NUMERICAL INVESTIGATION OF NATURAL CONVECTION HEAT TRANSFER FROM CIRCULAR CYLINDER INSIDE AN ENCLOSURE CONTAINING NANOFLUIDS

International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 12, December (2014), pp. 66-85 © IAEME
66
NUMERICAL INVESTIGATION OF NATURAL
CONVECTION HEAT TRANSFER FROM CIRCULAR
CYLINDER INSIDE AN ENCLOSURE CONTAINING
NANOFLUIDS
Omar M. Ali1
, Ramdhan H. S. Zaidky2
, Ahmed M. Saleem3
1
Department of Refrigeration and Air Conditioning, Technical Institute of Zakho, Zakho,
Kurdistan, Iraq,
2
Faculty of Petroleum Engineering, Zakho, Kurdistan, Iraq,
3
Department of Refrigeration and Air Conditioning, Technical college of Mosul, Mosul, Iraq,
ABSTRACT
In the present work, the enhancement of natural convection heat transfer utilizing nanofluids
as working fluid from horizontal circular cylinder situated in a square enclosure is investigated
numerically. The type of the nanofluid is the water-based copper Cu. A model is developed to
analyze heat transfer performance of nanofluids inside an enclosure taking into account the solid
particle dispersionrs on the flow and heat transfer characteristics. The study uses different Raylieh
numbers (104
, 105
, and 106
), different enclosure width to cylinder diameter ratios W/D (1.667, 2.5
and 5) and volume fraction of nanoparticles between 0 to 0.2. The work included the solution of the
governing equations in the vorticity-stream function formulation which were transformed into body
fitted coordinate system. The transformations are based initially on algebraic grid generation, then
using elliptic grid generation to map the physical domain between the heated horizontal cylinder and
the enclosure into a computational domain. The disecritization equation system are solved by using
finite difference method. The code build using Fortran 90 to execute the numerical algorithm.
The results display the effect of Raylieh number, enclosure width to cylinder diameter ratio,
and volume fractions of the nanofluids on the thermal and hydrodynamic characteristics. The results
were compared with previous numerical results, which showed good agreement. The Nusselt number
increases with increasing the Raylieh number for all cases. An enhancement in average Nusselt
number was found with the volume fraction of nanofluids for the whole range of Rayleigh number.
The results show that the isotherms are nearly similar when the volume fraction of nanoparticles is
increased from 0 to 0.2 for each Raylieh number and enclosure width to cylinder diameter ratio,
while, the streamlines are changed for same ranges.
INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING AND
TECHNOLOGY (IJMET)
ISSN 0976 – 6340 (Print)
ISSN 0976 – 6359 (Online)
Volume 5, Issue 12, December (2014), pp. 66-85
© IAEME: www.iaeme.com/IJMET.asp
Journal Impact Factor (2014): 7.5377 (Calculated by GISI)
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IJMET
© I A E M E

67
Keywords: Circular, Cylinder, Heat Transfer, Square Enclosure, Nanofluids.
NOMENCLATURE
Symbol Definition Unit
Nu Average Nusselt number, (h.D/k).
D Cylinder diameter. m
di,j Source term in the general equation, eqn. (18).
H Convective heat transfer coefficient. W/m2
.°C
J Jacobian.
K Thermal conductivity of the air. W/m.°C
P Pressure. N/m2
P Coordinate control function.
Pr Prandtl number, (ν/α).
Q Coordinate control function.
R maximum absolute residual value.
Ra Raylieh number, (gβ∆TD3
/να).
T Time. seconds
T Temperature. °C
u Velocity in x-direction. m/s
v Velocity in y-direction. m/s
W Enclosure Width. cm
W Relaxation factor.
x Horizontal direction in physical domain. m
X
Dimensionless horizontal direction in physical
domain.
Y Vertical direction in physical domain. m
Y Dimensionless vertical direction in physical domain.
Greek Symbols
∆T
Difference between cylinder surface temperature and
environmental temperature.
°C
µ Viscosity of the air. kg/m.s
β Coefficient of thermal expansion. 1/°C
η Vertical direction in computational domain.
ξ Horizontal direction in computational domain.
α Fluid thermal diffusivity m2
/s
Ψ Dimensionless stream function.
ω Vorticity. 1/s
ϖ Dimensionless vorticity.
υ Kinematic viscosity. m2
/s
θ Dimensionless temperature.
φ Dependent variable.
ϕ Volume fraction of nanofluid
ψ Stream Function. 1/sec.

