Mixed Convection of an Ag/Water Nanofluid in a Ventilated Square

Cavity Containing Cold Blocks of Different Shapes

MERYEM BRAHIMI1, 2, RAZIK BENDERRADJI1, 3, HAMZA GOUIDMI3

1Department of Physics, Faculty of Sciences,

University of M'sila,

ALGERIA

2Laboratory of Materials and Renewable Energy (LMER),

University of M'sila,

ALGERIA

3Laboratory of Renewable Energy and Sustainable Development (LERDD),

University of Constantine 1,

ALGERIA

Abstract: - This research presents the results of a numerical study on mixed convection in a ventilated cavity

with a central cold block of varying shapes. The direction of the forced flow of Ag/water nanofluid is

perpendicular to the transverse axis (y) of the central cold block. Mixed convection is induced by cooling at the

entrance of the ventilated cavity and uniformly heating its bottom wall. The governing equations for the flow of

an incompressible Newtonian nanofluid are assumed to be two-dimensional, steady, and laminar. The finite

volume method is employed for numerical simulations. A series of calculations are conducted to investigate the

effects of key influencing factors: Reynolds number (Re = 100), Richardson number (Ri = 1), and nanoparticle

volume fractions (0 ≤ ∅ ≤ 8%) on the enhancement of heat transfer. The impact of four different geometric

shapes of the cold obstacle (circular, square, triangular, and elliptical) on fluid flow and heat transfer rate is also

explored. The results indicate that an increase in nanoparticle volume fraction enhances the heat exchange rate

in the cavity only when the geometric shape of the cold obstacle is circular. This is followed by square and

triangular shapes, which approximately yield concordant results, and then the elliptical shape.

Key-Words: - Ventilated Square Cavity, Mixed Convection, Central cold Block, Nanofluid, Hybrid nanofluid

Outlet port location, Heat transfer, Richardson number.

Received: April 29, 2023. Revised: November 18, 2023. Accepted: January 16, 2024. Published: April 1, 2024.

1 Introduction

In recent years, various methods have been

employed to enhance mixed convection heat transfer

in enclosures, sparking significant interest in

numerous industrial applications such as process

cooling, electronic components, radiators, heat

exchangers, and more. One such method involves

the use of nanofluids, which consist of colloidal

suspensions of nanoscale solid particles (metallic or

non-metallic) in a base fluid (such as water, oil, or

ethylene glycol) to achieve improved thermal

conductivity.

Several studies in this field have utilized

different numerical methods. [1], aimed to review

all published studies on mixed convection of

nanofluids in enclosures. Papers were classified into

four main categories: square (and rectangular)

shapes, triangular shapes, trapezoidal shapes, and

non-conventional shapes. Most studies reported a

significant improvement in heat transfer with an

increase in nanoparticle concentration, Reynolds

number, and Richardson number, with the required

pumping power increasing when adding

nanoparticles to base fluids. Another numerical

study was conducted by [2], presenting an

investigation of mixed convection inside a

trapezoidal cavity filled with Cu-water nanofluid

under the influence of a constant magnetic field.

Mixed convection is induced by the action of the

inclined hot right-side wall in the direction of aiding

or opposing flow. The left inclined side wall is fixed

and maintained isothermal at a low temperature. The

upper and lower horizontal walls are fixed and

thermally insulated. The magnetic field is imposed

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horizontally. Results showed that the suppressing

effect of the magnetic field is more pronounced for

the aiding case than for the opposing case.

