the effect of various open cell aluminium foam geometrical ...€¦ · volume heat capacity ratio...

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 3, Issue 9, September 2013)

615

The Effect of Various Open Cell Aluminium Foam Geometrical

Shapes on Combined Convection Heat Transfer with Nanofluid Raed Abed Mahdi

1, H. A. Mohammed

2, K. M. Munisamy

3

1Department of Mechanical Maintenance, Doura Power Station, Ministry of Electricity, 10022 Al-doura quarter,

Almahdia place, Baghdad, Iraq 2Department of Thermo fluids, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia, 81310 UTM Skudai,

Johor Bahru, Malaysia 1,3

Department of Mechanical Engineering, College of Engineering, Universiti Tenaga Nasional, Jalan IKRAM-UNITEN,

43000 Kajang, Selangor, Malaysia

Abstract— Mixed convection heat transfer and fluid flow

through various an open cell aluminium foam around circular

heat source shapes with constant temperature inside

rectangular horizontal channel, filled with nanofluid is

numerically investigated. An open cell aluminium foam is

made of 6101-T6 alloy with pore densities (5, 10, 20, 40) PPI.

The nanoparticles aluminium oxide (Al2O3) with volume

fraction of (1-4) % and nanoparticle diameter of (15nm)

dispersed in water are used. Six models of open cell

aluminium foam shapes are employed around circular

cylinder surface as test sections: (model 1) aluminium foam

with angle (γ =90ο), (model 2) the aluminium foam with angle

(γ =85.71ο), (model 3) aluminium foam with angle (γ =81.47ο),

(model 4) the aluminium foam with angle (γ =77.32ο), (model

5) the aluminium foam with angle (γ =73.3ο) and (model 6) the

aluminium foam with angle (γ =69.44ο). In all models, the

aluminium foam length of (4cm) is used with Richardson

number range of (0.1-10). The governing equations continuity,

momentum and energy are solved by using the Finite-volume

method (FVM). The effects of aluminium foam angle,

nanofluid properties and Richardson number on the mixed

convection were investigated. The results have shown that

higher average Nusselt number is obtained with the use of

nanofluid and 40PPI aluminium foam pore density with

model (1). Average Nusselt number decreases with aluminium

foam angle decreases with increased aluminium pore density.

Average Nusselt number increased with nanoparticle volume

fraction and mixed convection parameter increased. Higher

mixed convection is obtained with the use of aluminium foam

angle γ =73.3ο.

Keywords— Mixed convection, open cell aluminium foam,

nanofluid, flow around cylinder

I. INTRODUCTION

Any material that consists of a solid matrix with an-inter

connected void is called porous media such as rocks and

open-cell aluminum foams [1]. There are two advantages of

porous media. First its dissipation area is greater than the

conventional fins that enhance heat convection.

Second the irregular motion of the fluid flow around the

individual beads mixes the fluid more effectively. A wide

range of porous media applications are found in many

practical situations, such as aluminum foams applications

in thermal management, including air-cooled condenser

towers. [1-7].

One of the ways to enhance the heat transfer is to

employ nanofluids. Nanofluids are fluids that contain

suspended nanoparticles such as metals and dioxides in the

base fluid. Thus, it does not cause an increase in pressure

drop in the flow field. Past studies showed that nanofluids

exhibit enhanced thermal properties, such as higher thermal

conductivity and convective heat transfer coefficients

compared to the base fluid [8-11].

Many studies have been done on heat transfer in porous

media with nanofluid. Sun and Pop [9] studied numerically

steady-state free convection heat transfer behaviour of

nanofluid inside a right-angle triangular enclosure filled

with a porous medium. The results have revealed that the

maximum value of average Nusselt number was obtained

by decreasing the enclosure aspect ratio and lowering the

heater position with the highest value of Rayleigh number

and the largest size of heater. Chamkha et al. [10] studied

numerically non-similar solution for natural convective

boundary layer flow over isothermal sphere embedded in

porous medium saturated with a nanofluid. The results

indicated that as buoyancy ratio and thermophoresis

parameter increased, the friction factor increased, whereas

the heat transfer rate and mass transfer rate decreased.

Bhadauria et al. [11] studied numerically the linear and

nonlinear thermal instability in a horizontal porous medium

saturated by a nanofluid heated from below and cooled

from above. The results revealed that when horizontal wave

number was small, onset of convection was obtained

through oscillatory mode. On increasing horizontal wave

number, the mode of convection for onset of thermal

instability became stationary.