68
Subscript
nf Nanofluid
p Particle
S Cylinder surface.
∞ Environment.
X Derivative in x-direction.
Y Derivative in y-direction.
ξ Derivative in ξ-direction.
D Circular cylinder diameter.
ψ Stream function.
T Temperature.
ω Vorticity
1. INTRODUCTION
Laminar buoyancy-driven convection in enclosed cylinders has long been a subject of interest
due to their wide applications such as in solar collector-receivers, cooling of electronic equipment,
aircraft cabin insulation, thermal storage system, and cooling systems in nuclear reactors, etc, Ali,
[1].
Nanofluids are relatively new class of fluids which consist of a base fluid with nano-sized
particles (1–100 nm) suspended within them. In order to improve the performance of engineered heat
transfer fluids, dispersion of highly-conductive nano-sized particles (e.g., metal, metal oxide, and
carbon materials) into the base liquids has become a promising approach since the pioneering
investigation by Choi [2]. Laminar steady-state natural convection of nanofluids in confined regions,
such as square/rectangular cavities, horizontal annuli and triangular enclosures, has been studied for
a variety of combinations of base liquids and nanoparticles [3]. It is noted that in most of the
numerical efforts the nanofluids were considered as a single phase such that the presence of
nanoparticles only plays a role in modifying the macroscopic thermophysical properties of the base
liquids. Therefore, a large number of studies have been dedicated to reveal the mechanisms of
thermophysical properties modification of nanofluids. Most of cooling or heating devices have low
efficiency because the working fluids have the low thermal conductivity. Many experiments have
been carried out in the past which showed tremendous increase in thermal conductivity with addition
of small amount of nanoparticles. However, very few mathematical and computational models have
been proposed to predict the natural convection heat transfer.
Zi-Tao Yu, et. al. [4], studied numerically the transient natural convection heat transfer of
aqueous nanofluids in a horizontal annulus between two coaxial cylinders is presented. The effective
thermophysical properties of water in the presence of copper oxide nanoparticles with four different
volume fractions are predicted using existing models, in which the effects of the Brownian motion of
nanoparticles are taken into consideration. It is shown that at constant Rayleigh numbers, the time-
averaged Nusselt number is gradually decreased as the volume fraction of nanoparticles is increased.
In addition, the time-averaged Nusselt number will be overestimated if the Brownian motion effects
are not considered. Eiyad Abu-Nada [5], investigated numerically the heat transfer enhancement in
horizontal annuli using variable properties of Al2O3–water nanofluid. Different viscosity and thermal
conductivity models are used to evaluate heat transfer enhancement in the annulus. It was observed
that the Nguyen et al. data and Brinkman model gives completely different predictions for Ra≥104
where the difference in prediction of Nusselt number reached 30%. However, this difference was
less than 10% at Ra = 103
.

69
Hakan, et. al. [6], studied the heat transfer and fluid flow due to buoyancy forces in a partially
heated enclosure using nanofluids using different types of nanoparticles. The flush mounted heater is
located to the left vertical wall with a finite length. The temperature of the right vertical wall is lower
than that of heater while other walls are insulated. The finite volume technique is used to solve the
governing equations. Calculations were performed for Rayleigh number (103
≤ Ra ≤ 5×105
), height
of heater (0.1 ≤ h ≤ 0.75), location of heater (0.25 ≤ yp ≤ 0.75), aspect ratio (0.5 ≤ A ≤ 2) and volume
fraction of nanoparticles (0 ≤ ϕ ≤ 0.2). Different types of nanoparticles were tested. An increase in
mean Nusselt number was found with the volume fraction of nanoparticles for the whole range of
Rayleigh number. Heat transfer also increases with increasing of height of heater. It was found that
the heater location affects the flow and temperature fields when using nanofluids. It was found that
the heat transfer enhancement, using nanofluids, is more pronounced at low aspect ratio than at high
aspect ratio.
The present work deals with numerical investigation natural convection heat transfer of the
water-based Cu nanofluid from circular horizontal cylinder situated in an enclosed square enclosure.
The work investigates the effect of nanofluids on the flow and heat transfer characteristics. The study
uses different Raylieh numbers, different enclosure width to cylinder diameter ratios W/D and
different volume fraction of nanoparticles.
2. MATHEMATICAL FORMLATION
The schematic diagram in figure (1), display the flow between the heated horizontal cylinder
and the enclosure. The fluid in the enclosure is a water based nanofluid containing copper Cu. The
governing equations of the flow based on the assumptions that the nanofluid is incompressible, and
the flow is laminar no internal heat sources, and two-dimensional. It is assumed that the base fluid
(water) and the nanoparticles are in thermal equilibrium and no slip occurs between them. The
thermophysical properties are given in table (1).The thermophysical properties of the nanofluids are
assumed to be constant and the flow is Boussinesq, Hakan et. al.[6].
Figure (1) Configuration of cylinder-enclosure combination
W
T∞
Te
W
Ts

70
Table (1): Thermophysical properties of fluid and nanoparticles, Hakan, et. al.[6].
Physical Properties Fluid phase (water) Nanoparticles (Cu)
Cp (J/kg.°K) 4197 385
ρ (kg/m3
) 997.1 8933
k (W/m.°K) 0.613 400
α×107
(m2
/sec) 1.47 1163.1
β×10-5
(m2
/sec) 21 1.67
The governing equations include the equation of continuity, momentum and the energy equation,
Hakan, et. al. [6]. These equations are presented below:
0=
∂
∂
+
∂
∂
y
v
x
u
(1)
The x –momentum equation is:
( )
Tgg
y
u
x
u
x
p
y
u
v
x
u
u
t
u
nf
nf
nf
nf
∆+