Meanwhile, the enhancement of the Nusselt number

due to the presence of Cu nanoparticles is more

significant for the opposing flow case. Heat transfer

by mixed convection in an asymmetrically heated

vertical channel filled with a mixture of water and

two types of nanoparticles (Cu, Al2O3) was

numerically investigated by [3]. Results indicated

that the nanoparticle volume fraction with forced

convection (induced by a heated channel wall) has a

significant effect on nanofluid velocities and the

average Nusselt number. Cu-water nanofluid

exhibited better thermal performance than Al2O3-

water nanofluid. Numerous studies have been

conducted on mixed convection in a square cavity

using nanofluids under various thermal and

kinematic boundary conditions. As an example, [4],

investigated stable conjugate mixed convection in a

double-lid square cavity containing an internal solid

body. The annulus is filled with Al2O3-water

nanofluid based on the Buongiorno two-phase

model. The upper horizontal wall is maintained at a

constant low temperature and moves to the right,

while the lower horizontal wall is maintained at a

constant high temperature and moves to the left. The

governing equations are numerically solved using

the finite element method. Key parameters used in

this study include internal solid body location,

nanoparticle volume fraction (0 ≤ φ ≤ 0.04),

Reynolds number (1 ≤ Re ≤ 500), Richardson

number (0.01 ≤ Ri ≤ 100), solid body size (0.1 ≤ D ≤

0.7), and solid body thermal conductivity (kw =

0.01, 0.045, 0.1, 0.76, and 1.95 W/m

℃

). Results

showed a notable increase in heat transfer with the

use of nanofluid in such a cavity. However, at low

Reynolds numbers, adding nanoparticles had a

negative effect on the Nusselt number when the

Richardson number was very high. It was also

observed that a large solid body could increase heat

transfer in cases of high Reynolds and Richardson

numbers. Additionally, several works have focused

on the convective phenomenon of nanofluid in a

ventilated square cavity. [5], numerically examined

laminar steady mixed convection in a ventilated

square cavity. The cavity is filled with different

nanofluids and contains two ports for inflow and

outflow. The right vertical wall is maintained at a

hot temperature, while the other walls are

considered adiabatic. Numerical simulations are

performed for pure water fluid and mixtures of this

basic fluid with nanoparticles (Ag and Cu) for

Richardson numbers ranging from 0.04 to 4 and

nanoparticle volume fractions between 0% and

10%. This study is dedicated to a dynamic

investigation with a fixed Grashof number of 104,

varying Reynolds numbers. The numerical results

obtained indicate an increase in heat transfer with an

increase in the volume fraction, and the

enhancement of the entropy generation and heat

transfer product increases significantly with an

increase in Reynolds number. The most effective

nanoparticles for increasing the heat exchange rate

are Ag, characterized by a high local Nusselt

number, indicating excellent heat transfer compared

to metallic Cu nanoparticles. In a similar context,

[6], conducted a numerical simulation of convection

in square-ventilated cavities containing insulated

parallel baffles. The left and right walls of the cavity

are maintained at a high temperature, while the

upper and lower walls of the cavity, as well as the

parallel baffles, are adiabatic. Opening slots are

positioned in the upper and lower corners of the hot

vertical walls. It was observed that the behavior of

ventilated cavities depends not only on the size and

position of the baffles but also strongly on the

ventilation cavity configuration. Additionally, flow

fields are limited by the larger baffle size, Sb =

0.75. However, few studies have addressed the case

of a partially open cavity with an internal heat

source, such as those by [7] , [8], which are related

to the study of the effect of the Richardson number,

the location of the exit orifice, and the nanoparticle

volume fraction. The obtained results indicate that

increasing the nanoparticle volume fraction and

reducing the Richardson number improve the heat

transfer rate. A study of forced, mixed, and natural

convection of nanofluid inside a ventilated cavity

with a central cold block of two different geometric

shapes (square and triangular) was conducted by [9].

They numerically studied the effect of the

Richardson number, different geometric shapes of

the obstacle, and nanoparticle types. The results

reveal that, in some cases, reducing the Richardson

number and the size of solid particles improves heat

transfer. It was also noted that there is an optimal

concentration of nanoparticles at which the

maximum average Nusselt numbers occur. This

study is divided into two main aspects:

1. The effect of nanoparticle concentration.

2. The effect of the geometric shape of a

central cold block.

Our results are presented in the form of

isotherms and streamlines. They are displayed in

temperature, velocity, local Nusselt number, and

average Nusselt number profiles for different

volume fractions.

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2 Computational Model

2.1 Configuration

The studied configuration is depicted in Figure 1. It

consists of a ventilated square-shaped cavity

containing a cold block occupying the center of the

cavity at a temperature (Tc). The block has different

geometric shapes (circular, elliptical, square, and

triangular) with the same peripheral surface area,

diameter, and size, each equal to D= 0.2H. The

cavity is uniformly heated; to a constant temperature

(Th) from the bottom wall, while the other walls are

all thermally insulated (adiabatic walls). Our

physical system is subjected to an external flow of

nanofluid (Ag/water), introduced into the cavity

with a velocity Ue and a temperature Te (Te = (Th +

Tc)/2), through an orifice placed at the upper level of

the left wall, with dimensions (h=0.1H), and an

outlet orifice of the same dimensions located at the

bottom of the opposite wall.

