International Journal of Mechanical Sciences 136 (2018) 200–219

Contents lists available at ScienceDirect

International Journal of Mechanical Sciences

journal homepage: www.elsevier.com/locate/ijmecsci

Conjugate natural convection of Al 2

O 3

–water nanofluid in a square cavity

with a concentric solid insert using Buongiorno ’s two-phase model

A.I. Alsabery

a , ∗ , M.A. Sheremet b , c , A.J. Chamkha

d , e , I. Hashim

a

a School of Mathematical Sciences, Universiti Kebangsaan Malaysia, UKM Bangi 43600, Selangor, Malaysia b Department of Theoretical Mechanics, Tomsk State University, Tomsk 634050, Russia c Institute of Power Engineering, Tomsk Polytechnic University, Tomsk 634050, Russia d Department of Mechanical Engineering, Prince Sultan Endowment for Energy and the Environment, Prince Mohammad Bin Fahd University, Al-Khobar 31952, Saudi

Arabia e RAK Research and Innovation Center, American University of Ras Al Khaimah, Ras Al Khaimah, United Arab Emirates

a r t i c l e i n f o

Keywords:

Natural convection

Thermophoresis

Brownian diffusion

Square cavity

Isothermal corner boundaries

Buongiorno model

a b s t r a c t

The problem of conjugate natural convection of Al 2 O 3 –water nanofluid in a square cavity with concentric solid

insert and isothermal corner boundaries using non-homogenous Buongiorno ’s two-phase model is studied numer-

ically by the finite difference method. An isothermal heater is placed on the left bottom corner of the square cavity

while the right top corner is maintained at a constant cold temperature. The remainder parts of the walls are kept

adiabatic. Water-based nanofluids with Al 2 O 3 nanoparticles are chosen for the investigation. The governing pa-

rameters of this study are the nanoparticle volume fraction (0 ≤ 𝜙≤ 0.04), the Rayleigh number (10 2 ≤ Ra ≤ 10 6 ),

thermal conductivity of the solid block ( 𝑘 𝑤 = 0 . 28 , 0.76, 1.95, 7 and 16) (epoxy: 0.28, brickwork: 0.76, granite:

1.95, solid rock: 7, stainless steel: 16) and dimensionless solid block thickness (0.1 ≤ D ≤ 0.7). Comparisons with

previously experimental and numerical published works verify good agreement with the proposed method. Nu-

merical results are presented graphically in the form of streamlines, isotherms and nanoparticles volume fraction

as well as the average Nusselt number and fluid flow rate. The results show that the thermal conductivity ratio

and solid block size are very good control parameters for an optimization of heat transfer inside the partially

heated and cooled cavity.

© 2017 Elsevier Ltd. All rights reserved.

1

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. Introduction

Natural convection heat transfer in cavities is a significant phe-

omenon in engineering systems due to its widespread applications in

perations of solar collectors, cooling of containment buildings, heat ex-

hangers, storage tanks, double pane windows, etc. Laminar fluid flow

as become a favorite aspect in heat storage applications, as there has

een a rise in research activities regarding natural convection heat trans-

er. Thermal fluids are very important for heat transfer in many indus-

rial applications. The low thermal conductivity of conventional heat

ransfer fluids such as water and oils is a primary limitation in enhanc-

ng the performance and the compactness of many engineering elec-

ronic devices. An innovative and new technique to enhance heat trans-

er is using solid particles in the base fluid (i.e. nanofluids) in the range

f sizes 10–50 nm. A nanofluid is defined as a smart fluid with sus-

ended nanoparticles of average sizes below 100 nm in conventional

eat transfer fluids such as water, oil, and ethylene glycol [1] . Due

o small sizes and very large specific surface areas of the nanoparti-

∗ Corresponding author.

E-mail address: alsabery_a@ukm.edu.my (A.I. Alsabery).

ttps://doi.org/10.1016/j.ijmecsci.2017.12.025

eceived 28 August 2017; Received in revised form 28 November 2017; Accepted 5 December

vailable online 21 December 2017

020-7403/© 2017 Elsevier Ltd. All rights reserved.

les, nanofluids have superior properties like high thermal conductivity,

inimal clogging in flow passages, longterm stability and homogene-

ty. Also, nanoparticles are used because they stay in suspension longer

han larger particles. Thus, nanofluid seems a good candidate for heat

emoval mechanisms in practical, thermal, fluid-based applications. The

hermal conductivity of nanoparticles is higher than that of traditional

uids. Thus, nanofluids can be used in a large industrial applications

uch as oil industry, nuclear reactor coolants, solar cells, construction,

lectronics, renewable energy and many others. The solid particles are

sually metal or metal oxides such as copper (Cu), copper oxide (CuO),

luminum oxide (Al 2 O 3 ), titanium (TiO 2 ) and silver (Ag).

A comprehensive work on natural convection in cavities that are

artially occupied by nanofluids was reported by Khanafer et al. [2] .

ou and Tzeng [3] considered natural convective heat transfer in

anofluids occupying a rectangular cavity. The numerical simulation of

he fluid and temperature distributions and the convective heat transfer

f the nanofluid could be classified by two main approaches, namely a

ingle-phase model (homogenous) or a two-phase model [4] . The single-

2017

A.I. Alsabery et al. International Journal of Mechanical Sciences 136 (2018) 200–219

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Nomenclature

C p Specific heat capacity

d Width and height of the inner solid block

d f Diameter of the base fluid molecule

d p Diameter of the nanoparticle

D Dimensionless thickness of the inner solid block,

𝐷 = 𝑑∕ 𝐿 D B Brownian diffusion coefficient

D B 0 Reference Brownian diffusion coefficient

D T Thermophoretic diffusivity coefficient

D T 0 Reference thermophoretic diffusion coefficient

g Gravitational acceleration

k Thermal conductivity

K r Wall to nanofluid thermal conductivity ratio, 𝐾 𝑟 =

𝑘 𝑤 ∕ 𝑘 𝑛𝑓 L Width and height of enclosure

Le Lewis number

N BT Ratio of Brownian to thermophoretic diffusivity

𝑁𝑢 Average Nusselt number

Pr Prandtl number

Ra Rayleigh number

Re B Brownian motion Reynolds number

T Temperature

T 0 Reference temperature (310 K)

T fr Freezing point of the base fluid (273.15 K)

v Velocity vector

V Normalized velocity vector

u B Brownian velocity of the nanoparticle

x, y and X, Y Space coordinates and dimensionless space coordi-

nates

Greek symbols

𝛼 Thermal diffusivity

𝛽 Thermal expansion coefficient

𝛿 Normalized temperature parameter

𝜃 Dimensionless temperature

𝜇 Dynamic viscosity

𝜈 Kinematic viscosity

𝜌 Density

𝜑 Solid volume fraction

𝜑 ∗ Normalized solid volume fraction

𝜙 Average solid volume fraction

Subscripts

c Cold

f Base fluid

h Hot

nf Nanofluid

p Solid nanoparticles

w Inner solid block

hase approach considers the fluid phase and the nanoparticles as being

n thermal equilibrium where the slip velocity between the base fluid

nd the nanoparticles is negligible. On the other hand, the two-phase

pproach assumes that the relative velocity between the fluid phase and

he nanoparticles may not be zero where the continuity, momentum and

nergy equations of the nanoparticles and the base fluid are handled us-

ng different methods. There are number of numerical studies used the

ingle-phase model for simulation of the nanofluids. Hu et al. [5] studied

xperimentally and numerically the natural convection heat transfer in

square cavity filled with TiO 2 –water nanofluids. They found that the

verage Nusselt number increased with the addition of nanoparticles.

arimipour et al. [6] reported a study on mixed convection in a shallow

nclined lid driven cavity filled with a Cu–water nanofluid using

201

he lattice Boltzmann method. Using the same method Karimipour

t al. [7] investigated the problem of laminar forced convection in

microchannel filled Cu–water nanofluids. Sheremet et al. [8] and

lsabery et al. [9] numerically investigated the natural convection heat

ransfer of nanofluid flow in different geometries. Ghalambaz et al.

