig. 3. Influence of Ri on (a) streamlines and (b) isotherms with Re = 100, γ = 0, Da = 10 −3, ϵ = 1 and ϕ = 0.04.

Fig. 4. Influence of Da on (a) streamlines and (b) isotherms with Re = 100, γ = 0, Ri = 1, ϵ = 1 and ϕ = 0.04.

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Fig. 5. Influence of ϵ on (a) streamlines and (b) isotherms with Re = 100, γ = 0, Ri = 1, Da = 10 −3 and ϕ = 0.04.

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Fig. 6. Influence of γ on (a) streamlines and (b) isotherms with Re = 100, Ri = 1, Da = 10 −3 and ϕ = 0.04.

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inlet flow velocity of the channel, ultimately increasing the shear forces. For Ri = 10, isotherms with maximum temperature occur near the left heated wall of the cavity and these cover the whole cavity travelling to the outlet of the channel. Increasing Re causes the reduction in the temperature of the heated area near the left cavity wall and due to temperature gradient, the isotherms with more temperature starts clustering to the right side of the cavity and ultimately for Re = 200, the right and bottom sides of the cavity, also, have isotherm

mentioned otherwise. Fig. 2 illustrates the effect of Reynolds number on the streamline contours and isotherm patterns. It is observed initially for Re = 10 that the flow along the upper side of the channel is strong and the flow circulation in the cavity is weak. Increasing Re, the strength of the flow travels to the middle of the cavity and an intense flow circulation encompassing almost the whole cavity is noticed for Re = 200. In fact, amplification in the strength of the flow occurs due to growth in the 205

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convective case (Ri = 10), the effect of the shear forces becomes weak and the maximum flow circulation occurs near the right side of the cavity. Eventually, this circulation of fluid becomes strong pushing more fluid to the lower right corner of the cavity at Ri = 50. As long as isotherm patterns are concerned, for Ri = 0.01 and Ri = 1, isotherms with more intensity are restricted to the left heated wall of the cavity. Increase in the Richardson number (Ri = 10), strong isotherms start travelling to the right and ultimately shifted to the area in the vicinity of right bottom corner of the cavity at Ri = 50. In fact, such happens

patterns with more temperature due to increase in the shear forces. Thus maximum heat transfer takes place for the large Reynolds number. The influence of Richardson number on the streamline contours and isotherm patterns is depicted in Fig. 3. For the forced convective dominated flow (Ri = 0.01), the maximum flow travels to the channel outlet, passing a small amount of fluid through the cavity. Since the buoyancy forces and the shear forces equally take part in the fluid flow for the mixed convective regime, the circulation of the fluid flow can also be seen in the cavity along with the channel. For the natural 206

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greater fluid resistance is experienced by the fluid through the porous matrix and consequently it suppresses the convective heat transfer. Initially, the isotherms with greater value are restricted to the left heated wall of the cavity. An increase in permeability causes these strong temperature lines to distribute to the right side of the cavity. In fact, as the Darcy number increases up to Da = 10 −3, the porous medium permeability grows, decreasing the resistance due to friction that results in an augmentation in the convection process. The influence of porosity on streamline contours and isotherm

due to the tremendous increase in buoyancy forces in the cavity. The impact of the Darcy number on the streamlines and temperature contours is portrayed in Fig. 4. It is observed that strong streamlines are found along the upper side of the channel whereas weak flow patterns can be seen in the cavity for low permeability. An increase in the Darcy number is meant for increasing the permeability of the porous layer, therefore, more nanofluid becomes penetrated into the porous layer, ultimately increasing the strength of the flow lines in the cavity, along with the channel. In fact, for the low Darcy number (Da = 10 −6), 207

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ε

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Fig. 10. Graphs of Nuavg, SHT,avg, SFF,avg, ST,avg and Be for different ϵ with Re = 100, Ri = 1, Pr = 6.2, γ = 0,ϕ = 0.04 and Da = 10 −3.

patterns is elucidated by Fig. 6. For γ = 0°, flow circulation in the cavity is weak as compared to the channel. On closer inspection, increasing the inclination angle, strength of the flow in the cavity is found to enhance gradually and it becomes maximum up to 135°. Beyond this inclination angle, the flow lines start weakening that indicates less convection. Initially, isotherms with higher magnitude are more clustered near the left heated wall of the cavity at γ = 0°. An increase in the inclination angle up to γ = 135°, maximum temperature gradient occurs and the temperature distribution is enhanced in the cavity that results an increase in the heat transfer. For an inclination angle greater than this value less heat transfer is observed.

