Analysis of natural convection in nanofluid-filled H-shaped cavity by entropy generation and heatline visualization using lattice Boltzmann method

Physica E: Low-dimensional Systems and Nanostructures 97 (2018) 347–362

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Analysis of natural convection in nanofluid-filled H-shaped cavity by entropy generation and heatline visualization using lattice Boltzmann method Alireza Rahimi a, Mohammad Sepehr b, Milad Janghorban Lariche c, Mohammad Mesbah d, Abbas Kasaeipoor e, Emad Hasani Malekshah f, * a

Faculty of Energy, University of Kashan, Kashan, I.R. Iran Department of Mechanical Engineering, Payame Noor University (PNU), P.O. Box, 19395-3697, Tehran, Iran c Abadan School of Medical Sciences, Abadan, Iran d Reservoir Studies Division, Department of Petroleum Engineering, Main Office Building, National Iranian South Oil Company (NISOC), Ahvaz, Iran e Faculty of Engineering, Department of Mechanical Engineering, University of Isfahan, Hezar Jerib Avenue, Isfahan 81746-73441, Iran f Department of Mechanical Engineering, Imam Hossein University, Tehran, Iran b

A R T I C L E I N F O

A B S T R A C T

Keywords: Lattice Boltzmann simulation Natural convection Local/total entropy generation Local/total Nusselt variation Heatline visualization H-shaped cavity

The lattice Boltzmann simulation of natural convection in H-shaped cavity filled with nanofluid is performed. The entropy generation analysis and heatline visualization are employed to analyze the considered problem comprehensively. The produced nanofluid is SiO2-TiO2/Water-EG (60:40) hybrid nanofluid, and the thermal conductivity and dynamic viscosity of used nanofluid are measured experimentally. To use the experimental data of thermal conductivity and dynamic viscosity, two sets of correlations based on temperature for six different solid volume fractions of 0.5, 1, 1.5, 2, 2.5 and 3 vol% are derived. The influences of different governing parameters such different aspect ratio, solid volume fractions of nanofluid and Rayleigh numbers on the fluid flow, temperature filed, average/local Nusselt number, total/local entropy generation and heatlines are presented.

1. Introduction Due to wide application of natural convection phenomenon in the engineering applications, many investigators analyzed the fluid flow and heat transfer of natural convection and influence of different parameters. Also, the effects of different configurations of cavities on the flow structure and heat transfer performance are studied [1–8]. The modern fluids with improved thermo-physical properties named nanofluid are being used by researchers [9–18]. To obtain the nanofluid with pronounced heat transfer performance, different metallic and nonmetallic nanoparticles such as Cu [19], CuO [20], Al2O3 [21,22], TiO2 [23,24], MgO [25], Fe2O3 [26], Fe3O4 [27], CNTs (carbon nanotubes) [28], SWCNTs (single wall carbon nanotubes) [29], DWCNTs (double wall carbon nanotubes) [30,31], MWCNTs (multi wall carbon nanotubes) [32–35] are added to the base fluids like water [36], oil [23] and ethylene glycol [37]. Also, the thermo-physical properties such as dynamic viscosity and thermal conductivity of nanofluids are analyzed by

many researchers. Rahimi et al. [38] used the analyzed the dynamic viscosity and thermal conductivity of DWCNTs-water nanofluid and developed the related correlations based on temperature solid volume fraction and temperature. They used these correlations in a numerical investigation of natural convection using lattice Boltzmann method. The natural convection fluid flow and heat transfer in cavities with different geometries, physical and thermal boundary conditions, operating fluids, and numerical methods are investigated in many works. In an experimental and numerical work, Malekshah and Salari [39] studied the heat transfer performances due to natural convection in a cuboid enclosure filled with two immiscible fluids of water and air. They considered the effects on liquid height and Rayleigh number on the temperature distribution and Nusselt number. They showed that the height of interface of water and air has pronounced effect on flow structure and temperature distributions. Sheremet et al. [40] studied the time-dependent natural convection within a wavy triangular enclosure filled with micropolar fluid. They considered many governing

* Corresponding author. E-mail addresses: [email protected] (A. Rahimi), [email protected] (M. Sepehr), [email protected] (M.J. Lariche), [email protected] (M. Mesbah), a.kasaeipoor@ gmail.com (A. Kasaeipoor), [email protected], [email protected] (E.H. Malekshah). https://doi.org/10.1016/j.physe.2017.12.003 Received 26 October 2017; Received in revised form 23 November 2017; Accepted 1 December 2017 Available online 5 December 2017 1386-9477/© 2017 Elsevier B.V. All rights reserved.