∂
∂
+
∂
∂
+
∂
∂
−=
∂
∂
+
∂
∂
+
∂
∂
ρ
βρ
ν
ρ
&
2
2
2
2
1
(2)
The y –momentum equation is:
( )
Tgg
y
v
x
v
x
p
y
v
v
x
v
u
t
v
nf
nf
nf
nf
∆+








∂
∂
+
∂
∂
+
∂
∂
−=
∂
∂
+
∂
∂
+
∂
∂
ρ
βρ
ν
ρ
&
2
2
2
2
1
(3)
The energy equation is:








∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
+
∂
∂
2
2
2
2
y
T
x
T
y
T
v
x
T
u
t
T
nfα (4)
With Boussinesq approximations, the density is constant for all terms in the governing
equations except for the buoyancy force term that the density is a linear function of the temperature.
( )To ∆−= βρρ &1 (5)
Where β is the coefficient of thermal expansion.
The stream function (ψ) and vorticity (ω) in the governing equations are defined as follows,
Anderson [7], and Petrovic [8]:
x
v
y
u
∂
∂
−=
∂
∂
=
ψψ
, (6)

71
y
u
x
v
∂
∂
−
∂
∂
=ω (7)
Or V
r
×∇=ω
the governing equations for laminar flow become:
Energy Equation:






∂
∂
∂
∂
+





∂
∂
∂
∂
=
∂
∂
∂
Ψ∂
−
∂
∂
∂
Ψ∂
+
∂
∂
yyxxyxxyt
θ
λ
θ
λ
θθθ
(8)
Momentum Equation:
( ) ( )
( )
( )
x
Ra
yx
yxxyt
s
f
f
s
s
f
f
s
∂
∂


















+
−
+
+
−
+







∂
∂
+
∂
∂






















+−−
=
∂
∂
∂
Ψ∂
−
∂
∂
∂
Ψ∂
+
∂
∂
θ
ρ
ρ
ϕ
ϕ
β
β
ρ
ρ
ϕ
ϕ
ϖϖ
ρ
ρ
ϕϕϕ
ϖϖϖ
1
1
1
1
1
1
Pr
11
Pr
2
2
2
2
25.0
(9)
Continuity Equation:
ϖ−=
∂
Ψ∂
+
∂
Ψ∂
2
2
2
2
yx
(10)
Where
λ =
ೖ೙೑
ೖ೑
ሺଵିఝሻାఝ
൫ഐ಴ು൯ೞ
൫ഐ಴ು൯೑
(11)
ߙ௡௙ =
௞೐೑೑
ሺఘ஼ುሻ೙೑
(12)
The effective density of the nanofluid is given as
ߩ௡௙ = ሺ1 − ߮ሻߩ௙ + ߮ߩ௦ (13)
The heat capacitance of the nanofluid is expressed as:
ሺߩ‫ܥ‬௉ሻ௡௙ = ሺ1 − ߮ሻሺߩ‫ܥ‬௉ሻ௙ + ߮ሺߩ‫ܥ‬௉ሻ௦ (14)
It is assumed the shape of the nanofluids is spherical, therefore, the effective thermal
conductivity of the nanofluid is approximated by the Maxwell–Garnetts model:
௞೙೑
௞೑
=
௞ೞାଶ௞೑ିଶఝ൫௞೑ି௞ೞ൯
௞ೞାଶ௞೑ାఝ൫௞೑ି௞ೞ൯
(15)

72
The viscosity of the nanofluid can be approximated as viscosity of a base fluid lf containing
dilute suspension of fine spherical particles and is given by Brinkman [9]:
ߤ௡௙ =
ఓ೑
ሺଵିఝሻమ.ఱ (16)
In the stream function-vorticity formulation, there is a reduction in the number of equations
to be solved in the ψ-ω formulation, and the troublesome pressure terms are eliminated in the ψ-ω
approach.
The dimensionless variables in the above equations are defined as:
D
x
X = ,
D
y
Y = ,
f
uD
U
α
= ,
f
vD
V
α
= ,
2
D
t fα
τ = ,
fα
ψ
=Ψ ,
f
D
α
ω
ϖ
2
= ,
∞
∞
Τ−Τ
Τ−Τ
=
c
θ (17)
The cylinder diameter D is the characteristic length in the problem. By using the above
parameters, the governing equations (8)-(10) transformed to the following general form in the
computational space:
(18)
Where φ is any dependent variable.
The governing equations represented by interchanging the dependent variable φ for three
governing equations as follow
φ φ bφ dφ
ψ 0 1 ω
ω 1 ( ) ( ) 







+−−
f
s
ρ
ρ
ϕϕϕ 11
Pr
25.0
( )
( )
( ) ( )[ ]ηξξη θθ
ρ
ρ
ϕ
ϕβ
β
ρ
ρ
ϕ
ϕ
yyRa
s
ff
s
s
f
−












+
−
+
+
−
1
1
1
1
1
1
Pr
T 1 λ 0
t∂
∂φ
represents the unsteady term.














