Two-phase modeling of the free convection of nanofluid inside the inclined porous semi-annulus enclosure

International Journal of Mechanical Sciences 164 (2019) 105183

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Two-phase modeling of the free convection of nanofluid inside the inclined porous semi-annulus enclosure Saber Yekani Motlagh∗, Ehsan Golab, Arsalan Nasiri Sadr Faculty of Mechanical Engineering, Urmia University of Technology (UUT), P.O. Box: 57166-419, Urmia, Iran

a r t i c l e Keywords: Buongiorno model Half-annulus enclosure Magnetic nanofluid Natural Convection Porous Medium

i n f o

a b s t r a c t In the present paper, the modeling of the free convection of two-phase nanofluid inside the inclined porous semi-annulus enclosure is considered. The cavity is filled with Fe3 O4 -water magnetic nanofluid. Buongiorno and Darcy models are used for modeling two-phase and porous media, respectively. The governing equations are discretized by finite volume method and SIMPLE algorithm. The effect of parameters such as inclination angle of cavity (0 ≤ 𝜃 ≤ 90), porous Rayleigh number (10 ≤ Rap ≤ 103 ), porosity number (𝜀 = 0.4 and 0.7), and volume fraction of nanoparticles (0 ≤ 𝜑Ave ≤ 0.04) on the flow pattern, temperature field, nanoparticle distribution, and Nusselt number are studied. In low porous Rayleigh numbers, Nusselt number is not the function of porosity number and the inclination angle of the enclosure. Also, Nusselt number increases by adding the volume fraction of nanoparticles. However, in high porous Rayleigh numbers, the heat transfer rate is reduced by increasing the inclination angle of the enclosure. Furthermore, with the increase in the porosity number, Nusselt number is increased.

1. Introduction Nanofluid is made up of base fluids, such as water and oil, and metal nanoparticles, which have many uses in all fields of engineering and medicine. Such as cooling of engines and electronics, nuclear and solar power plants, chillers and fridges, drag reduction, energy storage [1], heat exchangers, heat reservoirs, solar collectors [2], drying technology, photovoltaic cell (PVC) system, catalytic reactors, petroleum industries, loud speakers, lubricants and seals [3], sensors, and drug delivery [4]. Sheikholeslami et al. [5] studied the effects of the magnetic field on nanofluid natural convection in a half-annulus enclosure with the single-phase model. They used nanofluid consists of water and Cu. Numerical simulations of their work illustrated that in the presence of a magnetic field, Nusselt number is reduced. Sheikholeslami and Ganji [6] investigated the effect of adding Cu-water nanofluid in natural convection within a half-annulus enclosure with the single-phase model. They showed that the inclination angle of the enclosure changes the flow pattern, significantly. Bezi et al. [7] performed the numerical study on natural convection and entropy production in a half-annulus with different nanoparticles using the single-phase model. Their results clearly demonstrated that the average entropy production during heat transfer increases with increasing volume fraction and Rayleigh number. In addition, the average Nusselt magnitude has a direct correlation with increasing Rayleigh number and volume fraction.

∗

Garoosi and Talebi [8] investigated the mixed convection in a square enclosure with several hot and cold bars using the two-phase Buongiorno’s model. They found that the heat transfer rate increases with increasing Rayleigh number and thermal conductivity ratio. It was also observed that in all Rayleigh numbers and with increasing the volume fractions of nanoparticles, the total Nusselt number initially increased and then decreased. Yekani Motlagh et al. [9] studied the inclination angle effect on the natural convection of water-Al2 O3 nanofluid and distribution of nanoparticles in the square cavity using the Buongiorno’s twophase model. The results declared that in low Rayleigh numbers, the average Nusselt number remains constant with increasing the inclination angle of the enclosure. Yekani Motlagh et al. [10] investigated the natural convection inside a porous square enclosure filled with nanofluid with the two-phase model. Their numerical results showed that porosity plays an important role in the heat transfer, particularly, at the high Rayleigh numbers. Siavashi et al. [11] numerically solved the problem of natural convection of nanofluid of water-Cu in a cavity with two porous fins by a two-phase method. They predicted that high porosity fins improve the heat transfer, while low Darcy number weakens the convection. Miroshnichenko et al. [12] investigated on natural convection of alumina-water nanofluid in an open cavity having multiple porous layers. Izadi et al. [13] studied natural convective heat transfer of a magnetic nanofluid in a porous medium subjected to two variable magnetic sources. Mehryan et al. [14] modeled natural convection of magnetic

Corresponding author. E-mail address: [email protected] (S.Y. Motlagh).

https://doi.org/10.1016/j.ijmecsci.2019.105183 Received 1 May 2019; Received in revised form 16 August 2019; Accepted 19 September 2019 Available online 19 September 2019 0020-7403/© 2019 Elsevier Ltd. All rights reserved.