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Nomenclature Be CS eα FFI fkeq g g eq gy Gr Ge HTI H NuAvg NS Pr S_gen T u; v x; y W H K

Cp a b d D

Bejan number speed of sound discrete lattice velocity in direction fluid friction irreversibility equilibrium distribution internal energy distribution functions equilibrium internal energy distribution functions acceleration of gravity Grashof number ðGr ¼ gβ0TL3 =ν2 Þ Gebhart number ðGe ¼ gβH=CP Þ heat transfer irreversibility height/width of the cavity average Nusselt number dimensionless entropy generation Prandtl number total entropy generation fluid temperature velocity components Cartesian coordinates width of cavity height of cavity thermal conductivity

specific heat capacity length of internal heater width of internal heater length of partially heater and cooler location of partially heater and cooler

Greek symbols α thermal diffusivity ϕ solid volume fraction φ dimensionless viscous dissipation θ dimensionless temperature ðθ ¼ ðT  TC Þ=ðTH  TC ÞÞ ν kinematic viscosity ρ fluid density ψ stream function β thermal expansion coefficient μ dynamic viscosity Subscripts C H Avg nf f s

cold hot average nanofluid base fluid solid particles

volume fractions on the flow structure, temperature field, heatlines and entropy generation are studied comprehensively.

parameters such as the Prandtl number, dimensionless time, vortex viscosity parameter, and undulation number and their effects on fluid flow, heat transfer, vorticity isolines as well as mean Nusselt number at the wavy wall. Furthermore, the measured fluid flow rate within the enclosure. It was found that the heat transfer reduces and fluid flow attenuates with increasing of vortex vorticity. Koca et al. [41] studied the natural convection within a triangular enclosure filled with air ðPr ¼ 0:71Þ with localized heat source at the bottom of the enclosure and consider the effects of Prandtl number. At low Rayleigh number 103 < Ra < 104 , it was observed that the fluid rises from the middle of the enclosure and falls down from the sides. This kind of fluid flow caused to form two main circulation cells. Because of inclined wall at right section of the enclosure, fluid movements were weaker at this section, so the size of eddies became smaller at this part. The lattice Boltzmann method is one the new developed numerical method to solve the natural convection problems in different shapes of cavities with different boundary conditions. In this context, Zhou et al. [42] performed a three-dimensional lattice Boltzmann simulation for mixed convection for nanofluid-filled enclosure in presence of magnetic force. The influences of Rayleigh number, solid volume fraction of nanofluid, Hartmann number and Richardson number on the fluid flow and heat transfer are studied. They showed that the Richardson number has considerable effect on streamlines and temperature distribution. Ahrar and Djavareshkian [43] carried out lattice Boltzmann simulation of natural convection in a cavity filled with Cu-water nanofluid in presence of magnetic field. The results show that the direction of magnetic field has pronounced influence on high Rayleigh number and negligible effect in lower Rayleigh number. The natural convection fluid flow and heat transfer in a H-shaped cavity has been investigated using lattice Boltzmann method. The cavity is filled with SiO2-TiO2/Water-EG (60:40) hybrid nanofluid which its dynamic viscosity and thermal conductivity are measured experimentally in six solid volume fractions (φ ¼ 0.5%,1%,1.5%,2%,2.5% and 3%) and a temperature range of 30–80  C. Two sets of correlations for these parameters are developed and used in numerical simulations. The influences of different parameters such as four different arrangements of internal active bodies, four different Rayleigh numbers and six solid

2. Problem presentation and mathematical formulation 2.1. Problem presentation In Fig. 1, four different configurations of geometries and the related boundary conditions are depicted graphically. The width and height of the cavity are denoted by L and H and equal to unity. The length and height of gaps, which are represented by b in the cavity, are denoted by b ¼ 0:3L. Moreover, the dimensions of square internal bodies are equal to 1 3 b. It should be noted that the thermal boundary conditions of external surfaces are similar in all cases. The side walls have cold temperature represented by TC . Moreover, the surfaces of walls in the gaps have hot temperature of TH . The top and bottom walls are insulated. The no-slip condition is applied to the external surfaces of internal active bodies. Furthermore, the surfaces of the internal active bodies have constant hot and cold temperature which is equal to the temperature of side walls. It should be noted that the temperature of internal active bodies are represented by red and blue colors which show the hot and cold temperature, respectively. The no-slip boundary condition is applied at the surface of all internal surfaces of cavity. The considered nanofluid is SiO2- TiO2/Water-EG with experimental thermo-physical properties which is used as a single-phase fluid in the simulations.

2.2. Governing equations The flow of nanofluid within the enclosure is assumed to be laminar, Newtonian, and incompressible. The thermo-physical properties of the nanofluid are assumed constant except for a variation of the density which is determined based on Boussinesq approximation; as well as, the dynamic viscosity and the thermal conductivity which are based on experimental correlations. The continuity, momentum and energy equations for the laminar and steady state natural convection in two dimensional forms can be written as follows [44–46]: 348

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Fig. 1. The considered geometry for each case.