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Also that nonlinear finite amplitude was the preferred

mode of convection earlier to oscillatory mode. Hady et al.

[12] studied numerically the influence of yield stress on

free convective boundary-layer flow of a non-Newtonian

nanofluid past a vertical plate in a porous medium. The

results indicated that the reduced Nusselt and Sherwood

numbers were decreasing functions of the higher yield

stress parameter for each dimensionless numbers, power

index of non-Newtonian fluid and Lewis number, except

the reduced Sherwood number was an increasing function

of higher Brownian parameter for different values of yield

stress parameter.

Rashad et al. [13] studied numerically the effect of

uniform transpiration velocity on natural convection

boundary layer flow of a non-Newtonian fluid over a

permeable vertical cone embedded in a porous medium,

saturated with nanofluid. The results have shown that the

values of viscosity index increase led to an increase in both

the local Nusselt and Sherwood numbers. On the other

hand as the buoyancy ratio increased, both the local Nusselt

and Sherwood numbers decreased. Cheng [14] studied

numerically the natural convection boundary layer flow

over a truncated cone embedded in a porous medium

saturated by a nanofluid with constant wall temperature and

constant wall nanoparticle volume fraction. The results

showed that an increase in the thermophoresis parameter or

the brownian parameter tended to decrease the local

Nusselt number. The local Nusselt number increased as the

buoyancy ratio or the Lewis number is decreased. Hady et

al. [15] studied numerically the effect of heat generation

absorption on natural convective boundary layer flow from

a vertical cone embedded in a porous medium filled with a

non-Newtonian nanofluid. The results showed that the local

Nusselt number decreased as the heat generation absorption

parameter increased.

The local Nusselt number was predicted to decrease as a

result of increasing the values of the nanoparticles volume

fraction. Mahdy and Ahmed [16] studied numerically two-

dimensional steady laminar free convection over a vertical

wavy surface embedded in a porous medium saturated with

a nanofluid. The results showed that as the amplitude

wave-length ratio increased, the amplitude of local Nusselt

number and local Sherwood number increased. The heat

and mass transfer rates were decreased by increasing either

buoyancy ratio number or thermophoresis parameter.

Ghazvini and Shokouhmand [17] studied analytically

and numerically forced convection flow of (CuO-water)

nanofluid with (0-4) % volume fraction of nanoparticles,

having diameters of about (10nm) as a coolant through a

micro-channel heat sink with constant heat flux boundary

conditions. Two common analytical approaches were used:

the fin model and the porous media approach. The results

indicated that fin approach exhibited a higher value for

both dimensionless temperature for nanofluid and

dimensionless temperature for solid than porous media

approach. For both, fin and porous media approaches, an

increase in bulk temperature, channel aspect ratio led to

particle speed and Brownian motion increased and due to

that, a better heat transport would be possible. Chen and

Ding [18] studied numerically forced convection heat

transfer characteristics and cooling performance of a

microchannels heat sink with (water-γAl2O3) nanofluids

having different nanoparticle volume fraction. The results

showed that the temperature distribution of the channel

wall was practically not sensitive to the inertial effect,

while the fluid temperature distribution and the total

thermal resistance changed significantly due to the inertial

force effect. The effect of fluid inertia was to reduce the

total thermal resistance and the temperature difference

between the channel wall and the fluid phase. Nazar et al.

[19] studied numerically steady laminar mixed convection

boundary layer flow from an isothermal horizontal circular

cylinder embedded in a porous medium filled with a

nanofluid has been studied for both cases of a heated and

cooled cylinder.

Three different types of nanoparticles were considered,

Cu, Al2O3 and TiO2 with water as their base fluid. The

results revealed that an increase in the value of the

nanoparticle volume fraction led to decrease in the

magnitude of the skin friction coefficient, and an increase

in the value of mixed convection parameter. It was also

found that for any fixed values of volume fraction and

mixed convection parameter, the nanoparticle Cu gave the

largest values of the skin friction coefficient and heat

transfer enhancement followed by TiO2 and Al2O3.

Cimpean and Pop [20] studied numerically steady fully

developed mixed convection flow of three types of

nanofluids Cu, Al2O3 and TiO2 with water as their base

fluid in an inclined channel filled with a porous medium.