S.Y. Motlagh, E. Golab and A.N. Sadr

International Journal of Mechanical Sciences 164 (2019) 105183

Nomenclatures Cp k DB DT K kB g q′′ 𝐽⃗ Da Nu Le Pr NBT Raf Rap Sc V P T d r L x, y X, Y

specific heat, J kg−1 K − 1 thermal conductivity, W m − 1 K − 1 Brownian coefficient, kg m − 1 s − 1 thermophoresis coefficient, m2 s − 1 K − 1 permeability of the porous medium, m2 Boltzmann’s constant (=1.38066×10−23 ), J K − 1 gravitational acceleration vector, m s − 2 heat flux, Wm−2 mass flux vector, kg m − 2 s − 1 Darcy number ( = K/L2 ) Nusselt number on cold wall Lewis number (= kf /((𝜌Cp )p DB 𝜙ave )) Prandtl number ( = 𝜈 f /𝛼 f ) ratio of Brownian to thermophoretic diffusivity ( = (𝜙Ave DB0 𝛿)/DT0 ) base fluid Rayleigh number ( = gBf q′′L4 /(kf 𝛼 f vf )) porous Rayleigh number ( = gBf q′′KL2 /(kf 𝛼 f vf )) Schmidt number = (vf /DB0 ) velocity vector, m s − 1 pressure, N m − 2 temperature, K diameter of molecule, m radius, m the empty gap between the inner and outer walls, m Cartesian coordinates, m dimensionless Cartesian coordinates

Greek symbols 𝛼 thermal diffusivity, m2 s − 1 𝛽 thermal expansion coefficient, K − 1 𝜇 dynamic viscosity, kg m − 1 s − 1 𝜌 density, kg m − 3 𝜀 porosity 𝜈 kinematic viscosity, m2 s − 1 𝜑 volume fraction of the nanoparticles 𝜃 inclination angle of enclosure 𝜉 Polar angle 𝛾 constant parameter 𝛿 constant parameter Subscripts Ave c eq f h nf np loc T p out in B ∗

m ef

average cold equivalent base fluid hot nanofluid nanoparticle local thermophoresis porous outer wall (m) inner wall (m) Brownian motion Dimensionless unit mean effective thermal conductivity

hybrid nanofluid inside a double porous medium using a two-equation energy model. Several researchers have analyzed different geometric shapes, including; Sheikholeslami [15] assessed a numerical solution for magnetic nanofluid convective heat transfer in a porous curved enclosure. The

Fig. 1. Geometry of half-annulus enclosure with a porous medium.

motion of the nanoparticles decreases with increasing Hartmann number and it increases with the increase of the Darcy number. Also Sheikholeslami [16] showed temperature gradient enhances with the increase of Da in a numerical study of nanofluid free convection in the porous curved cavity. Taghizadeh et al. [17] modeled convective heat transfer and entropy generation in an inclined cavity with a circular porous cylinder is located at the center. The results declared that the effects of Darcy number and inclination angle on heat transfer depend on the Richardson number. Oğlakkaya and Canan Bozkaya [18] studied natural displacement inside Kuwait and in the presence of a magnetic field. According to the results, stream strength and Nusselt number increase with increasing Hartmann number and Rayleigh number. Hatami et al. [19] investigated the variable magnetic field (VMF) effect on the heat transfer of a half-annulus cavity filled by Fe3 O4 -water nanofluid under constant heat flux. Sheikholeslami and Ganji [20] examined the impact of external magnetic source on Fe3 O4 –water ferrofluid convective heat transfer in a porous cavity. Several researchers published articles about magnetic nanofluid or porous heat transfer [21-30]. In recent works, researchers have investigated the natural convection within the enclosures and at different inclination angles, but simultaneous using of the two-phase model, the porous medium and the distribution of nanoparticles is not studied. Besides, in the present work, the distribution of nanoparticles and the natural heat transfer rate have been thoroughly investigated with a two-phase Buongiorno’s model and Darcy model. Solutions are performed for different porous Rayleigh numbers (10 ≤ Rap ≤ 103 ), volume fractions of nanoparticles (0 ≤ 𝜑Ave ≤ 0.04)), porosity numbers (𝜀 = 0.4 and 0.7) and the inclination angle of the enclosure (0 ≤ 𝜃 ≤ 90). 2. Mathematical modeling 2.1. Problem statement The investigated porous half-annulus schematic is depicted in Fig. 1. The internal semicircle wall is under the effect of the constant heat flux q‫״‬, and it can be considered as a hot wall. Moreover, an outer semicircle wall is at a constant temperature of Tc as a cold wall. Also, the left and right bottom walls are insulated. The enclosure is filled with water- Fe3 O4 magnetic nanofluid. The inner and outer semicircle radius are rin = 0.025 (m), rout = 0.05 (m), and L = rout − rin = 0.025 (m). Gravity is in the downward direction. 2.2. Dimensional form of governing equations and boundary conditions The flow is assumed to be incompressible, steady, two-dimensional, and Darcy model is applied to model flow inside the porous medium. The

S.Y. Motlagh, E. Golab and A.N. Sadr

International Journal of Mechanical Sciences 164 (2019) 105183

two-phase model of Buongiorno applied for modeling the relative movement of nanoparticles inside the base flow. Nanoparticle and porous medium are in local thermal equilibrium and the same temperature. Nanoparticles shape is spherical. The density of fluid varies linearly with temperature in free convection by using Boussinesq approximation model. According to the above assumptions, mass, momentum, energy conservation, and nanoparticle distribution equations are as follows: Mass conservation: ∇.𝑽 = 0.