Fig. 1: The physical scheme of the problem and the

boundary conditions

The flow is assumed Newtonian, incompressible

in a steady laminar regime in a state of thermal

equilibrium. According to the Boussinesq

approximations, the density variation is neglected

everywhere, except in the buoyancy term. The

thermophysical properties with which we will work

are shown in Table 1

Table 1. Thermophysical properties of water and

nanoparticles with T=300 °K

997.1

4179

0.613

10500

235

429

2.2 Governing Equations

According to the assumptions mentioned above, the

equations governing the flow are the equation of

continuity, momentum and energy, which can be

written in the following dimensional form:

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(1)

(2)

(3)

(4)

The properties of the nanofluid are calculated

according to the following formulas, [10]:

The thermal diffusivity of nanofluid is:

(5)

The density of a nanofluid is:

(6)

The heat capacity of a nanofluid is:

(7)

The coefficient of thermal expansion of nanofluids

can be determined by:

(8)

According to Maxwell's model, the thermal

conductivity of a nanofluid is:

(9)

The dynamic viscosity of a nanofluid is given by

Brinkman (1952):

(10)

The boundary conditions of mixed convection are

listed in the following Table 2:

Table 2. Hydrodynamic and thermal boundary

conditions

Limit

Hydrodynamic

conditions

Thermal conditions

Bottom wall

Upper wall

Straight wall

Left wall

Hall

Exit

block

The reduced variables used during dimensioning

of equations (1-4) as well as the Reynolds, Grashof,

Prandtl and Richardson numbers are respectively

given by the following expressions:

(11)

(12)

By carrying the dimensionless quantities

defined above in the equations of the mathematical

model (1), (2), (3) and (4), we obtain:

(13)

(14)

(15)

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(16)

The dimensionless boundary conditions relating

to our physical domain are shown in the following

Table 3:

Table 3. Dimensionless hydrodynamic and thermal

boundary conditions

Limit

Hydrodynamic

conditions

Thermal conditions

Bottom wall

Upper wall

Straight wall

Left wall

Hall

Exit

block

The local Nusselt number (Nu) along the lower

hot wall can be expressed by:

(18)

The average value of the Nusselt number along

this wall is calculated by the following integral:

(19)

3 Numerical Method

To solve the equations governing this problem,

we employed the SIMPLE algorithm along with

the second-order scheme of the finite volume

method. The computations were carried out

using the Ansys‐Fluent 6.3 software.

3.1 Mesh Independence

A quadratic cell mesh is utilized for this study.

The mesh is meticulously designed to refine

near the block and adjacent walls, gradually

coarsening as it extends farther away from both.

This strategy aims to reduce the overall

computational cost while enhancing the

precision of the simulation results. As

established in the Table 4, four different meshes

are selected—100 x 160, 110 x 180, 120 x 200,

and 130 x 200 nodes—to analyze the effect of

the number of nodes, enabling the attainment of

highly accurate solutions without compromising

computational time.

Table 4. Flow characteristics for different grids

((Cu/water), φ=0.05, Re=100, Ri= 0.1)

3.2 Code Validation

To validate the code governing this simulation,

a comparison was made between the average

Nusselt values obtained in this work and those

calculated by [11]. This corresponds to cases of

a rectangular cavity with a central square-

shaped obstacle, maintained at a hot

temperature (Th = 310 °K). The cavity was

filled with a water-based nanofluid containing

Cu nanoparticles. The left and right vertical

walls of the cavity were cooled to constant

temperatures (Tc = 290 °K), while the upper

and lower walls were considered adiabatic.

The comparison is conducted for a specific

case with a Rayleigh number (Ra) of (104 and

106) and a nanoparticle volume fraction (φ) of

5%. Some values of this comparison are

gathered in Figure 2. The results are highly

compatible as they converge well, and the

margin of error is extremely minor. This

validation enhances confidence in the accuracy

of our numerical simulations and strengthens

the credibility of the results obtained in this

study.

Mesh

Nuavg

21.600

21.676

21.717

21.701

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0 1 2 3 4 5

Ra=104

Ra=106

Cu-water

Square Geometry

Average Nusselt number

j(%)

Present works

Boulahia

Present works

Boulahia

Fig. 2: Evolution of the average Nusselt number

along the hot wall. Comparison of the present

numerical study with the data processed by [11]

4 Results and Discussion

In this section, the numerical results are

presented in terms of stream function and

isotherm contours, average Nusselt number,

temperature profiles, and velocity profiles for a

range of volume fraction values (φ = 0, 2, 4, 6,

8%), Reynolds number Re = 100, and

Richardson number Ri = 1. The streamlines and

isotherms for some cases studied in this work

(φ= 4%, Re=100, Ri= 1.) are presented in

Figure 3. In general, it can be observed that the

nanofluid flow maintains the same structure for

all geometric shapes of the central obstacle,

where the flow is directed from the inlet port

upward and downward of the obstacle, then

moves toward the outlet port. The incoming

flow strikes the left side of the central obstacle

and distributes it to almost the same geometric

shape, producing recirculation zones that rotate

clockwise. The recirculation zones form near

the inlet port, and these vortices increase in size

when the obstacle is circular (Case B) compared

to other obstacles (Square (Case A), Triangle

(Case C), Ellipse (Case D)).