617

The walls of the channel were heated by a uniform heat

flux and a constant flow rate was considered through the

channel. The results showed that the nanofluid increased

the heat transfer, even for small additions of nanoparticles

in the base water fluid. Gorla et al. [21] studied numerically

two-dimensional mixed convective boundary layer flow

over a vertical wedge embedded in a porous medium

saturated with a nanofluid. The results indicated that as the

buoyancy ratio parameter and thermophoresis parameter

increased, the friction factor increased, whereas the heat

transfer rate and mass transfer rate decreased. As the wedge

angle increased, the heat and mass transfer rates increased

too. As Brownian motion parameter increased, the friction

factor and surface mass transfer rates increased, whereas

the surface heat transfer rate decreased. As Lewis number

increased, the heat transfer rate decreased, whereas the

mass transfer rate increased. Juncu [22] studied

numerically the forced convection heat transfer and steady

laminar flow around two isothermal circular cylinders in

tandem arrangement. The temperature inside the cylinders

was considered special uniform but not constant in time.

The Reynolds number was varied from (1-30) and fluid

phase Prandtl number was set at (0.1, 1, 10 and 100). The

results showed that the heat transfer from tandem cylinders

with uniform temperature had its own specific rules. The

average Nusselt numbers did not reach a frozen asymptotic

value. High heat transfer rates were obtained when the

interaction began and developed at high values of the

cylinders dimensionless temperature. High convection rate,

small gaps between cylinders and high values of the

volume heat capacity ratio led to high heat transfer rates.

Wu and Wang [23] studied numerically unsteady flow and

convection heat transfer for a heated square porous cylinder

in a channel. The results indicated that the average local

Nusselt number was augmented as the Darcy number

increased. The average local Nu number increased as

Reynolds number increased. Manay et al. [24] studied the

effect of the spacing between equilateral dual triangular

bodies symmetrically placed into the channel axis under

steady state conditions on heat transfer and fluid

characteristics by using artificial neural networks (ANN).

The results indicated that the local Nusselt number and skin

friction coefficient took a local maximum at the placed

position of the upstream body, and at the placed position of

the downstream body. The heat transfer enhances

especially for upstream flow region concerning with the

generation of vortices. Dhiman and Shyam [25] studied

numerically unsteady heat transfer from an equilateral

triangular cylinder in the unconfined flow regime.

The results showed that the time-averaged Nu number

increased monotonically with the increasing value of the

Reynolds number for the Prandtl number of 0.71.

It is obvious from the above literature review that there

are very limited data on mixed convection heat transfer

using aluminium foam with nanofluids. Thus, this paper

presents numerical simulations of mixed convection heat

transfer and fluid flow through an open cell aluminium

foam around circular heat source shape, with constant

temperature inside rectangular horizontal channel filled

with nanofluid. The purpose of the present study is to

clarify the effect of aluminium foam and nanofluid

properties on the mixed convection with six models in a

rectangular horizontal channel.

Nomenclature

Cf Forchheimer coefficient

C inertial resistance

Cp specific heat at constant pressure, (J/kg. K)

D viscous resistance

Dh hydraulic diameter, (m)

dp mean particle diameter, (m)

df equivalent diameter of a base fluid molecule,(m)

gravitational body force

Gr Grashof number, (Gr =( ρgβ∆TDh3)/µ

2 )

h heat transfer coefficient (W/m. K)

K permeability, (m2)

k thermal conductivity(W/m. K)

M molecular weight of the base fluid

N Avogadro number

Nu Nusselt number,(Nu =hDh/k)

P pressure

Pr Prandtl number, (Pr =μ CP/k)

Re Reynolds number, (Re =ρ u Dh/μ)

Ri Richardson number,(Ri=Gr/Re2 )

T temperature, (K)

u Darcy velocity, (m/s)

v physical velocity, (m/s)

1- 6 models numbers



618

Greek symbol

∆p pressure drop,(Pa)

ε porosity

τ shear stress, (Pa)

µ viscosity,(mPa s)

ρ density ,(kg/m3)

ϰ Boltzmann constant, (J/K)

γ foam angle (degree)

α thermal diffusivity

φ volume fraction, (%)

friction factor

Subscripts

bf base fluid

e effective

h hydraulic

nf nanofluid

p particle

s surface

ο reference, degree

∞ inlet

II. PHYSICAL DESCRIPTION OF THE PROBLEM AND

ASSUMPTIONS

A two dimensional problem of an open cell aluminum

foam inserted inside a rectangular horizontal channel and

surrounded by horizontal circular cylinder heat source

shape is used. The fluid enters the rectangular horizontal

channel with fully developed flow velocity (u∞) and

temperature (T∞ =300K) and all the channel walls are

considered thermally insulated.

