Bouyancy Driven in An inclind CIrcular Enclosure with

Journal of Mechanical Engineering Research and Developments

ISSN: 1024-1752

CODEN: JERDFO

Vol. 43, No. 7, pp. 62-74

Published Year 2020

62

Bouyancy Driven in An inclind CIrcular Enclosure with Elliptic

Heat Source

Ali Najim Abdullah Saieed†, Sanaa Turky Mousa Al-Mousawe‡, Auday Awad Abtan‡†, Laith

Jaafer Habeeb‡†

†Al Rafidain University College, Department of Air-Conditioning and Refrigeration Eng. Tech., Baghdad, Iraq.

‡University of Baghdad, Department of Reconstruction and Projects.

‡†University of Technology, Training and Workshops Center, Baghdad, Iraq.

*Corresponding Author Email: [email protected]; [email protected] ;

[email protected]; [email protected]

ABSTRACT: Numerical study of buoyancy driven in an inclined enclosure filled with Al2O3/water nanofluid

was carried out. The cold outer circular wall and hot inner elliptic wall of enclosure were maintained at constant

temperature TC and Th; respectively. The stream function–vorticity method was used to solve the prevailing

calculations which are discretized using the way of finite volume and then resolved via code of FORTRAN.

Validation was performed by comparison the current results with previous results and found to be in excellent

agreement. The study coved wide ranges of Rayleigh number (104 ≤Ra ≤ 106) and volume fraction were (0 ≤

𝜑 ≤0.2) with different angles of inclination 𝜙= 0o (horizontal position), 30o, 60o, and 90o (vertical position).

Results were presented in terms of streamlines, isotherms, local and average Nusselt numbers. The maximum

average Nusselt number is obtained by using nanofluids and it is more pronounced at high Rayleigh numbers.

Moreover, the heat transfer rates enhance at higher Rayleigh numbers as the angle of inner cylinder inclination

increases. While, at low Rayleigh numbers, there is no effect for changing angle of inclination on the heat

transfer process.

KEYWORDS: natural convection; buoyancy driven; circular; enclosure; cavity.

INTRODUCTION

Recent advances in nanotechnology have led to the development of fine nanomaterials with size range of 1-100

nm in carrier fluids and have higher thermal conductivity than the base fluid [1]. Using nanofluids in practical

applications have the potential to significantly increase heat transfer rates in a variety of areas such as thermal

storage systems, heat exchangers, biomedical applications, nuclear reactors, solar collectors, and cooling of

electronic devices. Different nanofluids were used in cavities such as Al2O3, Ag, Au, AgO, Cu, CuO, and TiO2

[2-17]. Obsthuizen and Fault (1991), [1] studied the natural convective flow in an inclined rectangular cavity

with partially heated side wall cold top wall over its entire surface at uniform temperatures whereas the

additional walls were adiabatic. The outcomes exhibited the governing parameters have significant effects on the

mean Nusselt number. Barakos et al. (1994), [2] used k--E model for turbulence modeling with and without

logarithmic wall functions to investigate the heat transfer characteristics in a square cavity. Comparisons with

experimental previous works for Nusselt number shows the limitations of the standard k--E model with

logarithmic wall functions, which give importance over predictions. Getachew et al (1998), [3] used non-

Newtonian fluid to study the double-diffusive convective heat transfer inside a rectangular porous enclosure with

a constant temperature and concentration of vertical walls. They showed that the overall heat and mass transfer

rates agree with the results produced by discrete numerical experiments. Mana and Saran (2006), [4] studied

the conjugate convection heat transfer in an inclined square enclosure containing a concentric square block. They

concluded that the wall thickness has a considerable effect on the heat transfer process. Hakan (2007), [5]

investigated heat transfer characteristics in a partially cooled and inclined rectangular cavity filled with saturated

porous medium with hot one side and partially cooled other side while the other walls were adiabatic. He showed

that inclination angle and aspect ratio are the dominant parameters on heat transfer and fluid flow. Ahmed