∂u ∂v þ ¼0 ∂x ∂y


u









(2)

fi ðx þ ci Δt; t þ ΔtÞ ¼ fi ðx; tÞ þ

(3)

½fieq ðx; tÞ  fi ðx; tÞ þ Δtci Fk

(5)

Δt

τC

½geq i ðx; tÞ  gi ðx; tÞ

(6)

where ci is the discrete lattice velocity in the direction of i, Δt presents the lattice time step, τν and τC represent lattice relaxation times for the flow and temperature fields respectively, Fk is the external force in the direction of the lattice velocity. The external force Fk in the natural convection problems is given by Eq. (7)

(4)

By introducing the following dimensionless parameters: X¼

τν

gi ðx þ ci Δt; t þ ΔtÞ ¼ gi ðx; tÞ þ



∂u ∂u ∂2 T ∂2 T þ v ¼ αnf þ ∂x ∂y ∂x2 ∂y2

Δt

For the temperature filed:



∂v ∂v 1 ∂p ∂2 u ∂2 u þv ¼ þ  þ μnf þ ðρβÞgðT  Tc Þ u ∂x ∂y ρnf ∂x ∂x2 ∂y2 u

Lattice Boltzmann equation). The general form of Lattice Boltzmann equation with external force is as follows [48–50]: For the flow filed:



∂u ∂u 1 ∂p ∂2 u ∂2 u þv ¼ þ  þ μnf ∂x ∂y ρnf ∂x ∂x2 ∂y2


(1)

x y uH vH pH 2 ;Y ¼ ;U ¼ ; V ¼ ;P ¼ H H αnf αnf ρnf α2nf

Fk ¼

} gβnf H 3 ΔT ðT  TcÞ νnf qH θ¼ ; Pr ¼ ; Ra ¼ ; ΔT ¼ ΔTθ knf αnf νnf αnf

ωk c2s

(7)

F⋅ck

where F in the total external body force. Moreover, the kinetic viscosity ðνÞ and thermal diffusivity ðαÞ are defined as a function of their respective relaxation times by Eq. (8)

2.3. The Lattice Boltzmann Method It is worth to mention that the lattice Boltzmann method has some pronounced advantages convincing the researchers to use this method, although it is computationally expensive respect to conventional methods such as FV and FD. The advantages of LBM can be mentioned as linear stability in parallel computing which makes this method more accurate due to locally collision calculation. Moreover, this method is able to handle complex geometry and simulate the fluid flow and heat transfer of incompressible flows accurately. Overall, The major advantages of LBM over other conventional CFD methods due to the fact that the solution for the particle distribution functions is explicit, easy for parallel computation and implementation of boundary conditions on complex boundaries is simple. Two distribution functions, f and g, are utilized by the thermal LB model for the flow and temperature fields, respectively [47]. The thermal Lattice Boltzmann Method applies modeling of the movement of the fluid particles in order to capture fluid quantities in macroscopic scale such as velocity, temperature and pressure. Moreover, the fluid region will be discretized to Cartesian cells uniformly. Also, the probability of finding each particle in a specific range of velocities and locations replaces tagging each particle as in the computationally-intensive molecular dynamics simulation approach. In Lattice Boltzmann Method, each uniform cell has a fixed and specific number of distribution functions presenting the number of the fluid particles movement in the discrete directions. In the present study, the D2Q9 model is employed which is shown in Fig. 2. In this model the values of parameters are assigned as: w0 ¼ 4=9 for jc0 j ¼ 0 for the static pffiffiffi particles, w14 ¼ 1=9 for jc14 j ¼ 1 and w59 ¼ 1=36 for jc59 j ¼ 2. It must be noted that the density and distributions functions (f and g) are calculated by discretizing the kinetic Boltzmann equation (solving the

ν ¼ Cs2 ðτν  1=2Þ; α ¼ Cs2 ðτC  1=2Þ

(8)

pffiffiffi where Cs is the lattice speed of sound and equal to Cs ¼ c= 3. It should be noted that the limitation τ > 0:5 must be satisfied for both relaxation timed in order to be ensure that the viscosity and thermal

Fig. 2. Discrete particle velocity vectors for D2Q9 model. 349

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Momentum : ρu ¼

diffusivity are positive. The type of simulated problem is determined by the local equilibrium distribution function. Also, it models the equilibrium distribution functions calculated with Eqs. (5) and (6) for flow and temperature fields, they are the local equilibrium distribution functions that have an appropriately prescribed functional dependence on the local hydrodynamic properties, respectively: " fieq ¼ wi ρ 1 þ

Ci : u 1 ðCi : uÞ2 1 u2 þ  Cs2 2 2 Cs2 Cs4

X

C i fi ;

(12)

i

Temperature :T ¼

X

gi :

i

#

2.4. Boundary conditions (9)


 Ci : u ¼ w T 1 þ geq i i Cs2

2.4.1. Fluid flow For all solid boundaries, bounce-back boundary conditions are applied which means that the incoming boundary populations are equal to outgoing populations after collision. It was proven by Chen at al [51].that the bounce-back boundary condition renders more accurate numerical results as the LBM approach is applied. As an example, the following conditions are considered for the east boundary:

(10)

Here ρ is lattice fluid density and wi is waiting factor. The force term in Eq. (5) is considered in the vertical y-direction to apply the buoyancy force in the model:   F ¼ 3wi ρ gy βðT  Tm Þ

f3;n ¼ f1;n ; f6;n ¼ f8;n ; f7;n ¼ f5;n

(11)

(13)