10] numerically studied the effects of the diameter and concentration

f nanoparticles on the natural convection of Al 2 O 3 –water nanofluids

onsidering the variable thermal conductivity around a porous medium.

hey found that the heat transfer rate decreased with an increase in the

olume fraction of nanoparticles or a decrease in the size of nanopar-

icles. Zaraki et al. [11] theoretically analyzed the natural convection

eat transfer of nanofluids for which various aspects of nanoparticles are

onsidered. Karimipour [12] developed a new correlation for Nusselt

umber for the problem of convective heat transfer in a microchannel

lled with three types of nanofluids by using lattice Boltzmann method.

Recently, Umavathi and Sheremet [13] numerically studied the ef-

ect of the temperature-dependent conductivity on natural convective

eat transfer in a vertical rectangular duct filled with a nanofluid using

he finite-difference method. They concluded that the heat transfer rate

ncreased at the left wall and decreased at the right wall as the aspect

atio increased, whereas the heat transfer rate increased at both of the

alls as the solid volume fraction increased. Karimipour et al. [14] con-

idered numerically the effect of indentation on flow parameters and

low heat transfer in in a rectangular micro channel filled with Ag–

ater nanofluid using the finite volume method. Sheikholeslami et al.

15] used the single phase model (Koo–Kleinstreuer–Li) of a nanofluid to

tudy the natural convection heat transfer in a square cavity where they

ound that the convection heat transfer is increased with the increase in

he volume fraction of nanoparticles. Most of these studies are used the

axwell-Garnett and Brinkman models to estimate the effective thermal

onductivity and viscosity of the nanofluid. However, the study of Cor-

ione [16] questions the validity of these models and tended to proposed

new model for estimating the effective thermal conductivity and vis-

osity of the nanofluid which appeared to be close to the experimental

ata. The results showed that the heat transfer rate enhanced with the

elative concentration of nanofluid. The experimental study of Wen and

ing [17] found that the slip velocity between the base fluid and par-

icles may not be zero. Thus, the two-phase nanofluid model observed

o be more accurate. Buongiorno [18] proposed a non-homogeneous

quilibrium model with the consideration of the effect of the Brownian

otion (movement of nanoparticles from high concentration site) and

hermophoresis (movement of nanoparticles from the high temperature

ite to the low temperature site) as a two important primary slip mecha-

isms in nanofluid. He tended to introduce in this study seven slip mech-

nisms between nanoparticles and the base fluid where he developed a

on-homogeneous two-component equations in nanofluids showing the

mportance of Brownian motion and thermophoresis compared to other

ransport mechanisms. Sheikholeslami et al. [19] used the two-phase

odel of the nanofluid to investigate the thermal management for nat-

ral convection heat transfer in a 2D cavity. Esfandiary et al. [20] and

otlagh and Soltanipour [21] investigated numerically the problem of

atural convection of nanofluids in a square cavity using the two phase

odel. The results of these studies indicated that the heat transfer rate

nhanced with the increasing of the concentration of the nanoparticles

p to 0.04.

Conjugate convective heat transfer for a regular fluid has very im-

ortant practical engineering applications in frosting practicalities and

efrigeration of the hot obtrusion in a geological framing. For exam-

le, modernistic construction of thermal insulators which are formed of

wo diverse thermal conductivities (solid and fibrous) materials can be

odeled by the partition length and conductivity model. Conjugate heat

ransfer (CHT) contains heat exchange that happens simultaneously by

onvection between a fluid and an adjacent surface, and by conduction

ver the solid. CHT is ubiquitous and very important in key applica-

ions that are predicted to be progressively relevant both in industrial

nd domestic environments. CHT is a predominating process in different

A.I. Alsabery et al. International Journal of Mechanical Sciences 136 (2018) 200–219

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Fig. 1. Physical model of convection in a square cavity together with the coordinate

system. (For interpretation of the references to colour in this figure legend, the reader is

referred to the web version of this article.)

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𝜌

(

xternal combustion motors, thermo-acoustic devices and other ma-

hines capable of utilizing immaculate and renewable energy styles like

olar, refuse and geothermal heat. Indeed, the prevalent and central ad-

antage of these devices is their ingrained dependence on internally

hermal fluid flows which result in CHT. However, developments to our

nderstanding of CHT and therefore, to the performance of numerous

f the above mentioned systems must come from exploring actual situ-

tions of CHT within a framework that can capture the varying inter-

lay between fluid flow and heat transfer, and account for the existence

f conjugate fluid–solid interactions. There are some excellent studies

onsidering the impact of partition length and conductivity on the heat

ransfer rate.

Viskanta and Kim [22] studied a rectangular cavity formed by finite

onductance walls of different void fractions and aspect ratios. They con-

idered the outer edge of the wall and the opposing vertical side were

sothermal at high and low temperatures, respectively, and the two hori-

ontal sides were insulated. They showed that for the high Grashof num-

er and decreasing wall conductivity, two dimensional effects on con-

uction in the wall were non-negligible. House et al. [23] investigated

he effect of a centered heat-conducting body on the natural convection

eat transfer in a square cavity. The two vertical walls were maintained

t two different constant temperatures and the horizontal walls were

diabatic. The results showed that the heat transfer decreased with the

ncrease of the solid body. Ha et al. [24] investigated the effect of un-

teady natural convection processes in similar vertical cavities with a

entered heat-conducting body. Zhao et al. [25] studied the effect of

centered heat-conducting body on the conjugate natural convection

eat transfer in a square enclosure. The results show that the thermal

onductivity ratio has strong influence on the flow within the square

avity. Saeid [26] investigated a differentially-heated vertical square

orous cavity where the conducting wall is next to the hot side. He

ound in most of the cases that either increasing the Rayleigh number

nd the thermal conductivity ratio or decreasing the thickness of the

ounded wall can increase the average Nusselt number.