Mezrhab et al. (2010), [6] studied the natural convection in a ‘‘T” form enclosure containing two symmetrically

identical isothermal blocks and was vented by two opening located in a vertical median axis at the top and the

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Bouyancy Driven in An inclind CIrcular Enclosure with Elliptic Heat Source

63

bottom parts of the enclosure. It is found that the radiation enhances linearly the heat transfer rates the Rayleigh

number increases, then it becomes more important when the solid blocks heights are large. Muneer et al. (2011),

[7] studied the natural convection heat transfer inside inclined square enclosure with two different boundary

conditions applied at the top and bottom walls. It is noticed that the vortex formation, size and flow behaviors are

greatly affected by the heat transfer mechanism for each boundary condition due to relatively high Rayleigh

number condition. Prakash and Satyamurty (2011), [8] studied the free in an anisotropic fluid filled porous

rectangular enclosure subjected to end to-end temperature difference. They concluded that heat transfer rates

increase with increase in permeability ratio depending on Darcy number, characterizing the Brinkman extended

non-Darcy flow. Sheikholeslami et al. (2014), [9] used CuO–water nanofluid and magnetic field to enhance

natural convection heat transfer in an enclosure has a sinusoidal hot wall. It is noticed that heat transfer rate

increases with increase nanoparticles volume fraction, dimensionless amplitude of the sinusoidal wall and

Rayleigh number and decrease Hartmann number. Mansour and Bakier (2015), [10] studied the effect of

inclined magnetic field on the natural convection inside enclosure filled with Cu–water nanofluid and subjected

to changeable thermal boundary conditions. It is concluded that the heat transfer process enhances with increase

in angle of inclination and the magnetic force pointed to horizontal trend. Ravnik and Škerget (2015), [11]

investigated the type of nanofluid and base fluid on the natural convection heat transfer inside a hot circular and

elliptical cylinder placed in a cooled cubic enclosure. They used Al2O3; Cu and TiO2 nanofluids with pure water

and air for validation purposes. The results show that the dominated conduction heat transfer gives highest rates

of heat transfer and efficiency than the convection dominated flow regime. Tahar and Ali (2017), [12]

examined the thermal and fluid flow characteristics by natural convection in an enclosure filled with Al2O3/water

nanofluid and Cu-Al2O3/water hybrid nanofluid based on water. It is noticed that the thermal and dynamic

performance of Cu-Al2O3/water hybrid nanofluid is better than Al2O3/water nanofluid. Yang Hu et al. (2017),

[13] investigated free convection in an eccentric annular enclosure filled with a Cu–water nanofluid based on

water. The results show that the presence of the nanoparticles with pure fluid enhances the behavior of flow

pattern. Mebarek (2018), [14] investigated the effects of some substances water base fluids, ethylene glycol, and

engine oil with Titanium nanoparticles on the behavior of heat transfer and fluid field in a cylindrical annulus. It

was concluded that the thermal efficiency relies on the volume fraction and the Rayleigh number of the

nanoparticles while the mediocre Nusselt number relies on on the type of the base fluid. Ishrat et al. (2018),

[15] studied the effect of magnetohydrodynamics (MHD) on conjugate natural convection flow in a rectangular

cavity filled with CO-H2O nanofluid. The results show that the heat transfer rate increases with increase in

Rayleigh number and decrease in Hartmann number. Tayebia et al. (2019), [16] investigated the free convection

heat transfer in a confocal elliptic annulus filled with CNT-water nanofluid containing hot inner cylinder and

cold outer cylinder. It was concluded that the increase of average Nusselt number occurs with increase in the

volume fraction of the nanoparticles particularly at high Rayleigh numbers. Suhail and Altamush (2020), [17]

investigated the convective heat transfer in a partially heated vertical annulus full with H2O-Al2O3 nanofluid

with high aspect ratio. For all models of nanofluids, Rayleigh number decreases with nanoparticle concentration.