Here n denotes the note of the lattice. where gy is gravitational acceleration, and β is thermal expansion coefficient. The Boussinesq approximation is applied for the natural convection simulation in the cavity. The characteristic velocity of fluid flow of natpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ural convection regime defined as Vnatural ¼ βgy 0TH must be small compared with the fluid speed of sound to be ensured that thermal Lattice Boltzmann Method code works well. For this purpose, the characteristic velocity is considered 0.1 of sonic speed. The macroscopic variables are calculated with the following formulations: Flow density :ρ ¼

X

2.4.2. Temperature The bounce-back boundary condition for thermal boundary condition representing adiabatic condition is applied for the north and south walls. As an example, the following boundary condition is applied at the north wall: g8;n ¼ g8;n1 ; g4;n ¼ g4;n1 ; g7;n ¼ g7;n1

(14)

It should be noted that the thermal boundary condition at the rectangular body and it's active walls are known. For this purpose, at the west wall the dimensionless temperature is θ ¼ 1, and the unknowns parameters are g3 , g6 , and g7 as D2Q9 is used:

fi ;

i

g3 ¼ TH ðwð3Þ þ wð1ÞÞ  g1 g6 ¼ TH ðwð6Þ þ wð8ÞÞ  g8

(15)

g7 ¼ TH ðwð7Þ þ wð5ÞÞ  g5 2.5. Entropy generation The total entropy generation rate is defined as follows [52]: Table 1 Properties of utilized nanoparticles and base fluids.

Fig. 3. The obtained nanofluids at different concentration of nanoparticles.

Material

Purity (%)

Color

Size (nm)

Density (kg/m3)

TiO2 SiO2 Water/Ethylene glycol

99 99.99 99.5

White Colorless Colorless

30–50 22 –

4230 2220 –

Fig. 4. TEM image of SiO2 and TiO2 nanoparticles. 350

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Fig. 5. KD2 pre thermal properties analyzer (Decagon devices, Inc., USA).

FFI ¼

μφ

(17)

T

And the Bejan number is defined as follows: Be ¼

HTI HTI þ FFI

(18)

The dimensionless form of total entropy generation, Ns, is defined as follows: H 2

S_gen k

Ω

Ns ¼

(19)

it can be extended it as follows: ∂ θ 2 Ns ¼ Fig. 6. Brookfield viscometer of Brookfield engineering laboratories of USA.

S_gen ¼ HTI þ FFI



1

Ω Ω 2

 2 ∂θ ∂y

þθ

2 þ

Geφ

RaΩ2 Ω1 þ θ

(20)

where Ra represents the Rayleigh number, and Ω is the non-dimensional temperature difference which is defined as follows:

(16)

Ω¼

where HTI represents the heat transfer irreversibility in the direction of finite temperature gradient and FTI denotes the fluid friction irreversibility. In terms of the basic variables, HTI and FFI are defined as follows: HTI ¼

þ

∂x

Th  TC TC

(21)

The dimensionless dissipation function which is used in Eq. (12) is as follows:

kðrT ⋅ rTÞ T2

φ¼2

  2   2 ∂u ∂v ∂u ∂v 2 þ þ2 þ ∂x ∂x ∂x ∂x

(22)

Fig. 7. Comparison between the present results and the mean Nusselt number in Ra ¼ 103 for different solid volume fractions done by Oztop and Abu-Nada [54] and isotherms of the present work (colored section) and experimental numerical study performed by Calcagni et al. (gray section) [55]. 351

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hNsi ¼ ∫ A

Ns dA A

(24)

It is clear that the entering heat energy will be equal to transferred heat energy from the cooled walls. As such, the total entropy generation can be written as follows:

   q00 1 1 S_gen ¼  H TC TH

(25)

It can be obtained by terms of Nusselt number as follows:

 k Ω2 S_gen ¼ 4Nu 2 H 1þΩ

(26)

For small value of Ω, such as Ω≪1, and by applying perturbation techniques, the above-mentioned equation can be written as follows:

 4Nu Ω2 ð1  ΩÞk S_gen  H2

(27)

The dimensionless form of total entropy generation can be obtained as follows: hNsi ¼ Fig. 8. Dynamic viscosity at different temperature as a function of solid volume fractions.

gβH CP

It must be noted that the nanofluid behavior is totally different, in both hydrodynamic and thermal properties, from pure fluids because of existence of inter-particle potentials and other common forces acting between nanoparticles and pure fluids. Using nanoparticles in the pure fluids cause higher rate of energy transport and efficiency. The governing equations are as same as Eqs. (1)–(6) as the pure fluid is the operating fluid in the problem. On the other hand, the governing equations must be modified when the nanofluid is being used because of different thermophysical properties such as density, thermal conductivity, heat capacitance and thermal expansion. The considered nanofluid in the present work is SiO2- TiO2/Water-EG. The simulations are performed in singlephase model, and the solid particles and the base fluid are in thermal equilibrium with no-slip condition between them. The local and average Nusselt numbers at the surface of heater are given by the following:

(23)