The mixed convective heat transfer and fluid flow

through an open cell aluminum foam filled with nanofluids

depend on several parameters such as, buoyancy force,

aluminum foam pore density (PPI), aluminum foam angle,

nanoparticles volume fraction, Richardson number are

investigated in this paper with:

1. Six test section models 1, 2, 3, 4, 5 and 6 are used

having variable aluminium foam shape, inside

rectangular horizontal channel as shown in Fig.1.

2. An open cell aluminium foam 6101-T6 alloy pore

densities (5, 10, 20, 40 ) PPI (pores per linear inch)

3. Nanofluid type (water+Al2O3) with volume fraction

(1-4) %.

4. Richardson number values 0.1, 5 and 10.

5. Reynolds number 400,600

The following assumptions are considered in the

numerical study to simplify the problem:

1. The flow is a steady state, two dimensional and

incompressible with fully developing laminar regime

as shown in Fig.2.

2. No chemical reactions and internal heat generation

occurred, and viscous dissipation is neglected.

3. The thermophysical properties of the fluid are

changed polynomial with temperature.

4. The aluminum foam is isotropic, homogeneous and

saturated with a single-phase fluid in local

equilibrium with the solid matrix.

5. The aluminium foam permeability and Forchheimer

coefficient values are considered constant for all

fluids used at each aluminium foam pore density

(PPI).

III. GOVERNING EQUATIONS

The flow was modeled by using Darcy-Forchheimer’s

model [1] to combine the inertia effect in the aluminum

foam region where:

(1)



619

Fig. 1 Test section models a) model (1) b) model (2) c) model (3)

d) model (4) e) model (5) f) model (6)

Fig. 2 fully developing profile at inlet channel for Re =400

Fluid domain was modeled by continuity, Navier–

Stokes, and energy equations such that

Continuity:

(2)

Momentum equation based on the Darcy velocity

formulation where:

X-Momentum:

(3) Y-Momentum:

(4) Energy:

(5)

Effective thermal diffusivity in energy equation depends

on porous media (aluminium foam) and nanofluid

properties. Kuznetsov and Nield [26] and commotional

software program FLUENT [27] presented the effective

thermal diffusivity in energy equation as shown below:

(6)

Momentum conservation equations to be solved by

FLUENT are [27]:



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(7)

where:

: external body forces and the gravitational body force

Within the porous medium, included viscous and

inertial fluid losses, and defined by FLUENT as:

(8)

where:

Fi: external body force term for the ith momentum

equation

D: viscous resistance

C: inertial resistance

The axial pressure drop in the homogeneous matrix with

only a steady flow and no internal axial body forces was

defined by FLUENT as:

(9) A comparison among equations Eq. 1 to Eq. 9, shows

that the viscous resistance term in Eq. 8 is related to the

permeability by:

(10)

Inertial resistance is related to the Forchheimer

coefficient and permeability by:

(11)

The best values for permeability and Forchheimer

coefficient were obtained from the experimental work of

Phanikumar and Mahajan [28] as shown in Table 1.

Table 1

Permeability and Forchheimer coefficient for aluminium foam

samples [28]

Sample PPI Porosity Permeability

[m2] ×10

-7

Inertial

coefficient

1 5 0.899 1.989 8.753e-02

2 10 0.9085 1.075 6.872e-02

3 20 0.92 1.063 1.023e-01

4 40 0.9091 0.5066 8.254e-02

IV. EFFECTIVE THERMOPHYSICAL PROPERTIES FOR

NANOFLUIDS

Base nanofluid properties have been published over the

past few years in literature. However, some data on

temperature-dependent properties have been provided, even

though they are only for nanofluid effective thermal

conductivity and effective absolute viscosity.

Density [29, 30]

(12)

Specific heat [31, 32]

(13)

Viscosity [33, 34]

(14)

where:

: Equivalent diameter of a base fluid molecule

(15)

M: Molecular weight of the base fluid

N: Avogadro number = 6.022×1023

1/mol

: Mass density of the base fluid calculated at

temperature T=293 K.