The present work is a numerical study of the fluid flow and thermal field characteristics of natural convection

heat transfer in a cold circular enclosure filled with Al2O3 /water nanofluid and having inclined hot elliptic

cylinder at different angles of inclination (𝜙= 0o, 30o, 60o, and 90o). The study covered a wide ranges of

Rayleigh number (104 ≤ Ra ≤ 106) and volume fraction (0 ≤ 𝜑 ≤0.1). The stream function–vorticity method

was used to solve the governing equations. The prevailing calculations are discretized by utilizing the way of

finite volume and resolved via code of FORTRAN.

MATHEMATICAL MODEL

Consider a cold circular enclosure with hot inner elliptic cylinder, as shown in Figure 1-a. The elliptic cylinder is

tilted at 𝜙 from the vertical. The inner and outer cylinder are kept at a constant higher and lower temperature Th,

and Tc; respectively. The angles of inclination are 0o, 30o, 60o, and 90o, respectively. The mathematical

formularization for the natural convection in an inclined arc-shape cavity was suggested by [18]. The

mathematical model of present work was employed to analyze the flow pattern and heat transfer characteristics

in a circular enclosure with an inclined elliptic heat source, and is briefly reviewed below.


64

a. Physical domain b. Grid generation

Figure 1. A schematic view and geometry characterization of the physical field.

Governing Equations

Figure 1 shows a schematic diagram of annulus enclosure with inner elliptic cylinder. The fluid in the enclosure

is a water-based nano-fluid comprising with various volume fraction of nanoparticles. The thermo-physical

features of the nano-fluid are given as shown in Table 1 [18]:

Table 1. Thermal physical features of water-𝐴𝑙2𝑂2 nanofluid [18]

Material 𝐶𝑝 (

𝐽

𝑘𝑔. 𝐾) 𝜌 (

𝑘𝑔

𝑚3) 𝑘 (

𝑊

𝑚. 𝐾) 𝛽 (

1

𝐾)

𝑊𝑎𝑡𝑒𝑟 4179 997.1 0.613 21*10-5

𝐴𝑙2𝑂2 765 3970 40 0.85*10-5

In this concern, the density 𝜌𝑛𝑓 , specific heat energy (𝜌𝐶𝑝)𝑛𝑓 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑛𝑎𝑛𝑜𝑓𝑙𝑢𝑖𝑑 and thermal expansion

coefficient (𝜌𝛽)𝑛𝑓are estimated as follows [18]:

𝜌𝑛𝑓 = (1 − 𝜑)𝜌𝑓 + 𝜑𝜌𝑠 (1)

𝛽𝑛𝑓 = (1 − 𝜑)𝛽𝑓 + 𝜑𝛽𝑠 (2)

(𝜌𝐶𝑝)𝑛𝑓 = (1 − 𝜑)(𝜌𝐶𝑝)𝑓 + 𝜑(𝜌𝐶𝑝)𝑠 (3)

Model of Maxwell-Garnetts was determined to compute the thermal conductivity of nanofluid [18].

𝑘𝑛𝑓 = 𝑘𝑓

(𝑘𝑠 + 2𝑘𝑓) − 2𝜑(𝑘𝑓 − 𝑘𝑠)

(𝑘𝑠 + 2𝑘𝑓) + 𝜑(𝑘𝑓 − 𝑘𝑠) (4)

The dynamic viscosity of nanofluid characterized as model of Brinkman [18] as follows:

𝜇𝑛𝑓 =𝜇𝑓

(1 − 𝜑)2.5 (5)

The viscous two-dimensional laminar air flow by natural convection in the cavity is governed by continuity,

momentum, and energy equations. The fluid properties are assumed to be constant except the density variation in

the buoyant force according to Boussinesq approximation. The way of stream function–vorticity was utilized to

resolve the prevailing calculations and the conversion of coordinates was prepared for plotting the wavy form

inside a rectangular computational field, as displayed in Figure 1b. The prevailing calculations are given down in

Equations (6–8) by utilizing dimensionless temperature (𝜃), stream function (𝛹), and dimensionless vorticity (𝛺)

which are founded on the body-fitted curvilinear coordinate (𝜉, 𝜂) as follows [19, 20]:


65

𝑐1

𝜕2𝛹

𝜕𝜂2+ 2𝑐2

𝜕2𝛹

𝜕𝜉𝜕𝜂+ 𝑐3

𝜕2𝛹

𝜕𝜉2+ 𝑐4

𝜕𝛹

𝜕𝜂+ 𝑐5

𝜕𝛹

𝜕𝜉= 𝐽𝛺 (6)

𝜕𝛺

𝜕𝜉

𝜕𝛹

𝜕𝜂−

𝜕𝛺

𝜕𝜂

𝜕𝛹

𝜕𝜉= 𝑃𝑟

𝜇𝑛𝑓

𝜇𝑓

𝜌𝑛𝑓

𝜌𝑓

(𝑐1

𝜕2𝛹

𝜕𝜂2+ 2𝑐2

𝜕2𝛹

𝜕𝜉𝜕𝜂+ 𝑐3

𝜕2𝛹

𝜕𝜉2𝑐4

𝜕𝛹

𝜕𝜂+ 𝑐5

𝜕𝛹

𝜕𝜉) −

𝑃𝑟𝛽𝑛𝑓

𝛽𝑓

𝑃𝑟 𝑅𝑎 {𝑠𝑖𝑛𝜙 (𝜕𝑌

𝜕𝜂

𝜕𝜃

𝜕𝜉−

𝜕𝑌

𝜕𝜉

𝜕𝜃

𝜕𝜂) − 𝑐𝑜𝑠𝜙 (

𝜕𝑋

𝜕𝜉

𝜕𝜃

𝜕𝜂−

𝜕𝑋

𝜕𝜂

𝜕𝜃

𝜕𝜉)} (7)

𝜕𝜃

𝜕𝜉

𝜕𝛹

𝜕𝜂−

𝜕𝜃

𝜕𝜂

𝜕𝛹

𝜕𝜉=

𝑘𝑛𝑓

𝑘𝑓

(𝜌𝐶𝑝)𝑛𝑓

(𝜌𝐶𝑝)𝑓

(𝑐1

𝜕2𝜃

𝜕𝜂2+ 2𝑐2

𝜕2𝜃

𝜕𝜉𝜕𝜂+ 𝑐3

𝜕2𝜃

𝜕𝜉2𝑐4

𝜕𝜃

𝜕𝜂+ 𝑐5

𝜕𝜃

𝜕𝜉) (8)

We can express the equations. (6–8) in terms Jacobian of the coordinate conversion from the rectangular

coordinates (𝑋, 𝑌) to the curvilinear coordinates (𝜉, 𝜂) as shown in Eq. (9); and is stated in Eq. (10). In this

research, the value of Prandtl number for water is fixed at 6.2.

𝛺 =𝜔𝐿2

𝛼𝑓 ; 𝛹 =

𝜓

𝛼𝑓 ; 𝜃 =

(𝑇−𝑇𝑐)

(𝑇ℎ−𝑇𝑐) (9)

𝐽 =𝜕𝑌

𝜕𝜉

𝜕𝑋

𝜕𝜂−

𝜕𝑌

𝜕𝜂

𝜕𝑋

𝜕𝜉 (10)

𝑐1 =1

𝐽[(

𝜕𝑋

𝜕𝜂)

2

+ (𝜕𝑌

𝜕𝜂)

2

] ; 𝑐2 = −1

𝐽(

𝜕𝑋

𝜕𝜉

𝜕𝑋

𝜕𝜂+

𝜕𝑌

𝜕𝜂

𝜕𝑌

𝜕𝜉) ; 𝑐3 =

1

𝐽[(

𝜕𝑋

𝜕𝜉)

2

+ (𝜕𝑌

𝜕𝜉)

2

] (11)

𝑐4 =1

𝐽[

𝜕

𝜕𝜂(

𝜕2𝑋

𝜕𝜉2+

𝜕2𝑌

𝜕𝜉2)] −

1

𝐽[

𝜕

𝜕𝜉(

𝜕𝑋

𝜕𝜉

𝜕𝑋

𝜕𝜂+

𝜕𝑌

𝜕𝜂

𝜕𝑌

𝜕𝜉)] (12)