The average value of dimensionless entropy generation is determined by hNsi. The angle brakets show an average taken at a specific area as follows: Table 2 Derived correlations for dynamic viscosity at different solid volume fractions. Solid volume fraction

Correlation

φ ¼ 0:5 vol% φ ¼ 1 vol% φ ¼ 1:5 vol% φ ¼ 2 vol% φ ¼ 2:5 vol% φ ¼ 3 vol%

R-squared

μ ¼ 5:0438  0:0865T þ 0:0005T μ ¼ 4:9161  0:0787T þ 0:0004T 2 μ ¼ 4:9801  0:0802T þ 0:0004T 2 μ ¼ 4:8547  0:0691T þ 0:0003T 2 μ ¼ 4:8509  0:0665T þ 0:0003T 2 μ ¼ 4:6687  0:0557T þ 0:0002T 2 2

(28)

2.6. Lattice Boltzmann model for nanofluid

where Ge denotes the Gebhart number defined as follows: Ge ¼

4 Nu 1þΩ

0.9993 0.9902 0.9907 0.9893 0.987 0.9707

Nu ¼

   Knf ∂T  1 x ¼ 0; 1 and Nuav ¼ ∫ 0 Nu dy Kf ∂x 

(29)

2.7. Heatline visualization The heat line can be used to visualize the path-line and intensity of heat flow which is similar to streamlines. The heatlines are applicable to visualize and identify the heat flow from heat sources to heat sinks in the cavities. The heat filed within a two-dimensional cavity for convective transport process was mathematically studied by Kimura and Bejan [53]. The heatlines are represented by heat functions (h) which can be applied to plot the heatlines in the cavity and also obtained from the conductive heat fluxes (∂∂Tx ; ∂∂Ty ) and convective heat fluxes (uT; νT). The heat Table 3 Derived correlations for thermal conductivity at different solid volume fractions.

Fig. 9. Thermal conductivity at different temperature as a function of solid volume fractions. 352

Solid volume fraction

Correlation

R-squared

φ ¼ 0:5 vol% φ ¼ 1 vol% φ ¼ 1:5 vol% φ ¼ 2 vol% φ ¼ 2:5 vol% φ ¼ 3 vol%

k ¼ 0:3939 þ 0:001T  2  106 T 2 k ¼ 0:4281  8  105 T þ 105 T 2 k ¼ 0:4346 þ 8  105 T þ 105 T 2 k ¼ 0:4421  5  105 T þ 105 T 2 k ¼ 0:437 þ 0:0004T  105 T 2 k ¼ 0:4531 þ 0:0002T  105 T 2

0.9975 0.9985 0.9951 0.9939 0.9884 0.9923

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Fig. 10. Flow structure for different arrangements of rigid bodies at Case A and φ ¼ 2 vol%.

To make dimensionless form of heatfunction Π, the above-mentioned dimensionless variables are used:

heatfunction parameter (h) satisfies the energy conservation equation for fluid media and nanofluids: 

u

∂T ∂T ∂2 T ∂2 T þν ¼α þ ∂x ∂y ∂x2 ∂y2



∂Π ∂θ ¼ Uθ  ; ∂Y ∂X

(30)

So,

∂Π ∂θ ¼ Vθ  ; ∂X ∂Y



∂h ∂T ¼ ρcp uðT  T0 Þ  k ∂y ∂x

(31a)

∂h ∂T ¼ ρνðT  T0 Þ  k ∂x ∂y

(31b)



(32a)

(32b)

Which can be written in a single equation as follows:

∂2 Π ∂2 Π ∂ ∂ ðUθÞ  ðVθÞ þ ¼ ∂X 2 ∂Y 2 ∂Y ∂X

Fig. 11. Flow structure for different arrangements of rigid bodies at Case B and φ ¼ 2 vol%. 353

(33)

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convection heat transfer. The SiO2 and TiO2 nanoparticles with high purity of 99%, obtained from US research nanomaterials, Inc, are dispersed in the base fluid. The base fluid is considered a mixture of pure water and ethylene glycol with volume ratio of 60:40. Different solid volume fractions are considered in this study such as 0.5%, 1%, 1.5%, 2%, 2.5%, and 3%. The obtained nanofluids at different concentration of nanoparticles and a TEM image of SiO2 and TiO2 nanoparticles are presented in Figs. 3 and 4, respectively. It should be noted that the color of obtained nanofluid is faded as the solid volume fraction reduces which is not clear in the picture. Furthermore, the properties of supplied nanoparticles and base fluids are presented in Table 1.

It should be noted that the counter clockwise circulation is represented by positive sign of Π, and the clockwise circulation is represented by negative sign of Π. 3. Material and methods 3.1. Nanofluid preparation The nanofluid preparing process in briefly presented here. In this study a modern nanofluid is selected for analyzing its thermo-physical properties and using in the experimental setup to study the natural

Fig. 12. Flow structure for different arrangements of rigid bodies at Case C and φ ¼ 2 vol%.

