NUMERICAL INVESTIGATION OF NATURAL CONVECTION HEAT TRANSFER FROM CIRCULAR CYLINDER INSIDE AN ENCLOSURE USING DIFFERENT TYPES OF NANOFLUIDS

International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),

ISSN 0976 – 6359(Online), Volume 5, Issue 5, May (2014), pp. 214-236 © IAEME

214

NUMERICAL INVESTIGATION OF NATURAL CONVECTION HEAT

TRANSFER FROM CIRCULAR CYLINDER INSIDE AN ENCLOSURE

USING DIFFERENT TYPES OF NANOFLUIDS

1Omar Mohammed Ali,

2Ghalib Younis Kahwaji

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

2Mechanical Department, RIT Institute, Dubai, UAE

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. Different types of nanoparticles were tested. The types of the nanofluids are Cu, Al2O3

and TiO3 with water as base fluid. 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), the enclosure

width to cylinder diameter ratio W/D is 2.5 and volume fraction of nanofluids is 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 comparisons between different types of the nanofluids based on the

effect of Raylieh number, and volume fractions on the thermal and hydrodynamic characteristics.

The results were compared with previous numerical results, which showed good agreement. For all

types of the nanofluids, the Nusselt number increases with increasing the volume fraction of the

nanofluids. The results show that the streamlines change with changing the type of the nanofluid,

while the isotherms remain unchanged. The Nusselt number of Cu nanofluids is more than the those

for other types of the nanofluids.

KEYWORDS: Circular Cylinder, Heat Transfer, Nanofluids, Numerical, Square Enclosure.

INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING

AND TECHNOLOGY (IJMET)

ISSN 0976 – 6340 (Print)

ISSN 0976 – 6359 (Online)

Volume 5, Issue 5, May (2014), pp. 214-236

© IAEME: www.iaeme.com/ijmet.asp Journal Impact Factor (2014): 7.5377 (Calculated by GISI) www.jifactor.com

IJMET

© I A E M E



215

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.

Subscript nf Nanofluid

p Particle

S Cylinder surface.

∞ Environment.

X Derivative in x-direction.



216

Y Derivative in y-direction.

ξ Derivative in ξ-direction.

D Circular cylinder diameter.

ψ Stream function.

T Temperature.

ω Vorticity

I. 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.

Ternik et., al. [4], performed numerical analysis to investigate the natural convection heat

transfer enhancement of Au, Al2O3, Cu and TiO3 water based nanofluids in an enclosure. The

Raylieh number range 103≤Ra≤10

5, and the nanofluid's volume fraction range is 0≤ϕ≤0.1. The

results indicate that the average Nusselt number is an increasing function of Raylieh number and

volume fraction of nanoparticles. The results display that low Raylieh numbers show more

enhancement compared to high Raylieh numbers.

Hakan, et. al. [6], studied heat transfer and fluid flow due to buoyancy forces in a partially

heated enclosure using different types of nanoparticles. They used a flush mounted heater. 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 (water as base fluid with nanoparticles Cu, Al2O3, and TiO3). An

increase in mean Nusselt number was found with the volume fraction of nanoparticles for the whole

range of Rayleigh number. It was found that the heat transfer enhancement, using nanofluids, is more

pronounced at low aspect ratio than at high aspect ratio. The increase in Nusselt number for Cu is

more than other nanofluids.

The present work deals with numerical investigation natural convection heat transfer using

different types of nanofluids from circular horizontal cylinder situated in an enclosed square

enclosure. The study uses different Raylieh numbers and different volume fraction of nanoparticles.



217

The work performed the comparison between three types of the nanofluids based on the flow and

heat transfer characteristics of the nanofluids.

II. MATHEMATICAL FORMLATION

The schematic diagram in figure (1), display the flow between the heated horizontal cylinder

and the enclosure. The different types of the nanofluids in the enclosure is a water based nanofluid

containing either Cu, Al2O3 or TiO3. 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

Table 1: Thermophysical properties of fluid and nanoparticles, Hakan, et. al.[6]

Physical Properties Fluid phase

(water) Cu

Al2O3 TiO3

Cp (J/kg.°K) 4197 385 765 686.2

ρ (kg/m3) 997.1 8933 3970 4250

k (W/m.°K) 0.613 400 40 8.9538

α×107(m

2/sec) 1.47 1163.1 131.7 30.7

β×10-5

(m2/sec) 21 1.67 0.85 0.9

W

T∞

Te

W

Ts



218

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)

y

u

x

v

∂

∂−

∂

∂=ω (7)

Or Vr

×∇=ω

The governing equations for laminar flow become:



219

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)

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]:



220

!�� "�

��#.% (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 bφ dφ

1 ω

( ) ( )

+−−

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

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



221

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

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].