The effective thermal conductivity of a nanofluid is

given by Vajjha et al. [35]

(16)



621

The static effective thermal conductivity of a nanofluid

is given by [36- 39]

(17)

The Brownian thermal conductivity of a nanofluid is

given by Koo and Kleinstreuer [40]

(18)

: Modelling function (Fraction of the liquid volume

which travels with a particle)

] (19)

(20)

V. EFFECTIVE THERMOPHYSICAL PROPERTIES FOR

ALUMINIUM FOAM 6101-T6

Methods for defining foam The Thermophysical

properties for aluminium foam alloy 6101-T6 are discussed

by Gibson and Ashby [41] and ERG [42] where:

(21)

(22)

[43] (23)

0.33 = coefficient representing the foam structure

geometric or "tortuosity" factor.

Thermophysical properties for nanoparticles material are

summarized in Table 2 and the thermophysical properties

for aluminium metal alloy 6101-T6 are summarized in

Table 3.

Table 2.

Thermophysical properties for nanoparticles material

Nanoparticles

material

Density

[kg/m3]

Specific

heat

[J/kg.k]

Thermal

conductivity

[W/m.k]

Al2O3 [44] 3970 765 40

Table 3

Thermophysical properties for aluminium alloy 6101-T6

Metal Density

[kg/m3]

Specific heat

[J/kg.k]

Thermal

conductivity

[W/m.k]

Aluminium

alloy 6101-

T6

2700

[42]

895

[40]

218

[24]

VI. GOVERNING PARAMETERS

The local Nusselt number at each angle is evaluated as

follows [45]:

(24)

is the Local heat transfer coefficient at each angle

on circular cylinder surface

(25)

The average Nusselt number at each circular cylinder

surface is calculated as follows [45]:

(26)

VII. BOUNDARY CONDITIONS

The following boundary conditions are used in this

study:

The flow is in steady state and incompressible with

fully developing laminar regimes imposed at the inlet.

Outflow condition is specified at the outlet.

External walls are thermally insulated.

Constant temperature is applied at the cylinder

surface.

VIII. COMPUTATIONAL DETAILS

a. Mesh Generation

The axisymmetric model shown in Fig.3 is created by

using a triangle-pave mesh for the fluid and aluminum

foam zones. A grid independence test is performed to

assess the effect of the meshes used in the study on the

results. Grid independent test results for the average

Nusselt number values for three sets of meshes are

generated, as shown in Table 4.



622

It is found that the average Nusselt number with

geometry 300 nods on cylinder surface confirms the grid

independence test with an error less than 2%. where the

mesh density around the variable cylindrical surface is

higher than those far away areas.

b. Modelling and numerical settings

The momentum and energy equations presented for the

above boundary conditions are solved by using FLUENT, a

commercial computational fluid dynamics package.

FLUENT handles steady governing equations efficiently by

using FVM in 2D and 3D geometries [27].

Fig. 3 Computational model with mesh generation

Table 4

Grid independent test convergence using average Nusselt number at

Re=600, Ts =303.6282, with 40PPI aluminum foam effect

Number of nodes around cylinder

surface

Average Nusselt

number

275 110.994

300 110.859

325 110.603

Numerical analysis by using FLUENT is performed to

understand the flow characteristics in all models. The

governing equations are a set of convection equations with

velocity and pressure coupling [46]. The COUPLED

algorithm is used to solve the problem of velocity and

pressure coupling. The pressure staggering option

(PRESTO) scheme is used to solve pressure equations. The

QUICK scheme is used to solve momentum and energy

equations. When the normalized residual values reach 10-6

,

the solutions are considered to be converged for all

variables.

IX. NANOFLUID THERMOPHYSICAL PROPERTIES

The thermophysical properties of nanofluid vary with

the temperature, for that piecewise-polynomial functional

relations are used to account for the temperature

dependence.

(27)

(28)

: Any property.

Second degree polynomial functional calculated by [47]

with residual sum of squares (rss), where: rss = (1×10-3

-

1×10-12

) are used Piecewise-polynomial functions constants

of nanofluid (water+Al2O3) with fraction factor (1-4) %

and practical diameter (15nm) for all properties and

temperatures ranges.

X. CODE VALIDATION

There are very limited studies close to the present work;

therefore the results have been compared with empirical

relations for fluid flow around circular cylinder surface,

and with flow through porous media in a horizontal

channel.