Fig. 13. Flow structure for different arrangements of rigid bodies at Case D and φ ¼ 2 vol%. 354

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laboratories of USA is utilized, shown in Fig. 6, which has high accuracy of 5%. The viscometer is calibrated by using distilled water before starting the measuring process. All of the measurements are performed at room temperature and repeated three times to be ensured that the results are reliable.

3.2. Stability analysis For all solid volume fractions for this nanofluid, the appropriate amounts of SiO2 and TiO2 nanoparticles are added to the base fluid. Afterwards, the particles and water are mixed with a magnetic stirrer for 2.5 h. After mixing of the nanoparticles and base fluid, the suspension is inserted to an ultrasonic processor (Hielscher Company, Germany) with the power of 400 W and frequency of 24 kHz for 5 h This process prevents the agglomeration between the nanoparticles and sedimentation. The stability of nanofluid is observed for at least one week with any sedimentation.

4. Results and discussion The natural convection fluid flow and heat transfer are investigated comprehensively using different approaches such as entropy generation analysis and heatline visualization. The lattice Boltzmann numerical method is employed to solve convective flow in H-shaped cavity filled with nanofluid. The dynamic viscosity and thermal conductivity of SiO2TiO2/Water-EG hybrid nanofluid are obtained experimentally, and two sets of correlations based on temperature and solid volume fraction are developed and used in numerical simulations. The influences of different governing parameters such as Rayleigh number, solid volume fraction of nanofluid, and different configurations of cavity on the fluid flow, heat transfer, volumetric and local entropy generation and heatline visualization are presented systematically. For validation analysis of the present code with previous work, two works are selected. The natural convection heat transfer analysis performed by Oztop and Abu-Nada [54] in a

3.3. Thermal conductivity and dynamic viscosity measuring In order to measure the thermal conductivity of nanofluid, the reliable and fast method of transient hot-wire (THW) technique is utilized. In this context, a KD2 pre thermal properties analyzer (Decagon devices, Inc., USA) is utilized shown in Fig. 5. The thermal conductivity analyzer device is calibrated with distilled water before starting the measurements, and the maximum error is measured equal to 5%. It should be noted that the each measurement is repeated three times to verify the obtained results. The Brookfield viscometer of Brookfield engineering

Fig. 14. Fluid friction irreversibility map for different aspect ratios and Rayleigh numbers of rigid bodies at φ ¼ 2 vol%. 355

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partially heated cavity filled with SiO2-TiO2/Water-EG nanofluid. The comparison of present results with the data of mentioned literature is shown in Fig. 7(A). Also, isotherms of the present numerical work (colored section) and experimental study performed by Calcagni et al. (gray section) [55] are compared in Fig. 7(B). The comparison, shown in Fig. 7, shows close agreement between the results.

between dynamic viscosity of nanofluid and temperature at all six different solid volume fractions are presented in Table 2.

4.2. Thermal conductivity analysis The influence of the temperature and solid volume fractions on the thermal conductivity is analyzed. The thermal conductivity is analyzed at six different solid volume fractions ðφ ¼ 0:5; 1; 1:5; 2; 2:5 and 3 vol%Þ and a temperature range of 30–80  C.The value of thermal conductivity ðkÞ with respect to temperature as a function of solid volume fraction is depicted in Fig. 9. It can be observed that the thermal conductivity increases considerably at high solid volume fractions since the number of collision between the solid particles augments, as a result of greater number of solid particles in a specific volume of base fluid and Brownian motion. On the other hand, the temperature increment boosts the kinetic energy of solid particles which has resulted in significant motions of solid particles and according collisions with other particles including base fluid and solid particles. As such, the temperature increment has more significant influence on the enhancing of effective thermal conductivity at high concentration. The experimental thermo-physical properties of SiO2- TiO2/Water-EG nanofluid are utilized in the numerical simulation. As such, it is needed to identify the relationship between the thermal

4.1. Dynamic viscosity analysis The influence of the temperature and solid volume fractions on the dynamic viscosity is analyzed. The dynamic viscosity is analyzed at six different solid volume fractions ðφ ¼ 0:5; 1; 1:5; 2; 2:5 and 3 vol%Þ and a temperature range of 30–80  C. The value of dynamic viscosity ðμÞ variation with respect to temperature as a function of solid volume fraction is depicted in Fig. 8. As it can be seen in Fig. 8, the dynamic viscosity of nanofluid reduces with increasing of temperature. It is due to the fact that the intermolecular forces reduce with augmenting of temperature in the fluid. As such, different layers of nanofluid can move easier as temperature increasing with respect to lower temperature because of lower shear stress between two layers. The experimental thermo-physical properties of SiO2- TiO2/Water-EG nanofluid are utilized in the numerical simulation. In this context, the correlations

Fig. 15. Heat transfer irreversibility map for different aspect ratios and Rayleigh numbers of rigid bodies at φ ¼ 2 vol%. 356