(1)

Momentum equation: 𝜌𝑛𝑓 𝜀2

𝑽 .∇𝑽 = −∇𝑃 −

𝜇𝑛𝑓 𝐾

( ) ( ) 𝑽 + ∇. 𝜇𝑛𝑓 ∇𝑽 + (𝜌𝛽)𝑛𝑓 𝑇 − 𝑇𝐶 𝒈.

(2)

Where V is velocity vector, P and T denote pressure and temperature, respectively. Moreover, in the above equation g is gravitational acceleration. The subscripts nf and 𝜀 denote nanofluid properties and porosity (fraction of fluid volume to total volume), respectively. Likewise, in Eq. (2) K refers to permeability of the porous medium. Volume fraction equation: ( ) 1 1 𝑽 .∇𝜑 = − ∇. 𝑱 𝑝 . 𝜀 𝜌𝑛𝑝

(3)

Two dominant nanoparticles mass fluxes introduced in Jp vector, which are Brownian motion (JB ) and thermophoresis effect(JT ). 𝑱 𝐵 = −𝜌𝑛𝑝 𝐷𝐵 ∇𝜑.

𝐷𝐵 =

(4)

𝑘𝐵 𝑇 . 3𝜋𝜇𝑓 𝑑𝑛𝑝

(5)

Mass flux due to Brownian motion conveyed by Eq. (4), where 𝜌np is the density of nanoparticles. Brownian diffusion coefficient (DB ) calculated by Einstein-Stokes’s Eq. (5). Thermophoresis effect obtained as: 𝑱𝑇

∇𝑇 = −𝜌𝑛𝑝 𝐷𝑇 . 𝑇

(6)

2.3. Properties of equivalent fluid The thermophysical properties of the base fluid, nanoparticles, and porous medium demonstrated in Table. 1. Nanofluid properties calculated from Eqs. (12-16) [31]: 𝜌𝑛𝑓 = (1 − 𝜑)𝜌𝑓 + 𝜑𝜌𝑛𝑝 . (

𝜌𝐶𝑃

𝜇𝑓 𝜌𝑛𝑝

𝜑.

𝑓 +𝑘𝑛𝑝

(8)

Finally, drift particles mass flux equal to Eq. (8), by substituting two mentioned mass fluxes in Eq. (3), the equation for volume fraction obtains as: ( ) 1 ∇𝑇 . 𝑽 .∇𝜑 = ∇. 𝐷 𝐵 ∇𝜑 + 𝐷 𝑇 (9) 𝜀 𝑇 (

Energy equation: ) ( ) ( ) 𝜌𝐶𝑃 𝑛𝑓 𝑽 .∇𝑇 = ∇. 𝑘𝑒𝑞 ∇𝑇 − 𝜀 𝐶𝑝 𝑛𝑝 𝑱 𝑝 .∇𝑇 .

+ 𝜑(𝜌𝐶𝑃 )𝑛𝑝 .

(13) (14)

Properties of equivalent fluid containing the base fluid, nanoparticles, and porous medium calculated as: ( ) (𝜌𝑐𝑝 )𝑒𝑞 = 𝜀(𝜌𝑐𝑝 )𝑛𝑓 + (1 − 𝜀) 𝜌𝑐𝑝 𝑝. (17) 𝑘𝑒𝑞 = 𝜀𝑘𝑛𝑓 + (1 − 𝜀)𝑘𝑝 .

(18)

The subscripts np, f, nf, eq, and p denote nanoparticles, fluid, nanofluid, equivalent properties, and porous medium, respectively. The permeability K of the solid matrix is calculated by the relation of Ergun [38]: 𝐾=

𝑑𝑏2 𝜀3 175(1 − 𝜀)2

.

(19)

Where db is the mean diameter of the porous medium. 2.4. Non-dimensional form of governing equations and boundary conditions

𝑋 =

(10)

(11)

𝑇 −𝑇 𝑦 𝜑 𝑥 𝑽𝐿 𝑃 𝐿2 , 𝑌 = , 𝑇 ∗ = 𝑘𝑓 ′′ 𝑐 , 𝑽 ∗ = , 𝑃∗ = , 𝜑∗ = , 𝐿 𝐿 𝑞 𝐿 𝑣𝑓 𝜑𝐴𝑣𝑒 𝜌𝑓 𝑣2𝑓

∇∗ = 𝐿∇, 𝛿 = 𝑘𝑓 𝐷𝐵0 =

𝜇𝑓 𝑇𝑐 𝐷 𝐷 ∗ , 𝐷𝐵 = 𝐵 , 𝐷𝑇∗ = 𝑇 , 𝐷𝑇 0 = 𝛾 𝜑 , 𝑞 ′′ 𝐿 𝐷𝐵0 𝐷𝑇 0 𝜌𝑓 𝐴𝑣𝑒

𝐾𝐵 𝑇𝑐 𝑞 ′′ 𝐿 , for 𝜑𝐴𝑣𝑒 = 0.02, 𝑇𝑚 = 𝑇𝑐 + . 3𝜋𝜇𝑓 𝑑𝑛𝑝 2𝑘𝑓

(20)