A. Validation with empirical correlations data

From the heat transfer point of view, the average Nusselt

number of a single isothermal circular cylinder surface in

cross flow of water is computed by using Computational

Fluid Dynamics package FLUENT and compared with the

average Nusselt number values obtained from the

correlations available in the literature:

Churchill and Bernstein [45]

(29)

Hilpert and Forsch [45]

(30)

The average Nusselt number computed from above-

mentioned correlations are comparatively summarized in

Table 5 for Re = 500, 1000 and 2000.



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The average Nu number for Re =500 is well approved

with the forecasts of all two correlations. The average

Nusselt number deviation results are about 8.6% and 17.5%

higher in values for Re =1000 and 2000, respectively.

Table 5

Validation with empirical correlations data for model (A) without

aluminium foam

Empirical

correlations name

Average

Nusselt

number at

Re=500

Average

Nusselt

number

at

Re=1000

Average

Nusselt

number at

Re=2000

Churchill and

Bernstein [43] 15 21.966 30

Hilpert and Forsch

[43] 14 19.28 26.63

Present work 15.161

±0.045

22.423±

0.086

33.7 ±

0.175

B. Validation with previous study data

The present results were compared with the results

obtained by Guerroudj and Kahalerras [48], who studied

mixed convective heat transfer in a parallel plate channel

with various porous block shapes, including rectangular (γ

=90°) and triangular (γ =50.1944°). The blocks were heated

from below and attached to the lower plate while the upper

plate was thermally insulated.

The global Nusselt numbers were compared with

numerical results reported by Guerroudj and Kahalerras

[48] at Darcy numbers of 10-2

and 10-3

. The comparison

shows that the present results have a 6% deviation, as

shown in Fig.4.

50 55 60 65 70 75 80 85 90

Porous media angle

4.25

4.75

5.25

5.75

6.25

4.00

4.50

5.00

5.50

6.00

6.50

Glo

ba

l N

uss

elt

nu

mb

er

Guerroudj at Da=1e-03

Guerroudj at Da=1e-02present study at Da=1e-03

present study at Da=1e-02

Fig. 4 Evolution of the average Nusselt number at each porous media

angle for various Darcy numbers with Re = 100 and thermal

conductivity ratio = 1.

XI. RESULTS AND DISCUSSION

The effects of aluminium foam and nanofluid properties

on mixed convection heat transfer, with six models in a

rectangular horizontal channel are presented in this section.

The nanoparticles Al2O3 with water as a base fluid are used

to perform the numerical simulations. Results are presented

for four different aluminium foam pore densities (5, 10, 20,

40) PPI and three different Richardson numbers in the

range of (0.1-10) with constant temperature in each model.

For evaluation of performance of these models, it is

necessary to understand the flow and heat transfer physics

over circular cylindrical surface and by analysing the

streamlines and heat transfer behaviour.



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Fig. 5 shows the cross water flow around circular

cylinder surface and behaviour of stream line contours in

the wake region at Re =400 at Ri = 0.0 (pure forced

convection), without an open cell aluminium foam. The

flow separates from the rear surface of the cylinder forming

a recirculation zone which has two symmetrical eddies and

the upstream and the downstream lines are symmetrical

because there is no buoyancy force effect.

In this paper, mixed convective heat transfer and fluid

flow through an open cell aluminium foam filled with

nanofluids depend on several parameters, such as buoyancy

force, aluminium foam pore density (PPI), nanoparticles

volume fraction and Richardson number. These effects are

studied separately and discussed in the following sections.

Fig. 5 Flow patron around circular cylindrical surface without

aluminium foam at Re =400, Ri= 0.0.

A. Effect of aluminium foam on buoyancy force

At Re =400 and Ri =10, the effect of the buoyancy force

on the stream lines contours around circular cylinder

without an open cell aluminium foam with water is shown

in Fig.6. The upstream and the downstream line are non-

symmetric due to buoyancy force effect and it affects the

upstream and downstream stagnation points.

Fig. 6 Effect of buoyancy force on fluid flow behaviour with water Re

=400, Ri =10

The streamlines for all models 1, 2, 3, 4, 5 and 6 with

40PPI an open cell aluminium foam pore density for Re

=400 and Ri =10 with nanofluid are shown in Fig.7. In

rectangular aluminium foam model γ =90ο, the upstream

and the downstream lines are symmetrical because the

irregular structure of the aluminium foam leads to linear

pressure drop around the circular cylinder surface and

inseparability of the boundary layer and wake region in

aluminium foam zone.