Mahmoodi and Sebdani [27] used the finite volume method to inves-

igate the conjugate natural convective heat transfer in a square cavity

lled with nanofluid and containing a solid square block at the center.

hey concluded that the heat transfer rate decreased with an increasing

f the size of the inner block for low Rayleigh numbers and increased

t high Rayleigh numbers. Mahapatra et al. [28] numerically used the

nite volume method to investigate the CHT and entropy generation

n a square cavity in the presence of adiabatic and isothermal blocks.

hey found that the heat transfer enhanced with the low Rayleigh num-

ers and for a critical block sizes. Chamkha and Ismael [29] studied the

ffect of conjugate natural convection heat transfer in a porous square

avity filled with nanofluids and heated by a thick triangular wall. Their

tudy showed that the heat transfer was significantly enhanced at low

ayleigh number with the increase of the nanoparticles volume fraction.

sing the finite volume method, Esfe et al. [30] investigated the prob-

em of natural convection in a 2D cavity filled with different types of

anofluids and containing a heated cylindrical block. Recently, Alsabery

t al. [31] used the finite difference method to study the unsteady natu-

al convective heat transfer in nanofluid-saturated porous square cavity

ith a concentric solid insert and sinusoidal boundary condition. Very

ecently, el malik Bouchoucha et al. [32] considered numerically the

roblem of natural convection and entropy generation in a nanofluid

quare cavity with a non-isothermal heating thick bottom wall. They

oncluded that an increasing of the thicknesses of the bottom solid wall

ended to reduce the heat transfer rate. Garoosi and Rashidi [33] used

he finite volume method to investigate the two phase model of conju-

ate natural convection of the nanofluid in a partitioned heat exchanger

ontaining several conducting obstacles. They found that the heat trans-

er rate was significantly influenced by changing the orientation of the

onductive partition from vertical to horizontal mode.

Based on the previously mentioned papers and to the authors ’ best

nowledge, there have been no studies of conjugate natural convection

202

f Al 2 O 3 –water nanofluid in a square cavity with a concentric solid in-

ert and corner heater using Buongiorno ’s two-phase model. Thus, we

elieve that this work is valuable. The aim of this study is to investi-

ate the conjugate natural convection of Al 2 O 3 –water nanofluid in a

quare cavity with a concentric solid insert and corner boundaries us-

ng Buongiorno ’s two-phase model. Solid inner blocks can be used to

ontrol heat transfer as passive element in various shaped cavities filled

ith nanofluids or pure liquids. This application can be seen in building

esign, electronic equipment, heat exchangers and solar energy systems

34] . The interaction between the shear-driven flow and natural con-

ection in closed enclosures is one of the most interesting topics which

an be used in design and analysis of many industrial heating or cool-

ng systems such as: indoor ventilation with radiators, nuclear reactors,

ooling of electronic components and heat exchangers and so on. The

onsidered geometry can be found in all of the above mentioned fields

here an internal solid block can be considered like an additional inter-

al system element (pipe, bracket, electrode, electronic element and so

n).

. Mathematical formulation

The steady two-dimensional natural convection problem in a square

avity with length L and with the cavity center inserted by a solid square

ith side d , as illustrated in Fig. 1 . The Rayleigh number range chosen

n the study keeps the nanofluid flow incompressible and laminar. The

orner heater and cooler have isothermal boundaries in both vertical

nd horizontal directions with length 0.4 L , which are shown by thick

ed and blue lines, respectively. While the remainder of these walls are

ept adiabatic. The boundaries of the annulus are assumed to be imper-

eable, the fluid within the cavity is a water-based nanofluid having

l 2 O 3 nanoparticles. The Boussinesq approximation is applicable, the

anofluid physical properties are constant except for the density. By

onsidering these assumptions, the continuity, momentum, energy and

olume fraction equations for the laminar and steady state natural con-

ection of incompressible flow can be written as follows [18] :

Continuity equation:

⋅ 𝑣 = 0 (1)

Momentum equation:

𝑛𝑓 𝑣 ⋅ ∇ 𝑣 = −∇ 𝑝 + ∇ ⋅(𝜇𝑛𝑓 ∇ 𝑣

)+ ( 𝜌𝛽) 𝑛𝑓 ( 𝑇 − 𝑇 𝑐 ) 𝑔 (2)

Energy equation:

𝜌𝐶 𝑝 ) 𝑛𝑓 𝑣 ⋅ ∇ 𝑇 𝑛𝑓 = ∇ ⋅(𝑘 𝑛𝑓 ∇ 𝑇 𝑛𝑓

)− 𝐶 𝑝. 𝐽 𝑝 ⋅ ∇ 𝑇 𝑛𝑓 (3)

A.I. Alsabery et al. International Journal of Mechanical Sciences 136 (2018) 200–219

𝑣

∇

w

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𝐽

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𝐽

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e

𝐷

J

d

𝐽

w

c

𝐷

a

(

T

𝛼

T

𝜌

T

m

(

T

p

f

T

b

w

𝑅

𝑢

w

0

e

𝑑

w

n

(

d

𝑑

N

𝑋

T

𝑉

w

𝐷

fi

i

d

𝑇

i

c

𝑈

𝑈

𝑈

Volume fraction equation:

⋅ ∇ 𝜑 = −

1 𝜌𝑝

∇ ⋅ 𝐽 𝑝 (4)

The energy equation of the inner solid wall is:

𝑇 𝑤 = 0 , (5)

here ∇ is the velocity vector, g is the acceleration due to gravity, 𝜑

s the local volume fraction of nanoparticles and J p is the nanoparticles

ass flux. Based on Buongiorno ’s model nanoparticles mass flux can be

ritten as:

𝑝 = 𝐽 𝑝,𝐵 + 𝐽 𝑝,𝑇 , (6)

here J p, B and J p, T represent the mass flux due to Brownian motion and

hermophoresis effect. J p, B is the drift flux due to the Brownian motion

hich can be defined as:

𝑝,𝐵 = − 𝜌𝑝 𝐷 𝐵 ∇ 𝜑, (7)

here the Brownian motion is described by the Brownian diffusion co-

fficient, D B , which is defined based on the model of Einstein–Stokes:

𝐵 =

𝑘 𝑏 𝑇

3 𝜋𝜇𝑓 𝑑 𝑝 , (8)

p, T represents the drift flux due to thermophoretic effects which can be

efined as:

𝑝,𝑇 = − 𝜌𝑝 𝐷 𝑇 ∇ 𝑇 , (9)

here the thermophoresis effect is described by the thermal diffusion

oefficient, D T , which is defined as:

𝑇 = 0 . 26 𝑘 𝑓

2 𝑘 𝑓 + 𝑘 𝑝

𝜇𝑓

𝜌𝑓 𝑇 𝜑. (10)

The thermo-physical properties of the nanofluid can be determined

s follows:

The heat capacitance of the nanofluids ( 𝜌C p ) nf is given as

𝜌𝐶 𝑝 ) 𝑛𝑓 = (1 − 𝜑 )( 𝜌𝐶 𝑝 ) 𝑓 + 𝜑 ( 𝜌𝐶 𝑝 ) 𝑝 . (11)

he effective thermal diffusivity of the nanofluids 𝛼nf is given as

𝑛𝑓 =

𝑘 𝑛𝑓

( 𝜌𝐶 𝑝 ) 𝑛𝑓 . (12)

he effective density of the nanofluids 𝜌nf is given as

𝑛𝑓 = (1 − 𝜑 ) 𝜌𝑓 + 𝜑𝜌𝑝 . (13)

he thermal expansion coefficient of the nanofluids 𝛽nf can be deter-

ined by:

𝜌𝛽) 𝑛𝑓 = (1 − 𝜑 ) ( 𝜌𝛽) 𝑓 + 𝜑 ( 𝜌𝛽) 𝑝 . (14)

he dynamic viscosity ratio of water–Al 2 O 3 nanofluids for 33 nm

article-size in the ambient condition was derived in reference [16] as

ollows:

𝜇𝑛𝑓

𝜇𝑓 = 1∕

(1 − 34 . 87

(𝑑 𝑝 ∕ 𝑑 𝑓

)−0 . 3 𝜑 1 . 03

). (15)

he thermal conductivity ratio of water–Al 2 O 3 nanofluids is calculated

y Corcione et al. model [16] is:

𝑘 𝑛𝑓

𝑘 𝑓 = 1 + 4 . 4 𝑅𝑒 0 . 4

𝐵 Pr 0 . 66

(

𝑇

𝑇 𝑓𝑟

) 10 (

𝑘 𝑝

𝑘 𝑓

) 0 . 03 𝜑 0 . 66 , (16)

here Re B is defined as

𝑒 𝐵 =

𝜌𝑓 𝑢 𝐵 𝑑 𝑝

𝜇𝑓 , (17)

𝐵 =

2 𝑘 𝑏 𝑇 𝜋𝜇𝑓 𝑑

2 𝑝

, (18)

203

here 𝑘 𝑏 = 1 . 380648 × 10 −23 (J/K) is the Boltzmann constant. 𝑙 𝑓 = . 17 nm is the mean path of fluid particles. d f is the molecular diam-

ter of water given as [16]

𝑓 =

6 𝑀

𝑁𝜋𝜌𝑓 , (19)

here M is the molecular weight of the base fluid, N is the Avogadro

umber and 𝜌f is the density of the base fluid at standard temperature

310 K). Accordingly, and basing on water as a base fluid, the value of

f is obtained:

𝑓 =

(

6 × 0 . 01801528 6 . 022 × 10 23 × 𝜋 × 998 . 26

) 1∕3 = 3 . 85 × 10 −10 m . (20)

ow, we introduce the following non-dimensional variables:

=

𝑥

𝐿 , 𝑌 =

𝑦

𝐿 , 𝑉 =

𝑣𝐿

𝜈𝑓 , 𝑃 =

𝑝𝐿 2

𝜌𝑛𝑓 𝜈2 𝑓

, 𝜑 ∗ =

𝜑

𝜙, 𝐷

∗ 𝐵 =

𝐷 𝐵

𝐷 𝐵0 ,

𝐷

∗ 𝑇 =

𝐷 𝑇

𝐷 𝑇 0 , 𝛿 =

𝑇 𝑐

𝑇 ℎ − 𝑇 𝑐 , 𝜃𝑛𝑓 =

𝑇 𝑛𝑓 − 𝑇 𝑐

𝑇 ℎ − 𝑇 𝑐 , 𝜃𝑤 =

𝑇 𝑤 − 𝑇 𝑐

𝑇 ℎ − 𝑇 𝑐 . (21)

his then yields the following dimensionless governing equations:

∇ ⋅ 𝑉 = 0 , (22)

𝑉 ⋅ ∇ 𝑉 = −∇ 𝑃 +

𝜌𝑓

𝜌𝑛𝑓

𝜇𝑛𝑓

𝜇𝑓 ∇

2 𝑉 +

( 𝜌𝛽) 𝑛𝑓 𝜌𝑛𝑓 𝛽𝑓

1 Pr 𝑅𝑎 ⋅ 𝜃𝑛𝑓 , (23)

⋅ ∇ 𝜃𝑛𝑓 =

( 𝜌𝐶 𝑝 ) 𝑓 ( 𝜌𝐶 𝑝 ) 𝑛𝑓

𝑘 𝑛𝑓

𝑘 𝑓

1 Pr

∇

2 𝜃𝑛𝑓 +

( 𝜌𝐶 𝑝 ) 𝑓 ( 𝜌𝐶 𝑝 ) 𝑛𝑓

𝐷

∗ 𝐵

Pr ⋅𝐿𝑒 ∇ 𝜑 ∗ ⋅ ∇ 𝜃𝑛𝑓

+

( 𝜌𝐶 𝑝 ) 𝑓 ( 𝜌𝐶 𝑝 ) 𝑛𝑓

𝐷

∗ 𝑇

Pr ⋅𝐿𝑒 ⋅𝑁 𝐵𝑇

∇ 𝜃𝑛𝑓 ⋅ ∇ 𝜃𝑛𝑓

1 + 𝛿𝜃𝑛𝑓 , (24)

𝑉 ⋅ ∇ 𝜑 ∗ =

𝐷

∗ 𝐵

𝑆𝑐 ∇

2 𝜑 ∗ +

𝐷

∗ 𝑇

𝑆𝑐 ⋅𝑁 𝐵𝑇

⋅∇

2 𝜃𝑛𝑓

1 + 𝛿𝜃𝑛𝑓 , (25)

∇ 𝜃𝑤 = 0 , (26)

here 𝐷 𝐵0 =

𝑘 𝑏 𝑇 𝑐

3 𝜋𝜇𝑓 𝑑 𝑝 is the reference Brownian diffusion coefficient,

𝑇 0 = 0 . 26 𝑘 𝑓

2 𝑘 𝑓 + 𝑘 𝑝

𝜇𝑓

𝜌𝑓 𝜃𝜙 is the reference thermophoretic diffusion coef-

cient, 𝑆𝑐 = 𝜈𝑓 ∕ 𝐷 𝐵0 is Schmidt number, 𝑁 𝐵𝑇 = 𝜙𝐷 𝐵0 𝑇 𝑐 ∕ 𝐷 𝑇 0 ( 𝑇 ℎ − 𝑇 𝑐 )s the diffusivity ratio parameter (Brownian diffusivity/thermophoretic

iffusivity), 𝐿𝑒 = 𝑘 𝑓 ∕( 𝜌𝐶 𝑝 ) 𝑓 𝜙𝐷 𝐵0 is the Lewis number, 𝑅𝑎 = 𝑔𝜌𝑓 𝛽𝑓 ( 𝑇 ℎ − 𝑐 ) 𝐿 3 ∕( 𝜇𝑓 𝛼𝑓 ) is the Rayleigh number for the base fluid and Pr = 𝜈𝑓 ∕ 𝛼𝑓 s the Prandtl number for the base fluid. The dimensionless boundary

onditions of Eqs. (22) –(26) are:

= 𝑉 = 0 , 𝜕𝜑 ∗

𝜕𝑛 = −

𝐷

∗ 𝑇

𝐷

∗ 𝐵

⋅1

𝑁 𝐵𝑇

⋅1

1 + 𝛿𝜃𝑛𝑓

𝜕𝜃𝑛𝑓

𝜕𝑛 , 𝜃𝑛𝑓 = 1

on the horizontal bottom wall ,

0 ≤ 𝑋 ≤ 0 . 4 , 𝑌 = 0 and on the vertical left wall ,

0 ≤ 𝑌 ≤ 0 . 4 , 𝑋 = 0 (27)

= 𝑉 = 0 , 𝜕𝜑 ∗

𝜕𝑛 = 0 ,

𝜕𝜃𝑛𝑓

𝜕𝑛 = 0 ,

(for the adiabatic parts of the remainder walls) (28)