222

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.



223

The Nusselt number Nu is a nondimensional heat transfer coefficient that calculated in the

following manner:

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.

III. 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.

Stability and Grid Independency Study The stability of the numerical method is investigated for the case Ra=10

5, 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



224

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

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.

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

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



225

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

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.

Figure (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 (4-11) display the streamlines and isotherms for W/D= 2.5. The Raylieh number

values are Ra=104, 10

5, and 10

6, volume fraction of the nanofluids are: ϕ=0.05, 0.1, 0.15, 0.2, and

the nanofluid types are: Cu, Al2O3, and TiO3 nanofluids with water as base fluid.



226

At ϕ=0.05, the flow patterns for Raylieh numbers Ra=104, 10

5, 10

6, and nanofluid types Cu,

Al2O3 and TiO3 are presented in figure (4). The maximum stream function value varies between

Ψmax=1.0744 at Ra=104 and Al2O3 nanofluid type to Ψmax=32.28 at Ra=10

6 and Cu nanofluid type.

For all Raylieh numbers and types of nanofluids, the flow is symmetrical about the vertical line

through the center of the circular cylinder. For Ra=104, the maximum stream function value is small

that means the flow circulation is weak. The flow patterns appear as a curved kidney-shaped contain

one kernel eddy. The densely packed of the flow for Cu nanofluid is relatively weak. The densely

package of the flow for Al2O3 and TiO3 nanofluid types becomes more and the flow region area

increases. The flow patterns for those types are nearly similar. Two eddies appear for each side rather

than single eddy for those of Cu nanofluid type. As Raylieh number increases to Ra=105 for Cu

nanofluid type, the flow circulation becomes stronger, it moves upward, the flow region area

becomes less and the densely package of the flow decreases as compared with Ra=104. A single

large eddy appears. The densely package of the flow for Al2O3 and TiO3 nanofluid types increase, the

flow region area become more and the size of the eddy decreases. The aspects of the flow patterns

for types Al2O3 and TiO3 nanofluid are nearly similar. At Ra=106, the aspects of the flow patterns

with types Al2O3 and TiO3 nanofluid are nearly similar. The flow circulation becomes stronger and

stronger and the stream function values increase. The flow move upward and the streamlines at the

upper of the enclosure appear as horizontal lines and the size of the eddies increase and appear as

triangular-shaped eddies. The stagnant area enlarges and the flow region area decreases. The flow

patterns of the various types are different. Tiny eddies appear inside the eddies. The single eddy size

for of the Cu nanofluid is larger than those of other types and it appear as longitudinal eddy. The

width of the eddies for Al2O3 nanofluid increase and the length of those decrease.

Figure (4): Effect of volume fraction of nanofluids on streamlines for ϕ=0.05 and (a) Ra = 104,

(b) Ra = 105, (c) Ra = 10

6

Cu Al2O3 TiO3

(c)

(b)

(a)



227

The characteristics of the temperature distributions are presented by means of isotherms in

figures (5, 7, 9, 11). 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. The isotherms are similar and independent of

volume fraction of nanofluids and types of the nanofluid for each Raylieh number. At Ra=104, 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. A thermal plume appear on the top of the cylinder. The isotherms appear as

horizontal and flat near the lower enclosure wall with very little distortion in the isotherms at this

region. At Ra=106, 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. It is noted that when the Raylieh number

increases the thermal boundary layer becomes thinner and thinner.


5, 10




6 and Cu nanofluid type. For

Cu Al2O3 TiO3

(c)

(b)

(a)

Figure (5): Effect of volume fraction of nanofluids on isotherms for ϕ=0.05 and (a) Ra = 104, (b)

Ra = 105, (c) Ra = 10

6



228

all Raylieh numbers and types of nanofluids, the flow is symmetrical about the vertical line through

the center of the circular cylinder. For Ra=104, the flow circulation is weak. The maximum stream

function value is small. The flow patterns appear as a curved kidney-shaped contain two kernel

eddies. The densely packed of the flow for Cu nanofluid type is strong and the flow region area is

large that lead to a decrease in the stagnant area. Two kernel eddies appear for each side for Cu

nanofluids. The aspects of the flow for Al2O3 and TiO3 nanofluid types are similar to Cu nanofluid

type, but the densely package becomes less and the flow region area decreases. The flow patterns for

those types are nearly similar. As Raylieh number increases to Ra=105 for Cu nanofluid type, the

flow moves upward, the flow region area becomes more and the densely package of the flow

decreases as compared with Ra=104. A single large eddy appears rather than two eddies for those of