𝑐5 =1

𝐽[

𝜕

𝜕𝜉(

𝜕𝑋

𝜕𝜉

𝜕𝑋

𝜕𝜂+

𝜕𝑌

𝜕𝜂

𝜕𝑌

𝜕𝜉) −

𝜕

𝜕𝜂(

𝜕𝑋

𝜕𝜉

𝜕𝑋

𝜕𝜂+

𝜕𝑌

𝜕𝜂

𝜕𝑌

𝜕𝜉)] (13)

𝑋 =𝑥

𝐿; 𝑌 =𝑦𝐿

(14)

𝑅𝑎 =𝑔𝛽𝑓(𝑇ℎ − 𝑇𝑐)𝐿3

𝜗𝑓𝛼𝑓 ; 𝑃𝑟 =𝜗𝑓

𝛼𝑓

(15)

The stream function and normalized vorticity are associated to the dimensionless velocities, is stated in Eqs. (16)

and (17) as follows:

𝛺 =1

𝐽[(

𝜕𝑉

𝜕𝜉

𝜕𝑌

𝜕𝜂−

𝜕𝑉

𝜕𝜂

𝜕𝑌

𝜕𝜉) +

𝜕𝑈

𝜕𝜉

𝜕𝑋

𝜕𝜂−

𝜕𝑈

𝜕𝜂

𝜕𝑋

𝜕𝜉] (16)

𝑈 =1

𝐽(−

𝜕𝛹

𝜕𝜉

𝜕𝑋

𝜕𝜂+

𝜕𝛹

𝜕𝜂

𝜕𝑋

𝜕𝜉) ; 𝑉 =

1

𝐽(−

𝜕𝛹

𝜕𝜉

𝜕𝑌

𝜕𝜂+

𝜕𝛹

𝜕𝜂

𝜕𝑌

𝜕𝜉) (17)

Boundary Conditions

The boundary conditions on the cold circular enclosure and the hot elliptic wall are given below in Eqs. (18) and

(19), respectively:

𝑈 = 0; 𝑉 = 0; 𝛹 = 0; 𝜃 = 0; 𝛺 = −1

𝐽(

𝜕𝑈

𝜕𝜂

𝜕𝑋

𝜕𝜉) (18)


66

𝑈 = 0; 𝑉 = 0; 𝛹 = 0; 𝜃 = 1; 𝛺 = −1

𝐽(

𝜕𝑉

𝜕𝜂

𝜕𝑌

𝜕𝜉+

𝜕𝑈

𝜕𝜂

𝜕𝑋

𝜕𝜉) (19)

GENERATION OF GRID

In this research, a methodical demonstration could obtain for the coordinate conversion from the physical field to

the computational field; all-geometrical factors could calculate accurately. The function of super elliptic become

as [20]:

(𝑥

𝑎)

2𝑛

+ (𝑦

𝑏)

2𝑛

= 1 (20)

Where:

a the elliptic lengths in the x direction

b the elliptic length in the y direction

n is a positive integer.

The conversion of coordinate for the current problematic can be accurately made, which is became as:

𝑥 = − 𝑠𝑖𝑛 𝑠𝑖𝑛 𝜉 ⋅ [𝑟𝑖 + (𝑟𝑜 − 𝑟𝑖)𝜂] (21)

𝑦 =𝑐𝑜𝑠 𝑐𝑜𝑠 𝜉 ⋅ [𝑟𝑖 + (𝑟𝑜 − 𝑟𝑖)𝜂] (22)

Where, ri is the radius of the equivalent radius of inner elliptic cylinder, and ro is the radius of the outer circular

cylinder and it is calculated as in reference one:

𝑟𝑖 =𝑏

(𝑐𝑜𝑠 𝑐𝑜𝑠 ( 𝜉)2𝑛 +𝑠𝑖𝑛 𝑠𝑖𝑛 ( 𝜉)2𝑛)1/2𝑛 (23)

The transformed computational domain to (ξ,η) plane is 0 ≤ 𝜂 ≤ 1 and 0 ≤ 𝜉 ≤ 2𝜋. A typical generated grid is

shown in Fig. (1-b), for 61×61 nodes.