∂
∂
−





∂
∂
ηξ
φ
ξ
ψ
φ
η
ψ
J
1
is the convective term.
( )φφ∇∇ b is the diffusion term.
In addition, φd is the source term.
( ) φφ
ηξ
φ φφ
ξ
ψ
φ
η
ψφ
db
Jt
a +∇∇=






















∂
∂
−





∂
∂
+
∂
∂ 1

73
2.1 Grid Generation
The algebraic grid generation method is used to generate an initial computational grid points.
The elliptic partial differential equations that used are Poisson equations:
( )ηξξξ ,Pyyxx =+ (19a)
( )ηξηη ,Qyyxx =+ (19b)
Interchanging dependent and independent variables for equations (19a, and b), gives:
( ) 0
2
2
=+
++−
ηξ
ηηξηξξ γβα
xQxPJ
xxx
(20a)
( ) 0
2
2
=+
++−
ηξ
ηηξηξξ γβα
yQyPJ
yyy
(20b)
Where 22
ηηα yx += ; ηξηξβ yyxx += ;
22
ξξγ yx +=
The coordinate control functions P and Q may be chosen to influence the structure of the
grid, Thomas et. al. [10]. The solution of Poisson equation and Laplace equation are obtained using
Successive over Relaxation (SOR) method with relaxation factor value equal to 1.4, Hoffman [11]
and Thompson [12]. The transformation of the physical domain into computational domain using
elliptic grid generation is shown in figure (2).
Figure (2) Transformations of the physical domains into computational domains using elliptic grid
generation.

74
2.2 Method of Solution
In the present study, the conversion of the governing integro-differential equations into
algebraic equations, amenable to solution by a digital computer, is achieved by the use of a Finite
Volume based Finite Difference method, Ferziger [13].
To avoid the instability of the central differencing scheme (second order for convective term)
at high Peclet number (Cell Reynolds Number) and an inaccuracy of the upwind differencing scheme
(first order for convective term) the hybrid scheme is used. The method is hybrid of the central
differencing scheme and the upwind differencing scheme.
( ) jijijijijiM
o
P
o
PSSNNWWEEPP
da
aaaaaa
,1,11,11,11,1 ++−−
+++++=
+−−+−−++ φφφφ
φφφφφφ
(21)
o
PSNWEP aaaaaa ++++= (22)
The resulting algebraic equation is solved using alternating direction method ADI in two
sweeps; the first sweep, the equations are solved implicitly in ξ-direction and explicitly in η-
direction. The second sweep, the equations are solved implicitly in η-direction and explicit in ξ-
direction. In first sweep, the implicit discretization equation in ξ-direction is solved by using Cyclic
TriDiagonal Matrix Algorithm (CTDMA) because of its cyclic boundary conditions. In second
sweep, the implicit discretization equation in η-direction is solved by using TriDiagonal Matrix
Algorithm (TDMA).
The solution of the stream function equation was obtained using Successive Over-Relaxation
method (SOR). The initial conditions of the flow between heated cylinder and vented enclosure are:
Ψ=0, θ = 0, ω = 0 For t = 0 (23)
The temperature boundary condition of the cylinder surface assumed as constant.
0=
∂
∂
m
η
θ
at enclosure wall (24a)
Using 2nd
order difference equation, the temperature at the enclosure surface becomes:
2,1,,
3
1
3
4
−− −= mimimi θθθ (24b)
Vorticity boundary conditions, Roache [14], are
( )1,,2
2
−−= mimi
J
ψψ
γ
ϖ at enclosure wall (25a)
( )2,1,2
2
ii
J
ψψ
γ
ϖ −= at cylinder surface (25b)
The stream function of the cylinder is assumed as zero because the cylinder is a continuous
solid surface and no matter enters into it or leaves from it. The stream function of the enclosure is
assumed as constant.
The Nusselt number Nu is a nondimensional heat transfer coefficient that calculated in the
following manner:

75
fk
hD
Nu =
(27)
The heat transfer coefficient is expressed as
h =
୯౭
୘ౄି୘ై
(28)
The thermal conductivity is expressed as
k୬୤ = −
୯౭
பθ ப୬⁄
(29)
By substituting Eqs. (24), (25), and (7) into Eq. (23), and using the dimensionless quantities, the
Nusselt number on the left wall is written as:
ζ
θπ
∂
∂
∂
−= ∫
2
0
nk
k
Nu
f
nf
(30a)
The derivative of the nondimensional temperature is calculated using the following formula, Fletcher
[15]:
( ) η
θγ
γθβθ
γ
θ
ηξ
η ∂
∂
=+−=
∂
∂
= JJn const
1
.
(30b)
θξ = 0 at cylinder surface
A computer program in (Fortran 90) was built to execute the numerical algorithm which is
mentioned above; it is general for a natural convection from heated cylinder situated in an enclosure.
3. RESULTS AND DISCUSSION
In the present study, the numerical work deals with natural convection heat transfer utilizing
nanofluids as working fluid from circular horizontal cylinder when housed in an enclosed square
enclosure. The Prandtl number is taken as 6.2. The cases for three different enclosure width to
cylinder diameter ratios W/D =1.67, 2.5 and 5, Rayleigh numbers of 104
, 105
, and 106
, and volume
fractions of nanofluid ϕ are 0, 0.05, 0.1, 0.15 and 0.2 were studied.
After numerical discretization by the Hybrid method, the resultant algebraic equations are
solved by the ADI method. The convergence criteria are chosen as RT<10-6
, Rψ<10-6
and Rω<10-6
for
T, ψ and ω respectively. When all the three criteria are satisfied, the convergent results are
subsequently obtained.
3.1 Stability and Grid Independency Study
The stability of the numerical method is investigated for the case Ra=105
, W/D=2.5, Pr = 0.7.
Three time steps are chosen with values 1×10-4
, 5×10-4
, 5×10-6
. The maximum difference between
the values of Nu with different time steps is 2%. The grid-independence of numerical results is
studied for the case with Ra=104
, and 105
, W/D =2.5, Pr = 6.2. The three mesh sizes of 96×25,
128×45, and 192×50 are used to do grid-independence study. It is noted that the total number of grid
points for the above three mesh sizes is 2425, 5805, and 9650 respectively. Numerical experiments
showed that when the mesh size is above 96×45, the computed Nu remain the same. The same

76
accuracy is not obtainable with W/D=5 and high Raylieh numbers, therefore; the mesh size 128×45
is used in the present study for all cases.
3.2 Validation Test
The code build using Fortran 90 to execute the numerical algorithm. To test the code
validation, the natural convection problem for a low temperature outer square enclosure and high
temperature inner circular cylinder was tested. The calculations of average Nusselt numbers and
maximum stream function ψmax for the test case are compared with the benchmarks values by
Moukalled and Acharya [16], for Prandtl number Pr=0.7, different values of the enclosure width to
cylinder diameter ratios (W/D=1.667, 2.5, and 5) with Rayleigh numbers Ra=104
and 105
as given in
table (2). From table 2, it can be seen that the present results generally agree well with those of
Moukalled and Acharya [16].
Table (2): Comparisons of Nusselt numbers and maximum stream function
L/D Ra
ψmax ܰ‫ݑ‬തതതത
Present
Moukalled and
Acharya [16]
Present
Moukalled and
Acharya [16]
5.0
104
2.45 2.08 1.7427 1.71
2.5 3.182 3.24 0.9584 0.97
1.67 5.22 5.4 0.4274 0.49
5.0
105
10.10 10.15 3.889 3.825
2.5 8.176 8.38 4.93 5.08
1.67 4.8644 5.10 6.23 6.212
3.3 Flow Patterns and Isotherms
The numerical solutions for four volume fraction of nanofluids ϕ were obtained. The volume
fraction values are: ϕ = 0.05, 0.1, 0.15, and 0.2 will be presented herein. Figure (3) shows a
comparison of streamlines and isotherms between Cu-water nanofluid (ϕ=0.1) and pure fluid (ϕ=0)
for W/D=2.5 with Raylieh number values Ra=104
, 105
, and 106
. At Ra=104
and 105
, the isotherms
for two cases are similar. There are some differences in isotherms between two cases for Ra =106
.
The disagreement appear at the upper region of the isotherms above the circular cylinder. The
plumes for pure fluid appear as more flat than those for ϕ=0.1. The streamlines are different between
two cases. The difference becomes more as Raylieh number increases. At Ra=104
, the conduction is
the dominant heat transfer, therefore, the disagreement between the two cases is very small. The
flow circulation for ϕ=0.1 is more than those for pure fluid. The maximum stream function value
ψmax=1.75 for ϕ=0.1 as compared with those for pure fluid ψmax=0.96. As Raylieh number increases
to Ra=105
, the flow circulation becomes more and the disagreement between two cases increases.
The general aspects of the flow patterns are similar except that a single small kernel eddy appears
rather than the dual kernel eddies for Ra=104
for pure fluid. The size of the kernel eddy increases for
ϕ=0.1 and the flow circulation is more than those for pure fluid. The maximum stream function
value ψmax=13.68 for ϕ=0.1, while; the maximum stream function value ψmax=8.05 for pure fluid. As
Raylieh number increases to Ra=106
, the flow becomes stronger and the maximum stream function
increases for two cases. The maximum stream function value ψmax=36.94 for ϕ=0.1, while; the
maximum stream function value ψmax=23.34 for pure fluid. The flow are symmetrical about the
vertical center line. As compared with the streamlines of the pure fluid, the streamlines of the ϕ=0.1
trends to the declination at the upper region of the enclosure and the flow region between the

