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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: [email protected] (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.)
o
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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
i
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w
𝐽
w
t
w
𝐽
w
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
𝑤207
= 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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