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arrangements of internal bodies are depicted graphically in Figs. 10–13. It should be noted that the difference in Rayleigh number caused by temperature difference which is main parameter which has considerable influence on flow structure. The nanofluid flow becomes stronger as Rayleigh number increases which can be concluded by compacted streamlines within the enclosure. Moreover, it can be observed that the nanofluid flow structure becomes irregular at higher Rayleigh number. It should be noted that the at higher Rayleigh number the regime of natural convection fluid flow will be changed from laminar to turbulent which can be concluded by irregular flow pattern as well.

conductivity with temperature at different solid volume fractions. In this context, the correlations between effective thermal conductivity of nanofluid and temperature at all six different solid volume fractions are presented in Table 3. 4.3. Flow structure The flow structure at different arrangements of internal active bodies and Rayleigh numbers are presented in Figs. 10–13. The fluid flow in the enclosure due to the natural convection is influenced by different parameters such as temperature difference, physical geometry of the enclosure, gravity acceleration, and thermo-physical properties of the operating fluid. The nanofluid stream ascends at the adjacent of internal hot bodies and hot walls due to reducing the density of nanofluid stream as the temperature enhances. On the contrary, the nanofluid stream descends along the side cold walls and internal cold bodies. As such, the thermal boundary conditions have considerable influence on the flow structure.

4.5. Influence of aspect ratio on flow structure The influence of different arrangements of internal bodies on the nanofluid flow structure for different Rayleigh numbers is presented in Figs. 10–13. It can be seen that the arrangements of internal active bodies have significant effect on the nanofluid flow structure. It is due to the fact that the thermal boundary conditions of the internal bodies are different in each case. The nanofluid flow pattern in case is more irregular with respect to other cases caused by symmetric thermal boundary condition. Moreover, the secondary eddies are formed in cases C and D due to reverse flow caused by hot and cold surfaces in some areas.

4.4. Influence of Rayleigh number on flow structure The nanofluid flow structures at different Rayleigh numbers and

Fig. 16. Local Nusselt variation map for different aspect ratios and Rayleigh numbers of rigid bodies at φ ¼ 2 vol%. 357

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4.6. Temperature field

pronounced. It can be observed that the most part of the nanofluid phase in the cavity of case A is in hot temperature. It is due to the fact that the internal bodies have hot temperature. As the Rayleigh number increases, the isothermal lines patterns become irregular with higher temperature gradient. It is due to the fact that the fluid flow becomes stronger, in high

The temperature fields for different arrangements of internal active bodies and Rayleigh numbers are depicted in Figs. 10–13. The influences of arrangements of the internal active bodies on the isothermal amps are

Fig. 17. Heatlines for different arrangements of rigid bodies at φ ¼ 2 vol% and 103 < Ra < 104 . 358

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specific solid volume fraction of φ ¼ 1 vol% are depicted in Fig. 15. Another main effective parameters constituting the total entropy generation is heat transfer irreversibility. The heat transfer irreversibility caused by the temperature gradient in the computational domain. As such, it can be concluded that the heat transfer irreversibility will have higher magnitude at the regions with high temperature gradient. It can be observed that the heat transfer irreversibility is dominant at the gaps of internal bodies and external walls. As an instance, in case A, the magnitude of heat transfer irreversibility between internal active bodies and side cold walls is higher than other region because of high temperature gradient at a limited space. As Rayleigh number increases, the value of the heat transfer irreversibility becomes uniform at the computational domain, and some isotherms are compacted at the adjacent of internal active walls.

Rayleigh number, which is able to transfer the heat energy. 4.7. Local fluid friction irreversibility (FFI) The local fluid friction irreversibility maps for different Rayleigh numbers and different arrangements of internal active bodies for one specific solid volume fraction of φ ¼ 1 vol% are depicted in Fig. 14. One of two main parameters constituting the total and volumetric entropy generation is the fluid friction irreversibility. The fluid friction irreversibility is caused by the velocity gradient at the computational domain. As such, the fluid friction irreversibility will have high value at the regions with high velocity gradient. It is worth to mention that the solid volume fraction has no considerable influence on the fluid friction irreversibility maps; as a result, one specific solid volume fraction is selected which its fluid friction irreversibly maps are fairly similar to other cases. It can be seen that the local fluid friction irreversibility is dominant at the adjacent of side cold walls and internal active bodies. It is due to the fact that the velocity magnitudes at these regions are higher than other regions. As Rayleigh number enhances, the fluid friction irreversibility maps are accumulated near the active walls, and the value of fluid friction irreversibility becomes uniform in computational domain.