77
cylinder and the enclosure becomes more than those for pure fluid, that means the stagnant area
decreases. The size of the kernel eddy becomes less and the densely packed becomes more.
Fig. (3) Streamlines (on the left) and Isotherms (on the right) for Cu-water nanofluids (- - -), pure
fluid (___
), W/D=2.5, (a) Ra = 104
, (b) Ra = 105
(c) Ra =106
.
Figures (4-6) display the streamlines and isotherms for W/D=1.67, 2.5, and 5, respectively.
The Raylieh number values are Ra=104
, 105
, and 106
for W/D=2.5, and 5, while, the Ra=104
, and 105
for W/D=1.67, because the flow velocity increases and becomes turbulent for nanofluids flow. At
W/D=1.67, that shown in figure (4), the circular cylinder diameter is relatively large and the physical
domain between the circular cylinder and the enclosure is small. The stream function values vary
with variation of the Raylieh number, and volume fraction of nanofluids. The maximum stream
function values between Ψmax=0.65 at Ra=104
and ϕ=0.05 to Ψmax=11.239 at Ra=105
and ϕ=0.2. The
flow is symmetrical about the vertical line through the center of the circular cylinder for all cases. At
Ra=104
, the flow circulation is very weak and the maximum stream function value is small. The flow
patterns appear as a curved kidney-shaped dual-kernel eddy. The aspects of the streamlines for all
cases are nearly similar and independent of volume fraction of the nanofluids with little difference in
the size of the kernel eddies. At Ra=105
, the flow circulation becomes stronger than Ra=104
and the
stream function value increases. Two tiny eddies appear near the vertical center line in addition to
the eddies around the cylinder. The aspects of the flow patterns are nearly similar to the cases of
Ra=104
.

78
ϕ=0.05 ϕ=0.1 ϕ=0.15 ϕ=0.2
Fig. (4) Effect of volume fraction of nanofluids on streamlines at W/D = 1.67 and (a) Ra = 104
, (b)
Ra = 105
.
The characteristics of the temperature distributions are presented by means of isotherms in figure (5).
The same arrangements as flow patterns are displayed in the figure with same volume fraction of
nanofluids and Raylieh numbers. The isotherms are symmetrical about the vertical line through the
center of the circular cylinder. At Ra=104
, the isotherms are similar and independent of volume
fraction of nanofluids for all cases. The isotherms display as rings around the cylinder. The shape of
the isotherms ensure that the mode of heat transfer is pure conduction and the effect of the
convection is very low. At Ra=105
, the temperature distributions have small distortions around the
cylinder due to the effect of the convection heat transfer. The same behaviors occur for all nanofluid
volume fractions.
ϕ=0.05 ϕ=0.1 ϕ=0.15 ϕ=0.2
Fig. (5) Effect of volume fraction of nanofluids on isotherms at W/D = 1.67 and (a) Ra = 104
, (b) Ra
= 105
.

79
At W/D=2.5, the flow patterns for Raylieh numbers Ra=104
, 105
, 106
, and nanofluid volume
fractions ϕ=0.05, 0.1, 0.15 and 0.2 are presented in figure (6). The circular cylinder diameter reduces
and the physical domain between the circular cylinder and the enclosure enlarges. The maximum
stream function value varies between Ψmax=1.3486 at Ra=104
and ϕ=0.05 to Ψmax=47 at Ra=106
and
ϕ=0.2. For all Raylieh numbers and volume fractions of nanofluids, the flow is symmetrical about
the vertical line through the center of the circular cylinder. At Ra=104
, the flow circulation is weak,
but it become stronger than the flow circulation for W/D=1.67. The maximum stream function value
is small. The flow patterns appear as a curved kidney-shaped contain single kernel eddy. The single
small kernel eddy appears and the densely packed of the flow is small. As volume fraction of
nanofluids increases to ϕ=0.1, the flow circulation increases. Two kernel eddies appear rather than
single kernel eddy for ϕ=0.05 and the densely packed of the flow becomes more and the stagnant
area reduces. The flow becomes stronger for ϕ=0.15, but the aspects of the streamlines remain
unchanged except the flow becomes more densely packed and elongate the kernel eddies below the
cylinder. The stagnant area of the flow region increases for ϕ=0.2, the flow circulation becomes
more and the densely packed becomes less. At Ra=105
, the strength of the flow circulation becomes
more, and the value of stream function increases. The flow patterns have less densely packed and
appears as single kernel eddy for ϕ=0.05. The shapes of the streamlines are nearly similar for all
volume fractions of the nanofluids, except that the densely packed of the flow increases with
increasing the volume fraction of the nanofluids. The kernel eddy size is constant for ϕ≤0.15, then
decreases the size of the kernel eddy for ϕ=0.2. As Raylieh number increases to Ra=106
, the flow
becomes stronger and the maximum stream function increases for all cases. The flow is symmetrical
about the vertical center line. The streamlines near the bottom enclosure wall move more and more
to the upward that lead to an increase in the stagnant area. The streamlines near the upper enclosure
wall are horizontal and flat. The kernel eddy elongates and its size becomes more. The flow region
becomes more and the stagnant area decreases with increasing the volume fractions of the
nanofluids. Also, the densely packed of the flow enlarges and the shape of the flow change. The
eddies transform from nearly triangle-shaped to strip-shaped.
ϕ=0.05 ϕ=0.1 ϕ=0.15 ϕ=0.2
Figure (6) Effect of volume fraction of nanofluids on streamlines at W/D = 2.5 and (a) Ra = 104
, (b)
Ra = 105
, (c) Ra = 106
.