Substituting the above parameters into the Eqs. (1), (2), (10) and (12) to (18), the non-dimensional form of governing equations can be obtained as: Dimensionless mass conservation equation: ∇∗ .𝑽 ∗ = 0. (

Boundary conditions on walls are non-slip for velocity, zero mass flux for volume fraction equation and already mentioned thermal boundary conditions. The mathematical form of boundary conditions are as follows: 𝑽 = 0 on all walls. ∇𝜑.𝑛 = 0 and ∇𝑇 .𝑛 = 0 on the horizontal bottom walls. 𝐷 𝑞 ′′ 𝐿 ∇𝜑.𝑛 = − 𝑇 ∇𝑇 .𝑛 and ∇𝑇 .𝑛 = on the inner semicircle wall. 𝐷𝐵 𝑘𝑓 𝐷𝑇 ∇𝜑.𝑛 = − ∇𝑇 .𝑛 and 𝑇 = 𝑇𝑐 on the outer semicircle wall. 𝐷𝐵

𝑓

There are many relations for the thermal conductivity of nanofluid [36]. Thermal conductivity coefficient of nanofluid calculated from Hamilton-crosser [37] model for spherical particles: ( ) 𝑘𝑛𝑓 𝑘𝑛𝑝 + 2𝑘𝑓 + 2𝜑 𝑘𝑛𝑝 − 𝑘𝑓 = (16) ( ) . 𝑘𝑓 𝑘𝑛𝑝 + 2𝑘𝑓 − 𝜑 𝑘𝑛𝑝 − 𝑘𝑓

).

𝑱𝑃 = 𝑱𝐵 + 𝑱𝑇 .

= (1 − 𝜑) 𝜌𝐶𝑝

By introducing the following dimensionless variables:

The constant magnitude of 𝛾 defines as a function of the base fluid and nanoparticles thermal conductivity(𝛾 = 0.26 𝑘

𝑛𝑓

(12)

)

For nanofluid viscosity calculation, Brinkman’s equation [35] used as below: 𝜇𝑓 𝜇𝑛𝑓 = . (15) (1 − 𝜑)2.5

(7)

𝑘𝑓

(

(𝜌𝛽)𝑛𝑓 = (1 − 𝜑)(𝜌𝛽)𝑓 + 𝜑(𝜌𝛽)𝑛𝑝 .

Thermophoresis coefficient showing by DT calculated as: 𝐷𝑇 = 𝛾

)

(21)

Dimensionless momentum equation: ) ( ) ( ) μ𝑛𝑓 μ𝑛𝑓 ∗ ∗ 1 𝜌𝑛𝑓 1 𝑽 ∗ .∇∗ 𝑽 ∗ = −∇∗ 𝑃 ∗ − 𝑽 ∗ + ∇∗ . ∇ 𝑽 𝜇𝑓 𝐷𝑎 𝜇𝑓 𝜀 2 𝜌𝑓 +

(𝜌𝛽)𝑛𝑓 𝜌𝑓 𝛽𝑓

1 𝑅𝑎𝑓 𝑇 ∗ .𝒆̂ . 𝑃𝑟

(22)

Dimensionless energy equation: ( ) (( ) ) 𝜌𝐶𝑃 𝑛𝑓 𝑘𝑒𝑞 1 ∗ ∇ . ∇∗ 𝑇 ∗ ( ) 𝑽 ∗ .∇∗ 𝑇 ∗ = 𝑃𝑟 𝑘𝑓 𝜌𝐶𝑃 𝑓 ( ) ∗ ∗ ∗ ∗ 1 1 1 ∗ ∗ ∗ ∗ ∗ ∗ ∇ 𝑇 .∇ 𝑇 +𝜀 𝐷 𝐵 ∇ 𝜑 .∇ 𝑇 + 𝐷 . (23) 𝑃 𝑟 Le 𝑁𝐵𝑇 𝑇 1+T∗∕ 𝛿

S.Y. Motlagh, E. Golab and A.N. Sadr

International Journal of Mechanical Sciences 164 (2019) 105183

Table 1 Properties of base fluid, nanoparticles, and porous medium.

Water Fe3 O4 Porous medium

𝝆( 𝒎𝒌𝒈3 )

𝒌( 𝒎𝑾𝑲 )

𝑪 𝒑 ( 𝒌𝒈𝑱𝑲 )

𝜷 × 10−5 ( 𝑲1 )

𝝁 × 10−6 ( 𝒎𝒌𝒈𝒔 )

dp (nm)

993 5200 –

0.628 6 100

4178 670 –

36.2 1.3 –

695 – –

0.384 25 –

Table. 2 Average Nusselt number at differrent gird. Mesh size in radial direction ×angular direction Average Nusselt number

𝑅𝑎 𝑓 =

𝑔 𝐵𝑓 𝑞 ′′ 𝐿4

𝑘𝑓

(24)