When the aluminum foam angle γ decreased, the

maximum values of velocity magnitude are surrounded by

the inclined aluminum foam surface and horizontal channel

wall because of the decrease in the cross sectional area of

the fluid zone as well as the low flow resistance of the

aluminum foam. The minimum velocity magnitude is

observed around the cylinder surface in the aluminum

region because of the high flow resistance of the aluminum

foam, thereby decreasing the cell Reynolds number, which

in turn increases the local mixed convection parameter

(Gr/Re2), for that the buoyancy force effect is greater when

the aluminum foam angle γ decreased, thereby the

upstream and downstream lines are non-symmetric.

Determine the local Nu number around circular heat

sources surface with constant temperature are one of the

main objectives in mixed convection heat transfer

calculations. The effect of the flow structure especially on

mixed convection heat transfer can be better observed by

analysing the local Nu number.

Fig.8a, b shows the upstream and the downstream local

Nu numbers values distributions for forced and mixed

convection cases along the perimeter angular points (Theta

θ ) over the circular cylinder surface, without and with

40PPI aluminium foam at Re =400 and Ri =10 and

nanoparticle volume fraction 3%.

Fig.8a shows the upstream and the downstream local Nu

numbers values without aluminium foam. The maximum

local Nu number occurs at the upstream stagnation point at

theta (θ) =0o while the minimum local Nu number is

spotted between the downstream stagnation point and the

boundary separation points at theta (θ) = ≈127o

and 232o.

The upstream and the downstream local Nu numbers are

symmetrical in forced convection and semi- symmetrical in

mixed convection due to buoyancy force effect which has

been taken into account in the calculation. Fig.8b shows the

upstream and the downstream local Nu numbers values

with rectangular and triangular aluminium foam models.



625

Fig. 7 Effect of aluminium foam on buoyancy force and fluid flow

behaviour with all models for Re =400 and Ri =10 with 40PPI

aluminium foam

The maximum local Nu number values occurs at the

upstream stagnation point at theta (θ) =0o, while the

minimum local Nu number values occurs at the

downstream stagnation point at theta (θ) =180o, because the

pressure gradually decreases along the front and rear half of

the circular cylinder surfaces resulting from the irregular

structure of the aluminium foam, consequently not to be in

the wake region. The upstream and the downstream local

Nu numbers in rectangular model are symmetrical in forced

and mixed convection because in Ri =10 and Re =400, the

low modified Grashof number is low, for that it’s have

negligible buoyancy force effects, due to nanofluid has

low thermal expansion property and high flow resistance

for secondary flow from aluminum foam structure. The

upstream and the downstream local Nu numbers in

triangular model are symmetrical in forced convection and

non-symmetrical in mixed convection because the

buoyancy force effect which seen very clear in this model

because of decreased the velocity magnitude in aluminium

foam zone around circular cylinder surface therefore cell

Reynolds number is decreased and leads to increase local

mixed convection parameter (Gr/Re2) thereby the upstream

and downstream local Nusselt number are non-symmetric.

B. Effect of nanoparticle volume fraction on average

Nusselt number

Fig. 9a shows the average Nu numbers values with

40PPI aluminium foam pore density, for both fluid (water

and nanofluid), at Re =400, and Ri =0.1 with and

nanoparticle volume fraction 3% with all models. The

average Nu numbers increases when nanofluid is used, due

to enhancement of the fluid thermophysical properties, and

with aluminium foam angle increased because of increased

the nanofluid amount which attached the circular surface.



626

Fig. 8 Effect of buoyancy force on local Nusselt disruption at Re =400,

Ri =10 and nanoparticle volume fraction 3% a) Without aluminium

foam b) with 40PPI aluminium foam

Fig.9b shows the effect of all nanoparticles

concentrations on the average Nu number values with

40PPI aluminium foam for rectangular model at Ts =310

for Re =600.

The average Nusselt number increased as the

nanoparticle concentration increased because of the

enhanced thermophysical properties. In same figure, the

average Nusselt number value became more accurate with

used variable nanofluid thermophysical properties with

temperatures in numerical analysis. Table 6 shows the

comparison with and without aluminium foam between

average Nusselt number values with used water + four

nanofluid volume fractions at constant and variable

properties with temperatures.