= 𝑉 = 0 , 𝜕𝜑 ∗

𝜕𝑛 = −

𝐷

∗ 𝑇

𝐷

∗ 𝐵

⋅1

𝑁 𝐵𝑇

⋅1

1 + 𝛿𝜃𝑛𝑓

𝜕𝜃𝑛𝑓

𝜕𝑛 , 𝜃𝑛𝑓 = 0

on the horizontal top wall ,

0 . 6 ≤ 𝑋 ≤ 1 . 0 , 𝑌 = 1 and on the vertical right wall ,

0 . 6 ≤ 𝑌 ≤ 1 . 0 , 𝑋 = 1 (29)

A.I. Alsabery et al. International Journal of Mechanical Sciences 136 (2018) 200–219

Fig. 2. Streamlines (a), Das and Reddy [35] (left), present study (right), isotherms (b), Das and Reddy [35] (left), present study (right) for 𝐾 𝑟 = 0 . 2 (top) and 𝐾 𝑟 = 5 (bottom) at 𝑅𝑎 = 10 6 , 𝜙 = 0 and 𝐷 = 0 . 5 .

204

A.I. Alsabery et al. International Journal of Mechanical Sciences 136 (2018) 200–219

Fig. 3. Comparison of the mean Nusselt number obtained from present numerical simu-

lation with the experimental results of Ho et al. [36] , numerical results of Sheikhzadeh

et al. [37] and numerical results of Motlagh and Soltanipour [21] for different values of

Rayleigh numbers.

𝑈

𝑈

w

t

w

w

𝑁

a

𝑁

F

w

𝑁

a

𝑁

3

g

(

F

𝜙

= 𝑉 = 0 , 𝜕𝜑 ∗

𝜕𝑛 = 0 ,

𝜕𝜃𝑛𝑓

𝜕𝑛 = 0 ,

(for the adiabatic parts of the remainder walls) (30)

𝜃𝑛𝑓 = 𝜃𝑤 , at the outer solid square surface , (31)

ig. 4. Streamlines (a), Sheikhzadeh et al. [37] (solid lines) (left), present study (right), isotherm

= 0 . 04 , 𝑑 𝑝 = 25 and 𝐷 = 0 .

205

= 𝑉 = 0 , 𝜕𝜑 ∗

𝜕𝑛 = −

𝐷

∗ 𝑇

𝐷

∗ 𝐵

⋅1

𝑁 𝐵𝑇

⋅1

1 + 𝛿𝜃𝑛𝑓

𝜕𝜃𝑛𝑓

𝜕𝑛 ,

𝜕𝜃𝑛𝑓

𝜕𝑛 = 𝐾 𝑟

𝜕𝜃𝑤

𝜕𝑛 , 𝑋, 𝑌 in

[ (1 − 𝐷)

2 , (1 + 𝐷)

2

] , (32)

here 𝐾 𝑟 = 𝑘 𝑤 ∕ 𝑘 𝑛𝑓 is the thermal conductivity ratio and 𝐷 = 𝑑∕ 𝐿 ishe aspect ratio of inner square cylinder width to outer square cylinder

idth.

The local Nusselt number evaluated at the left and bottom walls,

hich is defined by

𝑢 𝑥 = −

𝑘 𝑛𝑓

𝑘 𝑓

(

𝜕𝜃𝑛𝑓

𝜕𝑋

)

𝑋=0 , 𝑁 𝑢 𝑦 = −

𝑘 𝑛𝑓

𝑘 𝑓

(

𝜕𝜃𝑛𝑓

𝜕𝑌

)

𝑌 =0 , (33)

nd

𝑢 𝑛𝑓 = 𝑁 𝑢 𝑥 + 𝑁 𝑢 𝑦 . (34)

inally, the average Nusselt number evaluated at the left and bottom

alls which is given by:

𝑢 𝑥 = ∫0 . 4

0 𝑁 𝑢 𝑦 d 𝑋, 𝑁 𝑢 𝑦 = ∫

0 . 4

0 𝑁 𝑢 𝑥 d 𝑌 , (35)

nd

𝑢 𝑛𝑓 = 𝑁𝑢 𝑥 + 𝑁𝑢 𝑦 . (36)

. Numerical method and validation

An iterative finite difference method (FDM) is employed to solve the

overning equations (22) –(26) subject to the boundary conditions (27) –

32) . In the present paper, several grid testings are performed: 10 ×10,

s (b), Sheikhzadeh et al. [37] (solid lines) (left), present study (right) for 𝑅𝑎 = 3 . 37 × 10 5 ,

A.I. Alsabery et al. International Journal of Mechanical Sciences 136 (2018) 200–219

Fig. 5. Streamlines (a), Motlagh and Soltanipour [21] (left), present study (right), isotherms (b), Motlagh and Soltanipour [21] (left), present study (right) for 𝑅𝑎 = 10 2 (top) and 𝑅𝑎 = 10 6

(bottom) at 𝜙 = 0 . 02 and 𝐷 = 0 .

206

A.I. Alsabery et al. International Journal of Mechanical Sciences 136 (2018) 200–219

Fig. 6. Comparison of average Nusselt number with Motlagh and Soltanipour [21] for (a) 𝑅𝑎 = 10 2 and (b) 𝑅𝑎 = 10 6 at 𝐷 = 0 .

Table 1

Grid testing for Ψmin and 𝑁𝑢 𝑛𝑓 at different grid size for 𝑅𝑎 = 10 5 , 𝜙 = 0 . 02 , 𝑘 𝑤 = 0 . 76 and

𝐷 = 0 . 3 .

Grid size Ψmin 𝑁𝑢 𝑛𝑓

10 ×10 −1 . 3884 4.6524

20 ×20 −1 . 39 4.7126

40 ×40 −1 . 3908 4.7406

60 ×60 −1 . 3922 4.7579

80 ×80 −1 . 3907 4.7824

100 ×100 −1 . 3912 4.8572

120 ×120 −1 . 3913 4.9194

140 ×140 −1 . 3914 4.9197

160 ×160 −1 . 3914 4.9198

2

1

t

f

i

c

1

p

o

i

p

fi

n

a

w

a

s

i

N

n

a

d

a

c

t

h

n

a

Fig. 7. Comparison of average Nusselt number for different of Ra of Buongiorno ’s two-

phase model (BTPM) and homogeneous model (HM) for 𝜙 = 0 . 02 , 𝜙 = 0 . 04 , 𝑘 𝑤 = 0 . 76 and

𝐷 = 0 . 3 .

𝐷

s

t

m

c

u

t

m

e

p

s

[

fi

4

i

e

(

0

1

t

0 ×20, 40 ×40, 60 ×60, 80 ×80, 100 ×100, 120 ×120, 140 ×140 and

60 ×160. Table 1 shows the calculated strength of the flow circula-

ion ( Ψmin ) and average Nusselt number ( 𝑁𝑢 𝑛𝑓 ) at different grid sizes

or 𝑅𝑎 = 10 5 , 𝜙 = 0 . 02 , 𝑘 𝑤 = 0 . 76 and 𝐷 = 0 . 3 . The results show insignif-

cant differences for the 140 ×140 grids and above. Therefore, for all

omputations in this paper for similar problems to this subsection, the

40 ×140 uniform grid is employed.