𝑃𝑟 =

𝑣𝑓 𝛼𝑓

, 𝛼𝑓 =

𝑘𝑓 (𝜌𝑐𝑝 )𝑓

,

and Rap = Raf Da, and in energy equation 𝐿𝑒 =

𝑘𝑓 𝛼𝑓 𝑣𝑓

(𝜌𝐶𝑝 )𝑝 𝐷𝐵 𝜑𝑎𝑣𝑒

𝐾 , 𝐿2

, and 𝑁𝐵𝑇 =

𝜑𝑎𝑣𝑒 𝐷𝐵0 𝛿 𝐷𝑇 0

refer to Lewis number and the ratio

of Brownian motion to thermophoresis (motion caused by temperature 𝑣 gradient), respectively. The Schmidt number (𝑆𝑐 = 𝐷 𝑓 ) in volume frac𝐵0

tion equation presents the ratio of the momentum diffusivity to Brownian diffusivity. Furthermore, the dimensionless form of boundary conditions can be written as:

𝐷𝑇∗ ∇∗ 𝑇 ∗ .𝑛 ∗ 1+T ∕𝛿 ∗ ∗ ∗ 𝐷 − 𝑁1 𝐷𝑇∗ ∇ 𝑇 ∗ .𝑛 T 𝐵𝑇 𝐵 1+ ∕𝛿 𝐵𝑇

∇∗ 𝜑∗ .𝑛 =

∗ 𝐷𝐵

50×200 1.61948

70×280 1.61907

90×360 1.61907

and ∇𝑇 ∗ .𝑛 = 1 on the inner semicircle wall.

Object-oriented C++ code developed for numerical simulation of the governing equations and associated boundary conditions using the finite volume method. Diffusion and convective terms in equations are discretized using second-order central difference and upwind schemes, respectively. A collected grid system and SIMPLE algorithm are applied to handle the pressure-velocity coupling. 3.1. Mesh study The uniform mesh is applied for the simulations. The mesh study is performed for critical conditions: Rap = 1000, 𝜑Ave = 0.04, 𝜃 = 90°, and 𝜀 = 0.7. The results of average Nu for different meshes presented in Table 2, the mesh independent results achieved in the 50×200 grid. 3.2. Verification

𝑽 ∗ = 0 on all walls. ∇∗ 𝜑∗ .𝑛 = 0 and ∇𝑇 ∗ .𝑛 = 0 on the horizontal bottom walls. ∇∗ 𝜑∗ .𝑛 = − 𝑁1

30×120 1.62144

3. Numerical method, gird study and verification

Dimensionless volume fraction equation: ) ( 1 ∗ ∗ ∗ 1 1 ∇∗ 𝑇 ∗ ∗ ∗ ∗ 𝑽 .∇ 𝜑 = ∇∗ . 𝐷𝐵 ∇ 𝜑 + 𝐷𝑇∗ . ∗ 𝜀 𝑆𝑐 𝑁𝐵𝑇 1+T ∕𝛿 Where in momentum equation 𝒆̂ = 𝒈∕𝑔 , =

20×80 1.62484

(25)

and 𝑇 ∗ = 0 on the outer semicircle wall.

The average Nusselt number in the hot wall calculates by integrating the local Nusselt number as: 𝑁 𝑢𝑙𝑜𝑐 =

𝑘𝑛𝑓 1 . 𝑘𝑓 𝑇 ∗

(26)

𝑁 𝑢𝑎𝑣𝑒 =

1 𝜋 ∫ 𝑁 𝑢𝑙𝑜𝑐 (𝜉)𝑑𝜉. 𝜋 0

(27)

The developed code for this work has different sections, for validation of the porous medium modeling sector, the comparison of present work with study of Basak et al. [32] performed, in their work, the mixed convection of the porous lid-driven cavity investigated; bottom wall of the cavity is uniformly heated and the other two side walls are linearly heated and the upper wall is insulated. In Fig. 2, the variations of local Nusselt number presented at Re = 10, Gr = 105 , Da = 10−3 , Pr = 10, 𝜑Ave = 0, and inclination angle 00 . The results are in good agreement. Also, Fig. 3 shows the acceptable agreement of isotherms and streamlines with the study of Basak et al. [32]. For validation of Buongiorno two-phase model, the free convection of the Al2 O3 -water nanofluid in a cavity applied for the following conditions: 2 ≤ 𝚫T ≤ 10, (3.37 × 105 ≤ Ra ≤ 1.68 × 106 ), the average volume

Fig. 2. Comparison of local Nusselt number versus distance of Basak et al. [32] and current work. On (a) left wall and (b) right wall of the porous cavity.

S.Y. Motlagh, E. Golab and A.N. Sadr

International Journal of Mechanical Sciences 164 (2019) 105183

Fig. 3. Streamlines and isotherms comparison for Re = 10, Gr = 105 , Da = 10−3 , Pr = 10, 𝜑Ave = 0 and 𝛼 = 00 , (a) numerical results of Basak et al. [32] and (b) current work for porous cavity.