This can be explained by the higher

resistance of the square and triangular blocks to

fluid circulation within the ventilated cavity,

indicating that the circular geometric shape has

a remarkable effect on enhancing mixed

convection by improving heat transfer. For the

isotherms, the color representation varies from

blue to red. The blue color represents low-

temperature zones, while red corresponds to

high temperatures. The temperature field is

presented at the bottom of Figure 3, which

shows that high temperatures are localized in

narrow spaces near the hot wall, corresponding

to the thickness of the thermal boundary layer

(Red part in the mentioned figure). This

indicates good heat exchange through

convection (mixed convection). There is also an

increase in the cold zone, and the isotherms are

less tightly packed on the hot wall in case B

(circle obstacle), compared to the other cases

(A, C, D), due to the effect of the geometric

shape exchange of the cold obstacle

It is also noted that the cold obstacle at the

center of the cavity acts as a cold source and

leads to the concentration of high-temperature

regions near its walls. As long as the flow is

directed from the inlet port upwards to the

outlet port downwards, and during this path, it

strikes the left side of the central obstacle, the

isotherms move downward. Comparing the

streamlines in the figure, it can be seen that the

obstacle causes the appearance of vortices. This

dynamic phenomenon results in the

manifestation of wavelike isotherms inside the

cavity. The magnitude of the heat transfer rate,

deduced from the isotherms, is influenced by

both the geometry of the cold obstacle and the

inlet velocity of the nanofluid. Furthermore, the

position of these heat transfer fluctuations

depends on the inlet port's position (top,

bottom, or in the middle). Moreover, the

ventilation of the nanofluid inside the cavity

leads to the extension of the isotherms toward

the outlet port. This observation aligns with the

results obtained from the analysis of the Nusselt

number.

Figure 4 shows the effect of increasing the

volume fraction of nanoparticles in the nanofluid

(𝝋) on the average Nusselt number for four

geometric shapes of the obstacle. It is evident that

increasing the nanoparticle concentration enhances

the heat transfer rate, leading to a rising trend in the

average Nusselt number. This is attributed to the

improvement in the effective thermal conductivity

of the nanofluid with the addition of nanoparticles.

The agitation of particles near the obstacle promotes

heat transfer between the cold obstacle and the

nanofluid. Furthermore, it can be observed that the

values of the average Nusselt number for the

circular-shaped obstacle are higher than those for

other shapes. This is indeed due to the effect of the

resistance of nanofluid circulation within the

ventilated cavity, resulting from the geometric shape

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exchange of the central cold obstacle. The lowest

Nusselt value is, as expected, recorded when the

obstacle has an elliptical geometric shape.

0 2 4 6 8

19 Ag / Water, Ri=1, Re=100

Nu avg

j (%)

Square

Circular

Triangular

Elliptical

Fig. 4: Variation of the average Nusselt number

for different volume fractions (φ %) and

different geometric shapes of the cold obstacle

for (Ag/Water)

5 Conclusion

The heat transfer rate and flow structures of an

Ag/H2O nanofluid cooling a ventilated square

cavity containing a central cold block of various

shapes were numerically investigated, and the

impact of nanoparticle volume fractions on heat

transfer enhancement was examined.

Four different geometric shapes for the cold

obstacle (circle, square, triangle, and ellipse) were

considered. The results showed that:

1. Increasing nanoparticle volume fractions

had a positive impact on heat transfer,

leading to an increase in the average Nusselt

number. Circular obstacles consistently

exhibited higher Nusselt numbers compared

to other shapes, highlighting the influence

of geometric shapes on nanofluid

circulation and heat transfer.

2. The central cold obstacle acted as a cold

source, concentrating high-temperature

regions near its walls. The dynamic flow

around the obstacle caused the appearance

of vortices, resulting in wavelike isotherms

inside the cavity.

3. Isotherms, representing temperature

variations, showed that circular obstacles

facilitated better heat exchange, with less

tightly packed isotherms on the hot wall

compared to other shapes. The circular

shape contributed to enhanced mixed

convection and strengthened heat transfer.

The study provides insights into the role of

geometric shape and nanoparticle concentration in

improving heat transfer by mixed convection.

Practical applications could benefit from optimizing

Cas (a)

Cas (b)

Cas (c)

Cas (d)

The streamlines

The isotherms

Fig. 3: Streamlines and isotherms for different geometric shapes of the cold obstacle (Ag/Water), for φ=4%,

Re=100, Ri=1

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the design of cold obstacles and controlling

nanoparticle concentrations to achieve enhanced

heat exchange rates, offering valuable information

for applications in various industrial contexts, such

as cooling electronic components and industrial

processes.

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Contribution of individual authors to the

creation of a scientific article (ghostwriting

policy)

All authors equally contributed in the present

research.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare.

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(Attribution 4.0 International, CC BY 4.0)

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