80
The temperature distributions for W/D=2.5 are presented by means of isotherms in figure (7).
The same arrangements as flow patterns are displayed in the figure with same volume fractions of
the nanofluids and Raylieh numbers. The isotherms are symmetrical about the vertical line through
the center of the circular cylinder. As Raylieh number increases, the thermal boundary layer becomes
thinner and thinner. The isotherms do not change with changing the volume fractions of the
nanofluids for all Raylieh numbers.
At Ra=104
, the isotherms are similar and independent of volume fractions of the nanofluids.
The mode of heat transfer is the conduction with very little effect of convection heat transfer. The
isotherms display as circles around the cylinder. As Raylieh number increases to Ra=105
, the
temperature distributions are similar for all volume fractions of the nanofluids. The isotherms
distorts below the cylinder due to the effect of the convection heat transfer. A thermal plume appear
on the top of the cylinder. The isotherms are horizontal and flat near the lower enclosure wall with
very little distortion in the isotherms at this region. At Ra=106
, the isotherms are nearly similar and
independent of volume fractions of the nanofluids. The convection becomes the dominant mode of
heat transfer. A thermal plume impinging on the top of the enclosure. The thermal stratification
(horizontal and flat isotherms) are formed near the bottom region of the enclosure. Two thermal
plumes displayed on the top of the cylinder with about 60° from the vertical center line.
ϕ=0.05 ϕ=0.1 ϕ=0.15 ϕ=0.2
Fig. (7) Effect of volume fraction of nanofluids on streamlines at W/D = 2.5 and (a) Ra = 104
, (b) Ra
= 105
, (c) Ra = 106
.
The flow patterns for W/D=5 are presented herein by means of streamlines with ϕ= 0.05, 0.1,
0.15, and 0.2 and Ra=104
, 105
, 106
as shown in figure (8). The same arrangements that use for the
streamlines of the case W/D = 2.5 are used. The circular cylinder diameter is relatively small and the
physical domain between the circular cylinder and the enclosure enlarges. The maximum stream
function value varies between Ψmax=2.458 at Ra=104
and ϕ=0.05 to Ψmax=45.1 at Ra=106
and ϕ=0.2.

81
At Ra=104
, the flow circulation is weak, it become stronger than the flow circulation for previous
cases with Ra=104
. The flow is symmetrical about the vertical center line for all volume fractions of
the nanofluids. The flow patterns appear as a curved kidney-shaped contain one big kernel eddy. The
flow have less densely packed. As volume fraction of the nanofluids increases, the densely packed of
the nanofluids increase, the size of the kernel eddies decreases, and the flow region becomes more
that means decreasing in the stagnant area. At Ra=105
, the flow circulation becomes stronger, which,
the value of maximum stream function increases. The flow is symmetrical about vertical line through
circular cylinder. The streamlines near the bottom of the enclosure wall move upward toward the
cylinder and the stagnant area becomes more. As the volume fraction of the nanofluids increases, the
densely packed of the nanofluids increase, the size of the kernel eddies decreases, and the flow
region becomes more that means decreasing in the stagnant area. At Ra=106
, the flow circulation
becomes stronger and stronger, and the maximum stream function increases for all volume fractions
of the nanofluids. For ϕ=0.05, the streamlines near the bottom of the enclosure wall move upward to
reach the bottom of the cylinder and the stagnant area becomes more and more. The flow becomes
asymmetrical about vertical line through circular cylinder because a tiny eddy appears near the
vertical center line at the bottom of the enclosure. As volume fraction of the nanofluids increases, the
densely packed of the nanofluids increase, the size of the kernel eddies decreases, and the flow
region becomes more that means decreasing in the stagnant area. The shapes of the flow patterns
change, and the flow trend to become curvature specially below the cylinder.
ϕ=0.05 ϕ=0.1 ϕ=0.15 ϕ=0.2
Figure (8) Effect of volume fraction of nanofluids on streamlines at W/D = 5 and (a) Ra = 104
, (b)
Ra = 105
, (c) Ra = 106
.
The temperature distributions for W/D=5 are presented by means of isotherms as shown in
figure (8). The same arrangements as flow patterns are displayed in the figure with same Prandtl
numbers and Raylieh numbers. The isotherms are symmetrical about the y-axis line through the
center of the circular cylinder. As Raylieh number increases, the thermal boundary layer becomes
thinner. The isotherms are similar for each Raylieh number and independent of volume fractions of