Fig. 4. Average Nu number comparison at different Ra numbers obtained from the present study with the experimental results of Ho et al. [33] and numerical results of Sheikhzadeh et al. [31].

fraction 3% (𝜑Ave = 0.03), Pr = 4.623, and inclination angle 00 . The results of this investigation have compared with the experimental results of Ho et al. [33], and Sheikhzadeh’s [31] numerical results. Fig. 4 demonstrates the variations of Nusselt number for present work and the aforementioned experimental and numerical studies, the results indicate a fairly good agreement among present work and experimental results and numerical surveys. Moreover, Fig. 5 presents the comparison of the results of the isotherms and the nanoparticles’ distribution obtained from present work with numerical studies of Sheikhzadeh et al. [31] and Garoosi et al. [34]. The results show that the current work is consistent with other numerical studies.

ters such as porous Rayleigh number (10 ≤ Rap ≤ 1000), the volume fraction of nanoparticles (0 ≤ 𝜑Ave ≤ 0.04), the inclination angle of the enclosure (0 ≤ 𝜃 ≤ 90) and porosity (𝜀 = 0.4 and 0.7) have been investigated on the rate of heat transfer and nanoparticle distribution. In all cases the amount of the other parameters is set as: Pr = 4.623, Sc = 3.55 × 104 , Tc = 310◦ 𝐶, q′′ = 48.01 ( mw2 ), 1.71 × 105 ≤ Le ≤ 6.84 × 105 , Da = 10−3 , 𝛿 = 161, and NBT = 0.245. The effects of parameters mentioned above on the flow pattern, local distribution of nanoparticles, isotherms, and heat transfer are presented in the following sections.

4. Results and discussion

4.1. Effects of the porosity number and inclination angle on the flow pattern

Free convection of Fe3 O4 -water nanofluid inside the porous halfannulus cavity investigated. The distribution of nanoparticles has been simulated by employing a modified two-phase model of Buongiorno and Darcy model for porous medium. Effects of important parame-

The flow patterns are plotted for 𝜑Ave = 0.01 and porous Rayleigh numbers 10 and 1000 in Fig. 6. According to the figures, at zero inclination angle and for all the porosity and porous Rayleigh numbers, there are two counter-rotating vortices. Moreover, increasing the porosity and

S.Y. Motlagh, E. Golab and A.N. Sadr

International Journal of Mechanical Sciences 164 (2019) 105183

Fig. 5. Present results (a) isotherms and (b) contours of nanoparticles distribution, with the numerical results of Sheikhzade et al. [31] and Garoosi et al. [34].

Fig. 6. Streamlines for 𝜑Ave = 0.01, 𝜺 = 0.4 and 0.7, 𝜽 = 00 , 300 , 600 , and 900 at (a) Rap = 10 and (b) Rap = 1000.

porous Rayleigh numbers move the center of the vortex to upward in half-annulus. By increasing the inclination angle to 30° for all the porosity and porous Rayleigh numbers, the right-hand vortex shrinks and the lefthand vortex grows. Furthermore, at 𝜃 = 900 , the right vortex vanishes, and the left one grows and covers the entire enclosure. In addition, the

single-cell vortex center moves upward by increasing the porosity and porous Rayleigh numbers. In Fig. 7 the percentage of the move up or move down of the vortex presents for right and left vortices. Where the 𝜎 is the height of the center of the vortex to the bottom of the chamber (insulation surface), and the percentage of 𝜎 changes compared to the horizontal chamber in the

S.Y. Motlagh, E. Golab and A.N. Sadr

International Journal of Mechanical Sciences 164 (2019) 105183

Fig. 7. Percentage of 𝝈 variation at 𝜑Ave = 0.01 and different porosity and porous Rayleigh numbers.

Fig. 8. Dimensionless isotherms for 𝜑Ave = 0.01,𝜺=0.4 and 0.7, 𝜽=00 , 300 and 900 at (a) Rap = 10 and (b) Rap = 1000.

same Rayleigh and porosity numbers is: 𝜎_𝑉 𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛(%) = (𝜎 𝜃 − 𝜎 𝜃=0 ) × 100∕𝜎 𝜃=0 . For left vortex in the low Rayleigh number with low buoyancy and advection effects, porosity number does not have a significant effect on 𝜎_𝑉 𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 (%), and the inclination angle has the dominant effect, for

both porosity numbers these effects are similar. While, according to the high Rayleigh number, at the lower porosity number, the effect of the inclination angle on the vortex elevation becomes more apparent. By comparing the two porous Rayleigh numbers, the lower Rap is more influenced by the inclination angle. For the right vortex at all poros-

S.Y. Motlagh, E. Golab and A.N. Sadr

International Journal of Mechanical Sciences 164 (2019) 105183

Fig. 9. Volume fraction contour for 𝜑Ave = 0.02,𝜺 = 0.4 and 0.7, 𝜽 = 00 , 300 and 900 at (a) Rap = 10 and (b) Rap = 1000.

ity and porous Rayleigh numbers, the inclination angle has a similar effect. 4.2. Effects of the porosity and inclination angle on temperature field The temperature contour lines are depicted for 𝜑Ave = 0.01 and porous Rayleigh numbers 10 and 1000 in Fig. 8. Only the effect of conduction heat transfer is predominated at zero inclination angle, porous Rayleigh number 10 and for all porosity numbers. Therefore, semi-circle temperature contour lines can be observed between the inner and outer walls. Moreover, the effect of advection heat transfer grows by increasing the porous Rayleigh number (Rap = 1000) and porosity number. Furthermore, for Rap = 1000, at all inclination angles, in all porosity numbers, the density of isotherms increases at the upper portion of the top wall, and the left and right parts of the down wall due to the buoyancy effect. It demonstrates that heat transfer is improved in the same regions. As well as, the porosity number does not have any effect on the isotherms at low porous Rayleigh number (Rap = 10), because of the dominant effect of conduction heat transfer. Against, at the high porous Rayleigh number (Rap = 1000), due to the overcoming effect of advection, porosity number changes affected the advection mechanism of heat transfer, and consequently it can advance the heat transfer rate. In low porous Rayleigh number, the inclination angle, porosity and porous Rayleigh numbers do not affect on the temperature difference of the two walls. But, the temperature difference of the two walls has a direct relationship with porosity number and porous Rayleigh number.