C. Effect of mixed convection parameter (Gr/Re2) on

average Nusselt number

Fig.10 shows the variation of the average Nusselt

number with the aluminium foam angles γ at Re =400 and

40PPI aluminium foam pore density for different values of

the Richardson number (Gr/Re2). The buoyancy effects are

increased by increasing (Gr/Re2). By increasing (Gr/Re

2)

from (0.1-10), the rate of heat transfer enhancement

depends on the aluminium foam angles. At low Richardson

number (Gr/Re2) = 5, the rate of heat transfer enhancement

is between (2.02-3.3) % depending to the value of the

aluminium foam angle. However, at high Richardson

number (Gr/Re2) =10 the buoyancy effects are larger for

the trapezoidal model (enhancement rate around 5.5% for γ

=73.3ο) than for rectangular shape (enhancement rate

around 1.91% for γ =90ο).

D. Effect of aluminium foam pore density on average

Nusselt number

The pore density has an important influence on the

average Nu number value. The variation of the average Nu

number values for (5, 10, 20 and 40) PPI aluminium foam

pore densities with all aluminium foam angle γ at Re =400

and Ri =10 and nanoparticle volume fraction 3% is shown

in Fig.11.

b)

Theta

Lo

cal

Nu

sselt

nu

mb

er

0 45 90 135 180 225 270 315 3600

20

40

60

80

100

120

140

160

180

200

Forced convection with rectangularMixed convection with rectangularForced convection with triangularMixed convection with triangular

0

90

180

270

r

Frame 001 18 May 2013 Frame 001 18 May 2013

a

)



627

Fig. 9 Effect of nanofluid on average Nusselt number with 40PPI

aluminium foam a) Re= 400 and Ri= 0.1 with all models b) Re =600

and Ts =310K with all nanofluid volume fraction for rectangular

model

Table 6

Comparison with and without aluminium foam between average

Nusselt number values with used water + four nanofluid volume

fractions at constant and variable properties with temperatures

Status

Average Nu

number (Constant

properties)

Average Nu

number

(Variable

properties)

Water only 54.793224 55.986565

Water with aluminum foam 110.29799 111.49789

Nanofluid only

φ=1% 55.751755 56.819183

φ=2% 57.131615 58.194168

φ=3% 58.986954 59.987236

φ=4% 61.58865 62.595921

Nanofluid with

aluminum foam

φ=1% 111.42757 112.66758

φ=2% 113.47285 114.77147

φ=3% 116.57233 117.9501

φ=4% 121.10516 122.5919

At higher pore density, the higher average Nu number

values occur with rectangular model γ =90ο; this

corresponds with the results of Kurtbas and Celik [52]

because increasing the dissipation area would increase the

irregular motion of the fluid flow. With decreased

aluminium foam angle γ below 90ο, the average Nu number

values decrease with increased aluminum foam pore

density because of decreased the flow resistance near

horizontal channel walls and increased flow resistance

around circular cylinder surface.

XII. CONCLUSIONS

The problem of steady fully developed mixed

convection heat transfer and fluid flow through an open cell

aluminium foam around heat source surface subjected to

constant temperature inside rectangular horizontal channel,

filled with nanofluids (water+Al2O3) in six models is

numerically studied. The governing equations continuity,

momentum and energy are solved numerically by Finite

volume method with the aid of a commercial CFD package

FLUENT.



628

Fig. 10 Effect of mixed convection parameter on average Nusselt

number for Re =400 with 40PPI aluminium foam and nanoparticle

volume fraction 3%

Fig. 11 Effect of aluminium foam pore density on average Nusselt

number for Re =400 with Ri =10 and nanoparticle volume fraction

3%

The effects of aluminium foam and nanofluid properties

on mixed convection heat transfer with six models in a

rectangular horizontal channel are reported. The upstream

and the downstream local Nu numbers values are not

symmetrical due to the buoyancy force effect on

downstream stagnation point. The results reveal that the

average Nu numbers increases when nanofluid is used and

became more accurate with used variable nanofluid

thermophysical properties with temperatures in numerical

analysis. At low Richardson number (Gr/Re2) = 5, the rate

of heat transfer enhancement is between (2.02-3.3) %

depending to the value of the aluminium foam angle.

At higher pore density, the higher average Nu number

values occur with rectangular model γ =90ο but with

decreased aluminium foam angle γ below 90ο, the average

Nu number values decrease with increased aluminium foam

pore density. The main conclusion is that, model (1) is the

optimal shape due to its highest Nu at small values of pore

density.

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