The enforced boundary conditions aren’t slip and isothermal on external circular enclosure and internal elliptic

cylinder surfaces. So, the settings of boundary could be write as:

𝑈|𝜂=0,1 = 0, 𝑉|𝜂=0,1 = 0 (24𝑎)𝜓|𝜂=0 = 0 , 𝜓|𝜂=1 = 0 (24𝑏)𝜃|𝜂=0

= 1 , 𝜃|𝜂=1 = 0 (24𝑐)𝛺𝜂=0,1 =𝐶

𝐽

𝜕2𝜓

𝜕𝜂2|𝜂=0,1

=𝐶

𝐽

𝜕𝑈

𝜕𝜂|𝜂=0,1 (24𝑑)

Solution Procedure

The equations (6-8), and the boundary conditions (18 and 19), are discretized via the way of finite-volume. The

calculation of generation of grid is founded on the curvilinear coordinate system(𝜉, 𝜂). The functions of

coordinate conversion (i.e., 𝜉 = 𝜁(𝑋, 𝑌) and 𝜂 = 𝜂(𝑋, 𝑌)) were proposed by Thompson et al. [21] and adopted

by Chen, et al. [22]. The demonstrations of finite-volume for 𝛹, 𝛺 and 𝜃 can be conjointly resolved by utilizing

the way of iteration. The principal variance way is utilized for the way of discretization. The functions of

conversion 𝜉 = 𝜁(𝑋, 𝑌) and 𝜂 = 𝜂(𝑋, 𝑌) are gotten disjointedly via resolving the following equations of two

elliptic Poisson, as specified in Eq. (25):

𝜕2𝜉

𝜕𝑋2+

𝜕2𝜉

𝜕𝑌2= 𝑃(𝑋, 𝑌) (25 𝑎)

𝜕2𝜂

𝜕𝑋2+

𝜕2𝜂

𝜕𝑌2= 𝑄(𝑋, 𝑌) (25 𝑏)

Where:


67

P and Q are two arbitrary functions definite to regulate the limited density of the grids. The succeeding over-

relaxation way was assumed to raise computational accurateness. The assumed relaxation parameters for energy

equations, vorticity, and stream function, and are 0.1, 1, and 1.5. respectively. A complete description for this

solution method can be initiate in Cheng and Chen [23].

Nusselt number calculations

The local Nusselt number 𝑁𝑢𝑥 and the overall Nusselt number 𝑁𝑢 𝑎𝑣𝑒 are stated in Eqs. (26) and (27),

respectively:

𝑁𝑢𝑥 =𝑘𝑛𝑓

𝑘𝑓

𝜕𝜃

𝜕𝑛|𝑚𝑜𝑣𝑖𝑛𝑔 𝑙𝑖𝑑 (26)

𝑁𝑢𝑎𝑣𝑒 =1

𝑠∫

𝑠

0

𝑁𝑢𝑥 𝑑𝑠 (27)

Where:

𝑛 is the external coordinate usual to the wall,

𝑘𝑓 is the fluid thermal conductivity.

Code validation

The present numerical solution methodology used in the present code was validated by comparing the results of

local Nusselt number with the analytical results submitted by F.M. Mahfouz [24] as shown in Fig. 2 for the

problem of natural convection heat transfer in case of Mr=2.25, Ari=0.436, Ra= 3.72 × 105 and Pr=0.7. It can be

seen that the numerical model is in a good agreement with work [24].

Figure 2. Local Nusselt number distribution along inner and outer walls for Mr=2.25, Ari=0.436, Ra= 3.720

× 105 and Pr=0.70 and association with numerical outcomes of Mahfouz [24].