For the validation of data, the results are compared with previously

ublished numerical results obtained by Das and Reddy [35] for the case

f conjugate natural convection heat transfer in an inclined square cav-

ty with a concentric solid insert, as shown in Fig. 2 . In addition, a com-

arison of the average Nusselt number is made between the resulting

gure and the experimental results provided by Ho et al. [36] and the

umerical results provided by Sheikhzadeh et al. [37] and by Motlagh

nd Soltanipour [21] for the case of the natural convection of Al 2 O 3 –

ater nanofluid in a square cavity using Buongiorno ’s two-phase model

s shown in Fig. 3 . The comparison with Sheikhzadeh et al. [37] also

howed very good agreement between the maps of streamlines and

sotherms inside a square cavity filled by a nanofluid as shown in Fig. 4 .

ext, comparisons made between the present streamlines, isotherms,

anoparticles volume fraction and the average Nusselt number results

nd the numerical one obtained by Motlagh and Soltanipour [21] are

emonstrated in Figs. 5 and 6 . These results provide confidence to the

ccuracy of the present numerical method.

Before we move to the Results and Discussion section, we should

ompare the results using Buongiorno ’s two-phase model which show

he importance of the Brownian motion and thermophoresis with the

omogeneous model. Fig. 7 presents the various of the average Nusselt

umber for different of Ra of Buongiorno ’s two-phase model (BTPM)

nd homogeneous model (HM) for 𝜙 = 0 . 02 , 𝜙 = 0 . 04 , 𝑘 = 0 . 76 and

𝑤

207

= 0 . 3 . This figure shows that an optimal average volume fractions ob-

erve in the case of high Rayleigh number numbers using Buongiorno ’s

wo-phase model where the importance of Brownian motion and ther-

ophoresis can be observe, while the heat transfer rate is clearly in-

reased with the increment of Rayleigh number and nanoparticles vol-

me fraction by using the homogeneous model which clearly underes-

imated the real behaviour of the hear transfer compared to the experi-

ental results of Ho et al. [36] and the numerical results of Sheikhzadeh

t al. [37] and Motlagh and Soltanipour [21] . Also, Buongiorno ’s two-

hase model predicts a very good agreement with the experimental re-

ults of Ho et al. [36] and the numerical results of Sheikhzadeh et al.

37] and Motlagh and Soltanipour [21] as presented in the previous

gure ( Fig. 3 ).

. Results and discussion

In this section, we present numerical results for the streamlines,

sotherms and isoconcentrations with various values of the refer-

nce nanoparticle volume fraction (0 ≤ 𝜙≤ 0.04), the Rayleigh number

10 2 ≤ Ra ≤ 10 6 ), thermal conductivity of the conjugate square ( 𝑘 𝑤 = . 28 , 0.76, 1.95, 7 and 16) (epoxy: 0.28, brickwork: 0.76, granite:

.95, solid rock: 7, stainless steel: 16), dimensionless inner solid square

hickness (0.1 ≤ D ≤ 0.7), where the values of Prandtl number, Lewis

A.I. Alsabery et al. International Journal of Mechanical Sciences 136 (2018) 200–219

Fig. 8. Variation of the streamlines (left), isotherms (middle), and nanoparticle distribution (right) evolution by solid volume fraction ( 𝜙) for 𝑅𝑎 = 10 5 , 𝑘 𝑤 = 0 . 76 and 𝐷 = 0 . 3 .

208

A.I. Alsabery et al. International Journal of Mechanical Sciences 136 (2018) 200–219

Fig. 9. Variation of the streamlines (left), isotherms (middle), and nanoparticle distribution (right) evolution by Rayleigh number ( Ra ) for 𝜙 = 0 . 02 , 𝑘 𝑤 = 0 . 76 and 𝐷 = 0 . 3 .

209

A.I. Alsabery et al. International Journal of Mechanical Sciences 136 (2018) 200–219

Fig. 10. Variation of the streamlines (left), isotherms (middle), and nanoparticle distribution (right) evolution by thermal conductivity of the solid block ( k w ) for 𝑅𝑎 = 10 5 , 𝜙 = 0 . 02 and

𝐷 = 0 . 3 .

210

A.I. Alsabery et al. International Journal of Mechanical Sciences 136 (2018) 200–219

Fig. 11. Variation of the streamlines (left), isotherms (middle), and nanoparticle distribution (right) evolution by the length of the inner solid square ( D ) for 𝑅𝑎 = 10 5 , 𝜙 = 0 . 02 and

𝑘 𝑤 = 0 . 76 .

211

A.I. Alsabery et al. International Journal of Mechanical Sciences 136 (2018) 200–219

Fig. 12. Variation of local Nusselt number interfaces with n by different Ra for (a) 𝐷 = 0 . 1 and (b) 𝐷 = 0 . 6 at 𝜙 = 0 . 02 and 𝑘 𝑤 = 0 . 76 .

Fig. 13. Variation of local Nusselt number interfaces with n by different 𝜙 for (a) 𝑅𝑎 = 10 3 and (b) 𝑅𝑎 = 10 5 at 𝐷 = 0 . 3 and 𝑘 𝑤 = 0 . 76 .

n

s

𝐿

a

T

p

c

fl

i

Table 2

Thermo-physical properties of water with Al 2 O 3 nanoparticles at 𝑇 = 310 K [21,38] .

Physical properties Fluid phase (water) Al 2 O 3

𝐶 𝑝 (J∕kg K) 4178 765

𝜌 (kg/m

3 ) 993 3970

𝑘 (W m −1 K −1 ) 0.628 40

𝛽 ×10 5 (1/K) 36.2 0.85

𝜇 ×10 6 (kg/ms) 695 –

d p (nm) 0.385 33

umber, Schmidt number, ratio of Brownian to thermophoretic diffu-

ivity and normalized temperature parameter are fixed at Pr = 4 . 623 ,𝑒 = 3 . 5 × 10 5 , 𝑆𝑐 = 3 . 55 × 10 4 , 𝑁 𝐵𝑇 = 1 . 1 and 𝛿 = 155 . The values of the

verage Nusselt number are calculated for various values of 𝜙 and D .

he thermophysical properties of the base fluid (water) and solid Al 2 O 3

hases are tabulated in Table 2 . Streamlines, isotherms and nanoparti-

les volume fraction as well as the average Nusselt number and fluid

ow rate for different values of key parameters mentioned above are

llustrated in Figs. 8–22 .

212

A.I. Alsabery et al. International Journal of Mechanical Sciences 136 (2018) 200–219

Fig. 14. Variation of local Nusselt number interfaces with n by different 𝜙 for (a) 𝐷 = 0 . 1 and (b) 𝐷 = 0 . 6 at 𝑅𝑎 = 10 5 and 𝑘 𝑤 = 0 . 76 .

Fig. 15. Variation of local Nusselt number interfaces with n by different k w for (a) 𝑅𝑎 = 10 3 and (b) 𝑅𝑎 = 10 5 at 𝜙 = 0 . 02 and 𝐷 = 0 . 3 .