inclination angle, in porous Rayleigh number 10, and all the porosity numbers, the two open-mass boundary layers are formed near the left and right of half-annulus walls. In the 𝜀 = 0.4, the mass boundary layers remain unchanged when the porous Rayleigh number increases to 1000. However, in 𝜀 = 0.7, by the increment of Rayleigh number, the velocity of nanofluid is increased then it changes the mass boundary layer shape. The amount of accumulation of nanoparticles near the top wall is more than the down wall due to the lower temperature of the top wall and the thermophoresis effect. In Fig. 10 dimensionless distribution of nanoparticle is plotted for 𝜑Ave = 0.02 at 𝜀 = 0.4 and 0.7. As shown in Fig. 10a, for the low porosity number (𝜀 = 0.4), in the low porous Rayleigh number 10, with the variation of the tilt angle of the cavity, the local distribution of the nanoparticles on the inner and outer semicircle walls does not change. While in the high porous Rayleigh number (Rap = 1000), when the effects of the buoyancy force increase, the inclination angle variation causes the local distribution of the nanoparticles to change over the half-annulus walls. In addition, at Rap = 1000and 𝜃 = 900 , on the outer wall for 𝜉 < 600 and 𝜉 > 600 (angle of 𝜉 is shown in Fig. 1), the volume fraction of the nanoparticles decreases and increases, respectively. As well as, on the inner half-circle wall, for 𝜉 < 600 the volume fraction does not change on the wall, but the amount of nanoparticles on the wall decreases in 𝜉 > 600 . It is also observed by comparing Fig. 10a (𝜀 = 0.4) and b (𝜀 = 0.7), which porosity variation does not have a significant effect on the distribution of nanoparticles in both porous Rayleigh numbers. In Fig. 10a, the non-dimensional local distribution of the volume fraction (𝜑∗ = 𝜑 𝜑 ) is illustrated on the inner and outer semicircle walls 𝐴𝑣𝑒

4.3. Effects of the porosity and inclination angle on the distribution of nanoparticles The contour of nanoparticles distribution for 𝜑Ave = 0.02 and porous Rayleigh numbers 10 and 1000 are shown in Fig. 9. The black lines are the contour lines of volume fraction. According to the results, at zero

for the different averaged volume fractions. Given the result, it is obvious that the change in the average volume fraction does not affect the dimensionless local distribution of nanoparticles. Also, due to the thermophoresis effect, the accumulation of nanoparticles on the outer wall is higher than the inner wall, because the internal wall temperature is higher than the external one.

S.Y. Motlagh, E. Golab and A.N. Sadr

International Journal of Mechanical Sciences 164 (2019) 105183

Fig. 10. Local normalized volume fraction (𝝓∗ ) on inner and outer cylinders (versus angle of 𝝃) for 𝜑Ave = 0.02, Rap = 10and1000, and different inclination angle at (a) 𝜺 = 0.4 and (b) 𝜺 = 0.7.

Fig. 11. Local normalized volume fraction (𝝓∗ ) on inner and outer cylinder walls (versus angle of 𝝃) and 𝜃=30° (a) Rap = 10, 𝜺 = 0.7, and different volume fraction and (b) 𝜑Ave = 0.02 and different Rap and 𝜺.

In Fig. 11b, the effect of porous Rayleigh number on the local distribution of nanoparticles on the annulus walls is demonstrated for different porosity numbers. The results indicate that, firstly, in both porous Rayleigh numbers, porosity number does not affect the distribution of nanoparticles. Secondly, in both porosity numbers, by increasing Rayleigh number, 𝜙∗ decreases on the outer cylinder wall and increases on the inner cylinder wall. Also, 𝜙∗ on walls tends to the unit amount which represents a more uniform distribution of nanoparticles.

4.4. Effects of the different parameters on Nusselt number The variations of Nusselt number versus the inclination angle (𝜃) for different volume fractions are given in Fig. 12a for the low porosity number (𝜀 = 0.4) and in Fig. 12b for the high porous number (𝜀 = 0.7). Regarding Fig. 12a, in low porous Rayleigh number, due to the dominance of the conduction mechanism of the heat transfer, the change in the tilt angle, and consequently the alteration of the buoyancy force, does not affect Nusselt number. Also, increasing the volume fraction in-

S.Y. Motlagh, E. Golab and A.N. Sadr

International Journal of Mechanical Sciences 164 (2019) 105183

Fig. 12. Nu number versus inclination angle and different volume fractions at (a) 𝜺 = 0.4 and (b) 𝜺 = 0.7.