Ra=104. The densely package of the flow for Al2O3 and TiO3 nanofluid types increase, the flow

region area become more and the size of the eddy decreases. The aspects of the flow patterns for

Ra=106 with types Al2O3 and TiO3 nanofluid are nearly similar. The flow circulation becomes

stronger and stronger and the stream function values increase. The flow move upward and the

streamlines at the upper of the enclosure appear as horizontal lines and the eddies extended in the

longitudinal axis. The stagnant area enlarges and the flow region area decreases. The flow patterns of

the various types are different. The single eddy size of the Cu nanofluid is larger than those for other

types and it appear as longitudinal eddy.


(b) Ra = 105, (c) Ra = 10

6

Cu Al2O3

TiO3

(c)

(b)

(a)



229

Cu Al2O3 TiO3

Figure (7): Effect of volume fraction of nanofluids on isotherms for ϕ=0.1 and (a) Ra = 104,

(b) Ra = 105, (c) Ra = 10

6


5, 10




6 and Cu nanofluid type. For all

Raylieh numbers and types of nanofluids, the flow is symmetrical about the vertical line through the

center of the circular cylinder. For Ra=104, the flow circulation is weak. The maximum stream

function value is small. The flow patterns show that the densely package of the flow is strong. The

streamlines appear as a curved kidney-shaped contain two kernel eddies. The flow region area is

large that lead to a decrease in the stagnant area. The kernel eddies appear for each side for Cu

nanofluids. The aspects of the flow for Al2O3 nanofluid types are similar to Cu nanofluid type, but

the densely package becomes less and the flow region area decreases. The streamlines of the TiO3

becomes less densely package but the flow region area unchanged. Single longitudinal eddy appears

rather than two eddies for Al2O3 and Cu nanofluid types. At Ra=105 for Cu nanofluid type, the flow

circulation becomes more. The flow moves upward, the flow region area becomes less and the

densely package of the flow decreases as compared with Ra=104. A single large eddy appears rather

than two eddies for those of Ra=104. The densely package of the flow for Al2O3 and TiO3 nanofluid

types decrease, the flow region area become less and the size of the eddy decreases. As Raylieh

number increases to Ra=106, the flow circulation becomes stronger and stronger and the stream

function values increase. For Cu nanofluid type, the flow move upward and the streamlines at the

upper of the enclosure appear as nearly horizontal and flat lines near the upper surface of the

enclosure and the eddies extended in the longitudinal axis. Two tiny eddies appear near the vertical

center line at bottom of the enclosure. The flow patterns of the various types are different. For Al2O3

and TiO3 nanofluid types, the flatness becomes more and the densely package of the flow becomes

less. The two tiny eddies disappear. The single eddy size of the Cu nanofluid is larger than those for

other types and it appear as longitudinal eddy. The width of the eddies for Al2O3 nanofluid increase

and the length of those decrease.

(c)

(b)

(a)



230

Cu Al2O3 TiO3

Figure (8): Effect of volume fraction of nanofluids on streamlines for ϕ=0.15 and

(a) Ra = 104, (b) Ra = 10

5, (c) Ra = 10

6

Cu Al2O3 TiO3

Figure (9): Effect of volume fraction of nanofluids on isotherms for ϕ=0.15 and (a) Ra = 104,

(b) Ra = 105, (c) Ra = 10

6.

(c)

(b)

(a)

(c)

(b)

(a)



231


5, and 10

6, and nanofluid types

Cu, Al2O3 and TiO3 are presented in figure (10). The maximum stream function value varies between

Ψmax=1.39 at Ra=104 and Al2O3 nanofluid type to Ψmax=47 at Ra=10

6 and Cu nanofluid type. For all

Raylieh numbers and types of nanofluids, the flow is symmetrical about the vertical line through the

center of the circular cylinder. As volume fraction of the nanofluids increases, the strength of the

flow becomes more and the stream function increases. For Ra=104, the flow circulation is weak, the

maximum stream function values are small. The flow patterns appear as a curved kidney-shaped

contain single big kernel eddy. The densely packed of the flow for Cu nanofluid is very weak and the

flow region area is small that means the stagnant area enlarged. The aspects of the flow for Al2O3

and TiO3 nanofluid types are similar to Cu nanofluid type, but the densely package becomes more

and the flow region area increases. Two different sized eddies appear for Al2O3 nanofluid type rather

than single eddy fo Cu nanofluids type. The size of the eddies become same for TiO3 nanofluid type.