RESULTS AND DISCUSSION

Streamlines and isotherms

The present work comprises a numerical study using finite volume method and FORTRAN code to analyze the

natural convection heat transfer in an annulus full with Al2O3–water nanofluid and formed by external circular

cylinder and inner elliptic cylinder. The ratio amid the two main axes is measured to 3.5. The study covered wide

ranges of Rayleigh number and volume fraction as follows: 104 ≤ Ra ≤ 106 and 0 ≤ 𝜑 ≤0.2; respectively, with

Prandtl number equals to 6.2. The elliptic inner cylinder was tilted by angle 𝜙 from vertical axis for values equal

to 0o (horizontal position), 30o, 60o, and 90o (vertical position). The streamline and isotherm contours at Ra=105

and 106 are shown on the left and right of Figures (3 and 4); respectively, at four angles of inclination 𝜙=0o


68

(horizontal position), 30o, 60o, and 90o (vertical position). The solid line, dashed blue line, and dash dot red line

in these figures represent the behavior of flow and thermal fields at volume fraction ratio 𝜑=0, 0.12, and 0.2;

respectively.

It noticed that in horizontal position (𝜙 = 0𝑜), two alike circulating eddies devoid of worldwide flow movement

will be formed on the two sides of enclosure. These eddies consist of flow circulating rising lengthways the

internal elliptic heat source wall and descending lengthways the external circular cavity wall. The first eddy

rotates left to right on the right side and the other rotates counter clockwise on the left side. The two eddies are

separated by vertical line above elliptic cylinder represented the zero-stream function line. The external circular

wall is intersected by this line at two points; the top point represents the flow constant point on the wall, and the

bottom point represents the flow separation point from the wall. In horizontal position (𝜙 = 0𝑜), the center of

vorticity lies exactly on the horizontal axis of enclosure. The zero-stream function line is deviated away from the

wall in case of inclined cavity (𝜙 = 30𝑜 𝑎𝑛𝑑 60𝑜), leads to more mixing of flow due to increase the convection

currents. As a result, the wall streamline function indicating intensity of circulation will increase. The value of

wall stream function in the present work depends on significant parameters such as the inclination angle, volume

fraction of nanoparticles, and Rayleigh number.

It is shown that in Fig. (3) and Fig. (4), the global flow circulation depending on controlling parameters is not

generated. As Rayleigh number increases from 105 (Fig. 3) to 106 (Fig.4), the value of stream function increases

from (𝛹𝑚𝑎𝑥,𝑚𝑖𝑛 = 20, −20) to (𝛹𝑚𝑎𝑥,𝑚𝑖𝑛 = 184, −184) at (𝜙 = 0𝑜), and from (𝛹𝑚𝑎𝑥,𝑚𝑖𝑛 = 28, −28) to

(𝛹𝑚𝑎𝑥,𝑚𝑖𝑛 = 255, −255) at (𝜙 = 30𝑜), and from (𝛹𝑚𝑎𝑥,𝑚𝑖𝑛 = 35, −35) to (𝛹𝑚𝑎𝑥,𝑚𝑖𝑛 = 266, −282) at (𝜙 =

60𝑜), and from (𝛹𝑚𝑎𝑥,𝑚𝑖𝑛 = 37, −37) to (𝛹𝑚𝑎𝑥,𝑚𝑖𝑛 = 266, −282) at (𝜙 = 90𝑜); because of increasing the

convection currents. This means the heat transfer process rises with rise in Rayleigh number and angle of

inclination. In vertical position of inner elliptic cylinder (𝜙 = 90𝑜), the center of vorticity deviates towards

upper the horizontal axis on the left of enclosure, and lower the horizontal axis in the right of enclosure at

Ra=106 , while it stay at the horizontal axis at Ra=105 .

As shown in Fig. (3), at Ra=105, conduction is the principal mechanism of heat transfer associated to the

convection and the magnitudes of the stream functions are very weak. So, there is no effect of angle of

inclination on the isotherms lines which seem to be in the form of concentric elliptic curves parallel to the elliptic

wall boundary close the internal cylinder, and in the form of circular curves near the inner surface of outer

cylinder. While, the formation of thermal plume occurs at the upper of the inner elliptic cylinder wall at higher

Rayleigh number (Ra=106) due to the hot upward currents along the two sides of elliptic heat source wall. The

thermal boundary layer thickness near the internal hot surface rises because of increase the buoyancy driven and

temperature gradients indicating higher heat transfer rates. The isotherms will be distorted and the plume creeps

towards left hand side as the angle of inclination changes from horizontal𝜙 = 0𝑜, to inclined position𝜙 =

30𝑜 𝑎𝑛𝑑 60𝑜, which indicates the dominant convective heat transfer and the buoyancy forces will be strong to

overcome the viscous force. The thermal plume is symmetric about the vertical axis only for horizontal position

and becomes asymmetric in the path of the buoyancy-driven flow on the way to the higher portion of the inner

surface of external circular wall as the inclination angle moves from horizontal to vertical position.