Fig. 13. Percentage of Nu variation versus inclination angle for different volume fractions at (a) 𝜺 = 0.4 and (b) 𝜺 = 0.7.

creases Nusselt number only due to the enhance in the nanofluid conductivity. On the other hand, in high Rayleigh number (Rap = 1000), due to the dominance of the buoyancy force, the velocity of the fluid increases. Therefore, changing the inclination angle of the annulus affects the flow pattern, temperature field, and consequently Nusselt number. Then, by increasing 𝜃, in all volume fractions, Nusselt number changes and decreases. As well as, according to Fig. 12, at porous Rayleigh number 10, the volume fraction of nanofluid has a positive effect on conductivity and therefore Nusselt number. According to Fig. 12b, with the increase

of porosity number to 0.7, at porous Rayleigh number of 10, due to the dominance of the conduction, the amounts and variations of Nusselt number are similar to the porosity number of 0.4. However, in the high porous Rayleigh number (Rap = 1000), with increasing the porosity number, the flow rate in cavity increases, which increases Nusselt number in all volume fractions and tilt angles. In general, it is observed that in low Rayleigh number, the change of angle does not affect the results, but in high Rayleigh number, the increase of the tilt angle reduces the effects of the buoyancy force, and consequently Nu number decreases.

S.Y. Motlagh, E. Golab and A.N. Sadr

The percentage of variations of Nu number (𝑁𝑢_𝑉 𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 (%) = θ =0, θ=0 =0, θ=0 (𝑁 𝑢𝜑, − 𝑁 𝑢𝜑𝐴𝑣𝑒 ) × 100∕𝑁 𝑢𝜑𝐴𝑣𝑒 ) versus the inclination angle at 𝐴𝑣𝑒 different volume fractions and porous Rayleigh numbers, for the porosity numbers 0.4 and 0.7 are shown in Fig. 13a and b, respectively. According to the results, in the low porous Rayleigh number, as noted earlier, the conduction is dominant, and with the increase of the average volume fraction due to the increase of the conductivity, the percentage of Nu variation rises independently of the tilt angle and porosity number. In the high porous Rayleigh number, with the increase of the average volume fraction, the percentage of Nu variation rises. Also, in both porosity numbers 0.4 and 0.7, the percentage of Nu variation decreases by increasing the inclination angle of the semi-annulus due to the reduction of buoyancy effects. So that in the high porosity number, in volume fractions 0 and 0.01, at the enclosure tilt angle of 30 and 90°, the sign of the percentage of Nu variation is negative, which indicates the reduction of Nu number of nanofluid compared to the pure water at zero inclination angle. Accordingly, the positive effect of the porous medium on the heat transfer rate is clear. In general, the negative effect of the tilt angle on the Nu number variation, in the high porosity number (𝜀 = 0.7), is greater than the low porosity number (𝜀 = 0.4). 5. Conclusion In the present work, the two-phase numerical simulation of natural convection in a porous half-annulus cavity filled with a nanofluid has been studied. Buongiorno’s two-phase model along with the porous model of Darcy are applied for the modeling. The inner semicircle wall of half annulus is subjected to the constant flux and the outer semicircle wall is in constant temperature. Effects of different porous Rayleigh number, inclination angle of enclosure, volume fraction of nanoparticles and porosity number on the flow pattern, temperature field, nanoparticle distribution, and Nusselt number are investigated. The results showed that: - The concentration of nanoparticles on the wall with a constant temperature boundary condition is more than the wall with constant flux due to the thermophoresis effect. - The porosity number has a very small effect on the distribution of nanoparticles. - Distribution of nanoparticles in the high porous Rayleigh number is more uniform than the low porous Rayleigh number. - Local dimensionless distribution of nanoparticles on cavity walls is independent of average volume fraction magnitude. - Inclination angle in the low porous Rayleigh (Rap = 10) does not affect the average Nusselt number, because of the dominant effect of the conduction. However, in the high porous Rayleigh number (Rap = 1000), the Nusselt number is inversely proportional to the inclination angle. - Porosity in the low porous Rayleigh number does not affect the average Nusselt number. However, in the high porous Rayleigh number (Rap = 1000), the heat transfer rate is raised by increasing the porosity number. References [1] Saidur R, Leong KY, Mohammad H. A review on applications and challenges of nanofluids. Renew Sust Energy Rev 2011;15(3):1646–68. [2] Xu HJ, Xing ZB, Wang FQ, Cheng ZM. Review on heat conduction, heat convection, thermal radiation and phase change heat transfer of nanofluids in porous media: fundamentals and applications. Chem Eng Sci 2018;195:462–83. [3] Hajmohammadi MR. Assessment of a lubricant based nanofluid application in a rotary system. Energy Convers Manage 2017;146:78–86. [4] Sharifi A, Yekani Motlagh S, Badfar H. Numerical investigation of magnetic drug targeting using magnetic nanoparticles to the aneurysmal vessel. J Magn Magn Mater 2019;474:236–45. [5] Sheikholeslami M, Gorji-Bandpy M, Ganji DD, Soleimani S. Effect of a magnetic field on natural convection in an inclined half-annulus enclosure filled with Cu–water nanofluid using cvfem. Adv Powder Technol 2013;24(6):980–91. [6] Sheikholeslami M, Ganji DD. CVFEM for free convective heat transfer of CuO-water nanofluid in a tilted semi annulus. Alexandria Eng J 2017;56(4):635–45.

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