At Ra=105 for Cu nanofluid type, the flow region area becomes more and the densely package of the

flow becomes more as compared with Ra=104. A single tiny eddy appears rather than large eddy for

those of Ra=104. The densely package of the flow for Al2O3 and TiO3 nanofluid types decrease, the

flow region area become less and the size of the eddy increases. As Raylieh number increases to

Ra=106, the flow moves upward, the densely package of the flow increases. The flow region area

increase near the sides and upper walls of the enclosure, while; it decreases near the upper of the

enclosure surface. The aspects of the flow patterns for Ra=106 with types Al2O3 and TiO3 nanofluid

are nearly similar. The flow circulation becomes stronger and stronger and the stream function

values increase. The flow move upward and the streamlines at the upper of the enclosure appear as

horizontal and flat and the eddies appear as single triangular-shaped eddy. The densely package of

the flow becomes less.

Cu Al2O3 TiO3


(b) Ra = 105, (c) Ra = 10

6

(c)

(b)

(a)



232

Cu Al2O3 TiO3

Figure (11): Effect of volume fraction of nanofluids on isotherms for ϕ=0.2 and (a) Ra = 104, (b) Ra

= 105, (c) Ra = 10

6

Figures (4, 6, 8, and 10) show that the strongest flow occurs for Cu nanofluid type and the

weakest flow occurs for Al2O3 nanofluid type for all volume fractions of the nanofluids. The flow

strength becomes more as volume fractions of the nanofluid increases for all types of the nanofluids.

These figures display that the viscosity and thermal conductivity of the nanofluids have an effect on

the behavior of the thermal and flow characterstics.

Overall heat transfer Coefficient and correlations

The average Nusselt number is chosen as the measure to investigate the heat transfer from the

circular cylinder. Figures (12,13), display the effect of using different nanofluid types (Cu, Al2O3 and

TiO3 nanofluid types) on the Nusselt number, and 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 ratio

W/D=2.5, with different types of the nanofluids such as Cu, Al2O3 and TiO3 nanofluid types. 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 volume fraction of the nanofluids ϕ, for all Raylieh numbers

and nanofluid types. The effect of volume fraction of the nanofluids on the Nusselt number for

W/D=2.5 and different types of the nanofluids are shown in figure (12). The enhancement of the

Nusselt number with changing the nanofluids volume fractions increases with increasing Raylieh

number. 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 43% with type Cu

nanofluid, the maximum enhancement is around 48% for Ra= 105 with type Cu nanofluid, whereas

the maximum enhancement is around 46% for Ra= 106 with type Cu nanofluid. Figure (13) shows

the variation of the Nusselt number with volume fraction of the nanofluids using different nanofluid

types. The enhancement in the Nusselt number when the volume fraction of nanoparticles is

(c)

(b)

(a)



233

increased from 0 to 0.2 with Ra=104, is approximately 43% for Cu nanofluid type, 41% for Al2O3

nanofluid type, 37% for TiO3 nanofluid type. The maximum increase is around 48% for Ra=105 with

Cu nanofluid type, 37.5% for Al2O3 nanofluid type, 35% for TiO3 nanofluid type. The maximum

enhancement is around 46% with Ra=106 for Cu nanofluid type, 38% for Al2O3 nanofluid type, 35%

for TiO3 nanofluid type. This tells that the enhancement in heat transfer for Cu nanofluid type. The

heat transfer increase for Al2O3 is less than those for Cu nanofluid type, but, it is more those for TiO3

nanofluid type. This phenomena occurs because the thermal conductivity of the Cu nanofluid type

play a role in this behavior.

(a)

(b)

(c)

Figure (12): Effect of volume fraction of Nanofluids on the Nusselt number for each

Raylieh number, (a) Cu, (b) Al2O3, (c) TiO3



234

(a)

(b)

(c)

Figure (13): Effect of volume fraction of Nanofluids on the Nusselt number for each nanofluid type,

(a) Ra=104, (b) Ra=105, (c) Ra=106.

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 nanofluid types. The main conclusions of the present work can be

summarized as follows:



235

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, nanofluid type, 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 nanofluid type.

5. The streamlines are asymmetrical when the volume fraction of nanoparticles is increased

from 0 to 0.2 for each Raylieh number and nanofluid type.

6. The results display that the strongest flow occurs for Cu nanofluid type and the weakest flow

occurs for Al2O3 nanofluid type for all volume fractions of the nanofluids.

7. The flow strength becomes more as volume fractions of the nanofluid increases for all types

of the nanofluids

8. The average Nusselt number enhances gradually when the volume fraction of nanoparticles is

increased from 0 to 0.2 for each Raylieh number and nanofluid type.