patterns is shown in Fig. 5. Initially, for low porosity parameter (ϵ = 0.2), strong flow lines along upper side of the channel and weak flow patterns in the cavity are observed. Increasing the porosity parameter up to ϵ = 0.8, strong flow circulation in the cavity is also noticed. It is due to the fact that the increasing porosity causes more nanofluid to enter into the empty spaces of the porous media. Isotherms with large temperature value are mostly confined to the left heated wall of the cavity for low porosity parameter (ϵ = 0.2). On a closer examination, increase in the porosity leads to growth in temperature gradient in the cavity, hence, convection is enhanced. The effect of inclination angle on streamline contours and isotherm 208

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The effect of Re on the average Nu, average SHT, average SFF, average ST and Bejan number is portrayed in Fig. 7. It is observed that average Nu, average SHT, average SFF and average ST increase with an augmentation in Re that is more pronounced for the case of nanofluids (ϕ = 0.04). Bejan number, too, increases with Re but it is more amplified for the case of pure fluid (ϕ = 0). The effect of Ri on average Nu, average SHT, average SFF, average ST and Bejan number is portrayed in Fig. 8. It is observed that Nu, average SHT, average ST and Bejan number decrease with an increase in the buoyancy forces while the average SFF increases. The effect of inclination angle on average Nu, average SHT, average SFF, average ST and Bejan number is shown in Fig. 9. It is observed that γ = 135° is the critical angle. Average Nu, average SHT, average SFF, average ST and Bejan number enhance with an increase in the inclination angle and have maximum value at γ = 135°. The alteration of heat transfer mechanism with variation in inclination angle is also noticed earlier by the streamline contours and isotherm patterns in Fig. 6. An increase in the average Nu, average SHT, and average ST with an increase in the porosity parameter is observed in Fig. 10.

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5. Conclusions In this work, the entropy generation on mixed convective Al2O3water nanofluid flow in an inclined channel with cavity in porous medium, is investigated. The buoyancy forces due to the left heated wall of the cavity and shear forces are responsible for the motion of fluid. The Galerkin finite element approach is implemented for the discretization of the space variables. The discretized system of nonlinear equations is treated by the Newton method and the linear subproblems are solved by means of the Gaussian elimination method. It is worth mentioning to see the streamlines and isotherms in almost all the discussed cases that affect the inlet flow velocity in the channel and the cavity, for apex values of the pertinent parameters. Furthermore, the effect of the physical parameters on the fluid flow and heat transfer can be summarized in the following way. 1. Average Nu, average SHT, average SFF, average ST and Bejan number enhance with an increase in the inclination angle and have maximum values at γ = 135°. 2. An augmentation in average Nu, average SHT, and average ST is observed with growth in porosity parameter. 3. A reduction in average Nu, average SHT, average ST is noticed with a rise in Ri while an opposite effect is seen for average SFF. 4. An increase in the heat transfer is experienced for high Darcy number (Da = 10 −3) as seen from isotherms. 5. Increasing the value of Reynolds number up to Re = 200 makes the heat transfer to rise, significantly. 6. An enhancement in the heat transfer is examined when porosity parameter is increased from ϵ = 0.2 to ϵ = 0.8. 7. A growth in the Bejan number is seen with increasing Re and ϵ while an opposite effect is observed for the increasing values of Ri. Acknowledgments Calculations have been carried out on the LiDOng cluster at TU Dortmund. The support by the LiDOng team at the ITMC at TU Dortmund is gratefully acknowledged. We would like to thank the LiDOng cluster team for their help and support. We also used FeatFlow (www.featflow.de) solver package and would like to acknowledge the support by the FeatFlow team. References [1] A.F. Khudheyer, MHD mixed convection in double lid-driven differentially heated trapezoidal cavity, Int. J. Appl. Innov. Eng. Manag. 4 (2015). [2] G. Kafayati, M.G. Bandpy, H. Sajjadi, D.D. Ganji, Lattice Boltzmann simulation of MHD mixed convection in a lid-driven square cavity with linearly heated wall,

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