NUMERICAL INVESTIGATION OF NATURAL CONVECTION HEAT TRANSFER FROM CIRCULAR CYLINDER INSIDE AN...

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. Zaidky

2, Ahmed M. Saleem

3

1Department of Refrigeration and Air Conditioning, Technical Institute of Zakho, Zakho,

Kurdistan, Iraq, 2Faculty of Petroleum Engineering, Zakho, Kurdistan, Iraq,

3Department 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, 10

5, and 10

6), 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

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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 ≤ 5×10

5), 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(m

2/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

uv

x

uu

t

u

nf

nf

nf

nf

∆+

∂

∂+

∂

∂+

∂

∂−=

∂

∂+

∂

∂+

∂

∂

ρ

βρ

νρ

&

2

2

2

21

(2)

The y –momentum equation is:

( )Tgg

y

v

x

v

x

p

y

vv

x

vu

t

v

nf

nf

nf

nf

∆+

∂

∂+

∂

∂+

∂

∂−=

∂

∂+

∂

∂+

∂

∂

ρ

βρ

νρ

&

2

2

2

21

(3)

The energy equation is:

∂

∂+

∂

∂=

∂

∂+

∂

∂+

∂

∂2

2

2

2

y

T

x

T

y

Tv

x

Tu

t

Tnfα (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]:

xv

yu

∂

∂−=

∂

∂=

ψψ,

(6)



71

y

u

x

v

∂

∂−

∂

∂=ω (7)

Or Vr

×∇=ω

the governing equations for laminar flow become:

Energy Equation:

∂

∂

∂

∂+

∂

∂

∂

∂=

∂

∂

∂

Ψ∂−

∂

∂

∂

Ψ∂+

∂

∂

yyxxyxxyt

θλ

θλ

θθθ (8)

Momentum Equation:

( ) ( )

( )

( )

xRa

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

ρϕϕϕ

ϖϖϖ

11

1

11

1

Pr

11

Pr

2

2

2

2

25.0

(9)

Continuity Equation:

ϖ−=∂

Ψ∂+

∂

Ψ∂2

2

2

2

yx (10)

Where

λ =

��

��

(11)

�� =��

�� (12)

The effective density of the nanofluid is given as

�� = �1 − �� + �� (13)

The heat capacitance of the nanofluid is expressed as:

�� = �1 − �� + �� (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:

��

=��

�� (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]:

!�� ="�

��#.% (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

xX = ,

D

yY = ,

f

uDU

α= ,

f

vDV

α= ,

2D

t fατ = ,

fα

ψ=Ψ ,

f

D

α

ωϖ

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

ρϕϕϕ 11

Pr

25.0

( )( )

( ) ( )[ ]ηξξη θθ

ρ

ρ

ϕ

ϕβ

β

ρ

ρ

ϕ

ϕyyRa

s

ff

s

s

f

−

+−

+

+−

11

1

11

1Pr

T 1 λ 0

t∂

∂φ represents the unsteady term.

∂

∂−

∂

∂

ηξ

φξ

ψφ

η

ψ

J

1 is the convective term.

( )φφ∇∇ b is the diffusion term.

In addition, φd is the source term.

( ) φφ

ηξ

φ φφξ

ψφ

η

ψφdb

Jta +∇∇=

∂

∂−

∂

∂+

∂

∂ 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 22ηηα yx += ; ηξηξβ yyxx += ;

22ξξγ 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

oP

oPSSNNWWEEPP

da

aaaaaa

,1,11,11,11,1 ++−−

+++++=

+−−+−−++ φφφφ

φφφφφφ (21)

oPSNWEP 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=∂

∂

mη

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

Jψψ

γϖ at enclosure wall (25a)

( )2,1,2

2ii

Jψψ

γϖ −= 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

hDNu =

(27)

The heat transfer coefficient is expressed as

h = '()*�)+

(28)

The thermal conductivity is expressed as

k-. = − '(/θ /-⁄

(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:

ζθπ

∂∂

∂−= ∫

2

0nk

kNu

f

nf (30a)

The derivative of the nondimensional temperature is calculated using the following formula, Fletcher

[15]:

( )η

θγγθβθ

γ

θηξ

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, 10

5, and 10

6, 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 10

5, 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 10

5 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 123333

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, 10

5, and 10

6. At Ra=10

4 and 10

5, 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 = 10

5 (c) Ra =10

6.

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, 10

5, and 10

6 for W/D=2.5, and 5, while, the Ra=10

4, and 10

5

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=10

5 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=10

4 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, 10

5, 10

6, 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=10

6 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 = 10

6.



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 = 10

6.

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, 10

5, 10

6 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=10

6 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 = 10

6.

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 = 10

5, (c) Ra = 10

6.

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, 10

5, and 10

6 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= 10

6. 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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