82
the nanofluids for all volume fractions of the nanofluids. At Ra=104
, the mode of heat transfer is the
conduction with little contribution of convection heat transfer. The isotherms display as nearly
elliptical-shaped around the cylinder. As Raylieh number increases to Ra=105
, the isotherms distort
below the cylinder due to the effect of the convection heat transfer. A thermal plume appear on the
top of the cylinder. Two thermal plumes displayed on the top of the cylinder with about 60° from the
y-axis center line. The distortion of the isotherms become more around the cylinder due to the effect
of the convection heat transfer. At Ra=106
, the thermal boundary layer becomes thinner and thinner
and the convection becomes the dominant mode of the heat transfer. A thinner thermal plume
impinging on the top of the enclosure. Two plumes appear on top of the inner circular cylinder with
about 60◦ from the vertical centre line. The isotherms below the cylinder are wavy. Thermal plume
above top of the cylinder becomes thinner and increases its length. The isotherms below the cylinder
become more flat and horizontal as compared with those for Ra=105
.
Fig. (8) Effect of volume fraction of nanofluids on isotherms at W/D = 5 and
(a) Ra = 104
, (b) Ra = 105
, (c) Ra = 106
.
3.4 Overall heat transfer and correlations
The average Nusselt number is chosen as the measure to investigate the heat transfer from the
circular cylinder. The effect of volume fraction of the nanofluids on the average Nusselt numbers
with Ra=104
, 105
, and 106
for enclosure width to the cylinder ratios W/D=1.67, 2.5 and 5 are
presented in figures (10, 11). The volume fractions of the nanofluids ϕ in the present study are:0,
0.05, 0.1, 0.15, 0.2. The Nusselt number increases with increasing the Raylieh number for all ϕ and
all enclosure width to cylinder diameter ratios. The effect of volume fraction of the nanofluids on the
Nusselt number for each enclosure width to cylinder diameter ratio W/D are shown in figure (10).
The Raylieh number relation depend on the side length of the enclosure, therefore; the Nusselt

83
number value descending from W/D=1.67, 2.5 and 5 respectively. The variation of the Nusselt
number values with volume fractions of the nanofluids increases with increasing the enclosure width
to cylinder diameter ratio W/D for same Raylieh number. Also, the enhancement of the Nusselt
number with changing the nanofluids volume fractions increases with increasing Raylieh number.
Figure (10) shows the variation of the Nusselt number using different Raylieh numbers. The
maximum enhancement in the Nusselt number when the volume fraction of nanoparticles is
increased from 0 to 0.2, using Ra=104
, is approximately 46%, the maximum enhancement is around
48% for Ra= 105
, whereas the maximum enhancement is around 46% for Ra= 106
. Figure (11) shows
the variation of the Nusselt number using different enclosure width to cylinder diameter ratios. The
maximum enhancement in the Nusselt number when the volume fraction of nanoparticles is
increased from 0 to 0.2, using W/D =1.67, is approximately 46.4%, the maximum increase is around
48% for W/D=2.5, whereas the maximum increase is around 46% for W/D=5. This tells that the
enhancement in heat transfer, due to the presence of nanoparticles, is pronounced for all Raylieh
numbers and enclosure width to cylinder diameter ratios. The heat transfer enhances with increasing
the volume fraction of the nanofluids because more particles suspended and the effect of thermal
conductivity and viscosity of the nanofluids on the heat transfer.
(a)
(b)

84
(c)
Figure (11) Effect of volume fraction of Nanofluids on the Nusselt number for each enclosure width
to circular diameter W/D, (a) Ra=104, (b) Ra=105, (c) Ra=106.
4. CONCLUSIONS
Effect of the presence of the nanofluids on the natural convection heat transfer from circular
horizontal cylinder in a square enclosure was investigated numerically over a fairly wide range of Ra
with taking the effect of enclosure width. The main conclusions of the present work can be
summarized as follows:
1. The numerical results show that the Nusselt number increases with increasing the Raylieh
number for all cases.
2. The flow patterns and isotherms display the effect of Ra, enclosure width, and volume
fractions of the nanofluids on the thermal and hydrodynamic characteristics.
3. The Conduction is the dominant of the heat transfer at Ra=104
for all cases. The contribution
of the convective heat transfer increases with increasing the Raylieh number.
4. The results show that the isotherms are nearly similar when the volume fraction of
nanoparticles is increased from 0 to 0.2 for each Raylieh number and enclosure width to
cylinder diameter ratio.
5. The streamlines are asymmetrical when the volume fraction of nanoparticles is increased
from 0 to 0.2 for each Raylieh number and enclosure width to cylinder diameter ratio.
6. The average Nusselt number enhances gradually when the volume fraction of nanoparticles is
increased from 0 to 0.2 for each Raylieh number and enclosure width to cylinder diameter
ratio.
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NUMERICAL INVESTIGATION OF NATURAL CONVECTION HEAT TRANSFER FROM CIRCULAR CYLINDER INSIDE AN ENCLOSURE CONTAINING NANOFLUIDS

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NUMERICAL INVESTIGATION OF NATURAL CONVECTION HEAT TRANSFER FROM CIRCULAR CYLINDER INSIDE AN ENCLOSURE CONTAINING NANOFLUIDS