(

P

n

e

s

t

c

t

𝐷

R

c

r

l

a

w

a

f

c

p

c

The contour level legends define the direction of the fluid heat flow

clockwise or anti-clockwise direction) and also the strength of the flow.

ositive values of Ψ denotes the anti-clockwise fluid heat flow, whereas

egative designates the clockwise fluid heat flow. Ψmin represents the

xtreme values of the stream function. These values are important to

how the minimum change of the flow. Due to the fact that the nanopar-

icles are moving with the same direction with the flow which is in

lockwise direction and tend to take a negative values.

Fig. 8 shows streamlines, isotherms and distributions of nanopar-

icles volume fraction inside the cavity for 𝑅𝑎 = 10 5 , 𝑘 𝑤 = 0 . 76 and

= 0 . 3 and different values of reference nanoparticles volume fraction.

213

egardless of the reference nanoparticles volume fraction values one

an find a formation of two convective cells near the bottom left and top

ight corners of the internal solid body. The bottom convective cell il-

ustrates a clockwise nanofluid circulation, while the upper cell reflects

counter-clockwise nanofluid motion. In global circulation, alumina–

ater nanofluid rises near the left vertical wall that has a local heater

nd descends along the right vertical wall having a local cooler. There-

ore, temperature field characterizes a heating of the upper part and

ooling of the bottom one where the solid block is heated from upper

art and cooled from the bottom part. Distributions of nanoparticles

oncentration do not change with the reference nanoparticles volume

A.I. Alsabery et al. International Journal of Mechanical Sciences 136 (2018) 200–219

Fig. 16. Variation of local Nusselt number interfaces with n by different k w for (a) 𝐷 = 0 . 1 and (b) 𝐷 = 0 . 6 at 𝑅𝑎 = 10 5 and 𝜙 = 0 . 02 .

f

a

c

R

t

L

c

N

v

m

t

r

t

f

d

t

𝑅

r

t

s

w

c

(

t

c

a

c

t

c

t

i

l

c

a

c

w

n

m

0

p

t

t

g

m

p

i

R

F

0

i

F

d

r

n

A

l

i

t

D

n

p

t

o

fl

w

w

b

raction. At the same time, we can notice that an increase in 𝜙 leads to

weak attenuation of convective flow.

The effect of Rayleigh number on stream function, temperature and

oncentration fields is presented in Fig. 9 . It should be noted that

ayleigh number illustrates an influence of buoyancy force, therefore

his effect is more essential, especially for natural convection problems.

ow values of Ra such as 10 3 and 10 4 illustrate a domination of heat

onduction mode where isotherms are parallel to the heater and cooler.

anofluid circulation is weak. One can find a formation of four con-

ective cells near each corner of the solid block due to changing of the

otion direction. At the same time, at 𝑅𝑎 = 10 3 heating and cooling of

he central block occur from the left bottom corner and right upper one,

espectively, while at 𝑅𝑎 = 10 3 one can find more intensive circulation

hat leads to heating of the block from left upper corner and cooling

rom the right bottom corner. Further increase in Ra ( Fig. 9 c) leads to a

issipation of vortexes near the left upper and right bottom corners of

he solid block, while the rest circulations are enhanced. In the case of

𝑎 = 10 6 ( Fig. 9 d) two global circulations are formed near the left and

ight sides of the solid block. Isotherms reflect more essential heating of

he block from the upper part and cooling from the bottom one. At the

ame time, the thermal boundary layers thickness decreases with Ra ,

hile nanoparticles distribution is more uniform for high values of Ra .

An increase in the thermal conductivity of solid block leads to

hanges in temperature and nanoparticles concentration distributions

Fig. 10 ). High values of k w illustrate intensive heating of the solid block

herefore, for k w > 1.95 there are no isotherms inside the block with the

onsidered temperature step. As a result the main differences in temper-

ture field occur inside the solid block, while an intensity and shape of

irculations do not change. It is worth noting that solid block of high

hermal conduction allows to reduce the zones of high nanoparticles

oncentration. As a results we can conclude that nanoparticles distribu-

ion for high k w is more uniform.

The effect of square solid block size on distributions of streamlines,

sotherms and isoconcentrations is shown in Fig. 11 . An increase in D

eads to a reduction and displacement of two convective cells sizes. Four

onvective cells appear near the block corners at D ≥ 0.5. This appear-

nce of additional recirculations leads to modification of the heating and

214

ooling direction inside the solid block. For the case of 𝐷 = 0 . 7 ( Fig. 11 d)

e have narrow nanofluid circulation zones with additional vortexes

ear the cavity walls. At the same time, nanoparticles concentration is

ore essential inside the narrow zones for high values of D .

Profiles of local Nusselt number along the heater from the point (0,

.4) ( 𝑛 = 0 ) to the point (0.4,0) ( 𝑛 = 0 . 8 ) for different values of governing

arameters are presented in Figs. 12–16 . First of all, we have to notice

hat a decrease in y -coordinate along the vertical part of the heater leads

o a reduction of local Nusselt number due to a decrease in temperature

radient and thickening of the boundary layer. At point (0, 0) we have a

inimum value of Nu nf . An increase in x -coordinate along the horizontal

art of the heater leads to a growth of local Nusselt number due to an

nteraction between hot and cold temperature waves. An increase in

ayleigh number characterizes a growth of local Nusselt number (see

ig. 12 ). It should be noted that for high size of central solid block ( 𝐷 = . 6 ) an increase in Ra from 10 3 till 10 4 does not lead to essential change

n local Nusselt number values.

Effect of reference nanoparticles volume fraction is demonstrated in

igs. 13 and 14 for different values of Rayleigh number ( Fig. 13 ) and

ifferent values of solid block size ( Fig. 14 ). An increase in 𝜙 leads to

ise of Nu nf and this behavior is more essential for low values of Rayleigh

umber when heat conduction is a dominating heat transfer mechanism.

t the same time, high value of Ra illustrates significant increase in

ocal Nusselt number along the horizontal part of the heater due to more

ntensive interaction between hot and cold temperature waves. The heat

ransfer enhancement along the heater occurs for D < 0.6. An increase in

> 0.6 leads to the heat transfer rate reduction with D due to narrowing

anofluid flow effects.

An influence of thermal conductivity on Nu nf for different Ra and D is

resented in Figs. 15 and 16 . An increase in k w for low values of Ra leads

o a raise of local Nusselt number, while for high Ra one can find a lack

f an influence of k w on Nu nf . At the same time for intensive convective

ow ( 𝑅𝑎 = 10 5 ) at low values of D local Nusselt number does not change

ith k w , while for high values of D ( 𝐷 = 0 . 6 ) we have a reduction of Nu nf

ith k w .