9. The enhancement of the heat transfer for Cu nanofluid type is more than other types, the heat

transfer enhancement for Al2O3 nanofluid type is the weakest.

REFERENCES

[1] Ali O. M. (2008), “Experimental and Numerical Investigation of Natural Convection Heat

Transfer From Cylinders of Different Cross Section Cylinder In a Vented Enclosure,” Ph. D.,

Thesis, College of Engineering, University of Mosul.

[2] S.U.S. Choi, (1995), "Enhancing thermal conductivity of fluids with nanoparticles, in: D.A.

Siginer, H.P. Wang (Eds.)," Developments and Applications of Non-Newtonian Flows,

ASME, New York, 231(66), 99–105.

[3] Zi-Tao Yu, Xu Xu, Ya-Cai Hu, Li-Wu Fan, Ke-Fa Cen, (2011), " Numerical study of

transient buoyancy-driven convective heat transfer of water-based nanofluids in a bottom-

heated isosceles triangular enclosure", International Journal of Heat and Mass Transfer 54,

526–532.

[4] Tinker, P., Rudolf, R., (2012), "Heat transfer enhancement for natural convection flow of

water-based nanofluids in a square enclosure," International journal of simul mdel 11, 29-39.

[5] E. Abu-Nada, (2009), "Effects of variable viscosity and thermal conductivity of Al2O3–water

nanofluid on heat transfer enhancement in natural convection," Int. J.Heat Fluid Flow 30,

679–690.

[6] Hakan F. Oztop, Eiyad Abu-Nada, (2008), "Numerical study of natural convection in

partially heated rectangular enclosures filled with nanofluids," International Journal of Heat

and Fluid Flow 29, 1326–1336.

[7] John D. Anderson Jr., (1995), "Computational Fluid Dynamics, the Basics with

Applications," McGraw–Hill Book Company.

[8] Petrović Z., and Stupar S., (1996), "Computational Fluid Dynamics, One," University of

Belgrade.

[9] Brinkman, H.C., (1952), "The viscosity of concentrated suspensions and solutions," J. Chem.

Phys. 20, 571–581.

[10] Thomas P. D., and Middlecoff J. F., (1980), "Direct Control of the Grid Point Distribution in

Meshes Generated by Elliptic Equations," AIAA Journal, 18, 652-656.

[11] Hoffmann K. A., (1989), "Computational Fluid Dynamics For Engineers," Engineering

Education System, USA.



236

[12] Thompson J. F., Warsi Z. U. A. and Mastin C. W., (1985), "Numerical Grid Generation:

Foundations and Applications," Mississippi State, Mississippi.

[13] Ferziger J. H. and Peric M., (2002), "Computational Methods for Fluid Dynamics," Springer,

New York.

[14] Roache, P., J., (1982), "Computational Fluid Dynamics," Hermosa publishers.

[15] Fletcher C.,A.,J., (1988), "Computational Techniques for Fluid Dynamics 2," Springer,

Verlag.

[16] Moukalled F., Acharya S., (1996), "Natural convection in the annulus between concentric

horizontal circular and square cylinders," Journal of Thermophysics and Heat Transfer, 10(3),

524 –531.

[17] C. Shu; and Y. D. Zhu, (2002), "Efficient computation of natural convection in a concentric

annulus between an outer square cylinder and an inner circular cylinder," International

Journal for Numerical Methods in Fluids, 38, 429-445.

[18] S. Bhanuteja and D.Azad, “Thermal Performance and Flow Analysis of Nanofluids in a Shell

and Tube Heat Exchanger” International Journal of Mechanical Engineering & Technology

(IJMET), Volume 4, Issue 5, 2013, pp. 164 - 172, ISSN Print: 0976 – 6340, ISSN Online:

0976 – 6359.

[19] Dr.N.G.Narve and Dr.N.K.Sane, “Experimental Investigation of Laminar Mixed Convection

Heat Transfer in the Entrance Region of Rectangular Duct”, International Journal of

Mechanical Engineering & Technology (IJMET), Volume 4, Issue 1, 2013, pp. 127 - 133,

ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359

[20] S.K. Dhakad, Pankaj Sonkusare, Pravin Kumar Singh and Dr. Lokesh Bajpai, “Prediction of

Friction Factor and Non Dimensions Numbers in Force Convection Heat Transfer Analysis of

Insulated Cylindrical Pipe”, International Journal of Mechanical Engineering & Technology

(IJMET), Volume 4, Issue 4, 2013, pp. 259 - 265, ISSN Print: 0976 – 6340, ISSN Online:

0976 – 6359.