4.9. Local Nusselt variation The local Nusselt variation maps for different Rayleigh numbers ð103 < Ra < 106 Þ and different arrangements of internal active bodies for specific solid volume fraction of φ ¼ 1 vol% are depicted graphically in Fig. 16. The Nusselt number is a dimensionless parameter which shows the share of each heat transfer mechanism of conduction and convection in the natural convection heat transfer. As Nusselt number increases, it can be concluded that the share of convection heat transfer augments with respect to conduction mechanism. It can be observed that the arrangements of internal active bodies have significant influence on the

4.8. Local heat transfer irreversibility (HTI) The local heat transfer irreversibility maps for different Rayleigh numbers and different arrangements of internal active bodies for one

Fig. 18. Average Nusselt number with respect to Rayleigh number as a function of Rayleigh number for different aspect ratios at the surface of left hot wall. 359

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0:5; 1; 1:5; 2; 2:5 and 3 vol%Þ for four different arrangements of internal active bodies are presented in Fig. 18. The average Nusselt number enhances as the Rayleigh number increases due to stronger velocity magnitude in the computational domain. It should be noted that it can be concluded that the heat transfer coefficient augments as the average Nusselt number increases. Moreover, the rate of increment in average Nusselt number enhances in higher solid volume fraction because of existence of nanoparticles. As the number of collision of nanoparticles with other nanoparticles and base-fluid enhances in higher Rayleigh number, the heat transfer coefficient and corresponded average Nusselt number increase. On the other hand, the value of average Nusselt number augments as the solid volume fraction of nanofluid increases due to improved thermo-physical properties of nanofluid.

structure of local Nusselt number maps. Moreover, the Nusselt number has high magnitude at the adjacent of internal active bodies and side walls due to strong nanofluid flow at these regions. As Rayleigh number enhances, the structure of local Nusselt variation becomes irregular as a result of strong fluid flow. Furthermore, the isotherms of Nusselt variation are compacted near the active walls. 4.10. Heatline visualization In order to detect the heat energy pathway from heat sources to heat sinks, the heatline visualization can be employed. The heatlines inside the cavity for two different Rayleigh numbers ð103 < Ra < 104 Þ and different aspect ratios are depicted in Fig. 17. It can be observed that the pathway of heat energy is considerably determined by the thermal boundary condition in each case as the heat energy is transferred from heat sources to heat sinks with specific pattern at each case. As Rayleigh number enhances, the heatline patterns become irregular with different circulations. It is due to the fact that nanofluid flow transfers the heat energy as the strength of nanofluid flow augments. On the contrary, in lower Rayleigh number, the heat energy move from heat sources to heat sinks without any strong distortion.

4.12. Entropy generation The total entropy generation causes by two parameters of heat transfer irreversibility and fluid friction irreversibility. The values of total entropy generation with respect to Rayleigh number ð103 < Ra < 106 Þ as a function of solid volume fraction ðφ ¼ 0:5; 1; 1:5; 2; 2:5 and 3 vol%Þ for different aspect ratios are presented in Fig. 19. The value of average Nusselt number enhances as Rayleigh number increases. It is due to the fact that the value of both heat transfer irreversibility and fluid friction irreversibility enhances due to higher temperature gradient and velocity gradient, respectively. Also, the nanoparticle concentration causes changing of the value of total entropy generation. As the solid volume

4.11. Average Nusselt number The values of average Nusselt number with respect to Rayleigh number ð103 < Ra < 106 Þ as a function of solid volume fractions ðφ ¼

Fig. 19. Total entropy generation with respect to Rayleigh number as a function of solid volume fraction for different aspect ratios. 360

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Fig. 20. Bejan number with respect to Rayleigh number as a function of solid volume fraction for different aspect ratios.

experimentally at six different solid volume fractions of 0.5%, 1%, 0.05%, 1.5%, 2%, 2.5% and 3% and temperature range of 30–80 ( C). Two sets of correlations for thermal conductivity and dynamic viscosity based on temperature and solid volume fraction are developed and used in the numerical simulations. The influences of different governing parameters such as Rayleigh number, solid volume fraction and different arrangements of internal active bodies on the fluid flow, heat transfer, average/local Nusselt number, total/local entropy generation are presented comprehensively. The results can be listed as:

fraction of nanofluid increases, the total entropy generation reduces due to weaker velocity magnitude of nanofluid stream and reduction in the temperature gradient. 4.13. Bejan number The share of each parameter of fluid friction irreversibility and heat transfer irreversibility on forming the total entropy generation can be identified by Bejan number. For this purpose, the values of Bejan number with respect to Rayleigh number as a function of different solid volume fractions of nanofluid for different aspect ratios of the cavity are presented in Fig. 20. As Rayleigh number increases, the value of Bejan number reduces because of increasing value of fluid friction irreversibility. In addition, the Bejan number has direct relationship with the solid volume fraction. It is due to the fact that the fluid flow becomes weaker and heat transfer performance improves as the solid volume fraction augments.

 The average Nusselt number has direct relationship with the Rayleigh number.  The average Nusselt number has direct relationship with the solid volume fraction.  The total entropy generation has direct relationship with the Rayleigh number.  The total entropy generation has reverse relationship with the solid volume fraction of nanofluid.  The arrangements of internal active bodies have significant on the local entropy generation and Nusselt number.  The lattice Boltzmann numerical method renders more accurate results compared with conventional methods.

5. Conclusions The natural convection fluid flow and heat transfer within the Hshaped cavity filled with nanofluid is investigated using lattice Boltzmann numerical method. The heatline visualization and entropy generation analysis are employed to analyze this problem comprehensively. The produced nanofluids is SiO2- TiO2/Water-EG (60:40) hybrid nanofluid which its thermal conductivity and dynamic viscosity are measured

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