Taking into account the presented profiles of local Nusselt num-

er, it is possible to conclude about the behavior of average Nusselt

A.I. Alsabery et al. International Journal of Mechanical Sciences 136 (2018) 200–219

Fig. 17. Variation of average Nusselt number with 𝜙 for different Ra at 𝐷 = 0 . 3 and 𝑘 𝑤 = 0 . 76 .

n

n

t

𝑁

e

f

f

f

h

w

D

(

s

r

o

o

c

o

t

l

h

r

(

t

h

e

s

t

𝑅

t

b

t

t

t

f

f

umber, presented in Figs. 17–22 . In the case of heat conduction domi-

ated regime ( 𝑅𝑎 = 10 3 ) average Nusselt number is an increasing func-

ion of 𝜙. In the case of convective heat transfer regime development

𝑢 𝑛𝑓 is a non-linear function of 𝜙, where one can optimize the consid-

red heat transfer process. It is interesting to note that an increase in Ra

rom 10 4 till 10 6 leads to a growth of reference nanoparticles volume

raction value for maximum 𝑁𝑢 𝑛𝑓 (see Fig. 17 ).

Figs. 18 and 19 present the heat transfer enhancement with k w or low values of Ra and moderate values of D ( Fig. 18 a) or for

igh values of Ra and D ( Fig. 19 b); and heat transfer reduction

ith k w for high values of Rayleigh number and moderate values of

( Fig. 18 b and c) or for high values of Ra and low values of D

Fig. 19 a).

The abovementioned nonlinearity for 𝑁𝑢 𝑛𝑓 with Ra and D is pre-

ented in detail in Figs. 20–22 . In the case of heat conduction dominated

egime ( 𝑅𝑎 = 10 3 ) the average Nusselt number is an increasing function

f D . While for moderate values of Ra one can find a non-linear behavior

f with respect to D . Such behavior can be used for optimization of the

onsidered process using the optimal value of solid block size. In the case

215

f high intensive convective flow ( 𝑅𝑎 = 10 6 ) one can find also the heat

ransfer enhancement with D . At the same time, high convective circu-

ation inside the cavity ( 𝑅𝑎 = 10 6 in Fig. 21 b) illustrates a growth of the

eat transfer rate with nanoparticles volume fraction for the considered

ange of solid block size. While in the case of low convective intensity

𝑅𝑎 = 10 4 in Fig. 21 a) we have a non-linear dependence of 𝑁𝑢 𝑛𝑓 on

he nanoparticles volume fraction for D < 0.5 and the heat transfer en-

ancement for D > 0.5. It is worth noting that, the obtained non-linear

ffect for D < 0.5 has been shown in Figs. 17, 18 b and 19 a. The rea-

on for such behavior is an interaction between heat conduction and

hermal convection regimes inside the cavity. It is well known that for

𝑎 = 10 3 the heat conduction is a dominating mode while for 𝑅𝑎 = 10 5 hermal convection is a major regime and 𝑅𝑎 = 10 4 is a transition regime

etween heat conduction and thermal convection. Taking into account

he internal solid body where energy is transferred by the heat conduc-

ion it is possible to conclude that the size of this body characterizes

he domination of the specific heat transfer regime for 𝑅𝑎 = 10 4 . The ef-

ect of thermal conductivity ratio on 𝑁𝑢 𝑛𝑓 characterizes the heat trans-

er enhancement with an increase in k w in the case of heat conduction

A.I. Alsabery et al. International Journal of Mechanical Sciences 136 (2018) 200–219

Fig. 18. Variation of average Nusselt number with 𝜙 for different k w for (a) 𝑅𝑎 = 10 3 , (b) 𝑅𝑎 = 10 4 and (c) 𝑅𝑎 = 10 5 at 𝐷 = 0 . 3 .

Fig. 19. Variation of average Nusselt number with 𝜙 for different k w for (a) 𝐷 = 0 . 1 and (b) 𝐷 = 0 . 6 at 𝑅𝑎 = 10 4 .

216

A.I. Alsabery et al. International Journal of Mechanical Sciences 136 (2018) 200–219

Fig. 20. Variation of average Nusselt number with D for different Ra at 𝜙 = 0 . 02 and 𝑘 𝑤 = 0 . 76 .

Fig. 21. Variation of average Nusselt number with D for different 𝜙 for (a) 𝑅𝑎 = 10 4 and (b) 𝑅𝑎 = 10 6 at 𝑘 𝑤 = 0 . 76 .

d

(

D

a

t

i

t

i

ominated regime ( 𝑅𝑎 = 10 3 in Fig. 22 a). Moreover, low value of k w epoxy) illustrates a weak increase in the average Nusselt number with

, while high values of the thermal conductivity ratio (granite, solid rock

nd stainless steel) reflect an essential growth of with D . At the same

217

ime, moderate convective heat transfer regime ( 𝑅𝑎 = 10 5 in Fig. 22 b)

llustrates a reduction of the heat transfer rate with the thermal conduc-

ivity ratio for the considered range of solid block size. This reduction

s significant for D > 0.5.

A.I. Alsabery et al. International Journal of Mechanical Sciences 136 (2018) 200–219

Fig. 22. Variation of average Nusselt number with D for different k w for (a) 𝑅𝑎 = 10 3 and (b) 𝑅𝑎 = 10 5 at 𝜙 = 0 . 02 .

5

t

w

a

f

m

n

a

a

b

A

(

t

R

e

o

R

[

[

[

[

[

[

[

[

. Conclusions

In the present study, the finite difference method (FDM) is employed

o analyze the steady natural convection of an alumina–water nanofluid

ithin a square cavity with a centered heat-conducting solid block

nd corner heater and cooler. Governing equations in dimensionless

orm have been formulated using the two-phase Buongiorno nanofluid

odel. The detailed computational results for the flow, temperature and

anoparticles volume fraction fields within the cavity for laminar flow,

nd the local and average Nusselt numbers are shown graphically for

wide range Rayleigh number, thermal conductivity ratio, and solid

lock size. The important conclusions in the study are provided below:

1. A solid block with high thermal conduction allows to reduce the

zones of high nanoparticles concentration. As a result, we can ob-

serve that nanoparticles distributions for a high solid thermal con-

ductivity are more uniform.

2. When the heat conduction is dominated (at low Ra numbers), the

heat transfer rate is clearly increased with the increment of the

nanoparticles volume fraction. While an optimal average volume

fractions observe in the case of high Ra numbers with the extreme

heat transfer rate.

3. In the case of heat conduction dominated regime or low value of Ra ,

higher thermal conductivity of solid block shows more enhancement

on the heat transfer rate, while a solid block with low thermal con-

ductivity allows more heat to transfer for the intensive convective

flow case or high values of Ra .

4. The average Nusselt number is an increasing function of the Rayleigh

number; nanoparticles volume fraction for the case of heat conduc-

tion regime ( 𝑅𝑎 = 10 3 ) and intensive convection regime ( 𝑅𝑎 = 10 6 );thermal conductivity ratio in the case of heat conduction regime

( 𝑅𝑎 = 10 3 ); and heat-conducting solid block size for the case of

heat conduction regime ( 𝑅𝑎 = 10 3 ) and intensive convection regime

( 𝑅𝑎 = 10 6 ). 5. The thermal conductivity ratio and solid block size are very good

control parameters for an optimization of heat transfer inside the

partially heated and cooled cavity.

cknowledgments

The work was supported by the Universiti Kebangsaan Malaysia

UKM) research grant DIP-2017-010 . Also M.A. Sheremet acknowledges

he financial support from the Grants Council under the President of the

ussian Federation (MD-2819.2017.8). We thank the respected review-

218

rs for their constructive comments which clearly enhanced the quality

f the manuscript.

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