Furthermore, in these figures, the presence of nanoparticles affects the isotherms and streamlines. As can be

shown from these figures that the strength of flow increases with increase in the volume fraction of the

nanoparticles for all angles of inclination. In fact, this increasing in Ra=106 is much higher than that in Ra=105.

The dynamic domain is made via a couple of symmetrical vortices at vertical and horizontal positions with

anticlockwise circulation in the left side and clockwise circulation in the right side for all volume fraction values.

The existence of nanoparticles aids to accumulate of the isotherms close the hot elliptic wall, which shows

enhancement in the rate of heat transfer. It is concluded that using a nanofluid is more active associated to pure

water only.


69

𝜙 = 0𝑜 (horizontal position)

𝜙 = 30𝑜 (inclined position)



70

𝜙 = 90𝑜 (vertical position)

Figure 3. Effect of nanoparticle concentration on Streamline (left)and isotherm counter (right) for Ra=105, and

ϕ=0o (horizontal position), 30o, 60o , and 90o (vertical position).

𝜙 = 0𝑜 (horizontal position)



71


𝜙 = 90𝑜 (vertical position)

Figure 4. Effect of nanoparticle concentration on Streamline (left)and isotherm counter (right) for Ra=106, and

ϕ=0o (horizontal position), 30o, 60o , and 90o (vertical position).

Local Nusselt number

The behavior of thermal domain close to the hot wall is calculated by finding local Nusselt number variation

around the wall. The angular variation of local Nusselt number for the hot inner ellipse cylinder at Ra=106 at

four angles of inclination and different nanoparticles volume fractions is shown in Fig. 5. The figure shows that

the local Nusselt number increases with increase in nanoparticles volume fraction and has the same trend at any

certain angle of inclination. The values of local Nusselt number are negative because the flow is previously

heated for the duration of its rising movement lengthways the hot internal wall and it hits the cold outer wall

leads to creating great negative temperature grade. The maximum value of local Nusselt number of inner hot

ellipse cylinder occurs at horizontal axis of hot cylinder (at angular position,𝛾 = 90 𝑎𝑛𝑑 270𝑜, side points) for

𝜙 = 0𝑜 (horizontal position). The increasing of local Nusselt number is due to minimum resistance to

conduction and advanced velocity grade (higher heat convection coefficient) close to the walls, Whereas the

smallest magnitudes of local Nusselt number are located in the top of hot cylinder (angular position𝛾 =

0𝑜 𝑎𝑛𝑑 360𝑜, the plume region) for all angles of inclination and nanoparticles volume fractions. The maximum

value of local Nusselt number stays at 𝛾 = 270𝑜 only (right-hand side of hot cylinder) on the horizontal axis as

the angle of inclination moves from horizontal to vertical position for all nanoparticles volume fractions of fluid.

The strong fluid motion close to the hot walls causes great temperature grade at the walls and gives great local

Nusselt number. The convection currents get much faster with increase of volume fraction, causing active flow

movement and more advanced thermal plume.


72

(a) 0𝑜 (horizontal position) (𝑏) 𝜙 = 30𝑜 (inclined position)

(𝑐) 𝜙 = 60𝑜 (inclined position) (𝑑) 𝜙 = 90𝑜 (vertical position)

Figure 5. Angular variation of local Nusselt number on hot cylinder at Ra=106 for inclination angle 0o

(horizontal), 30o, 60o, and 90o (vertical).

Average Nusselt number

The variation of average Nusselt number versus Rayleigh number for the inner hot ellipse cylinder at various

values of fluid nanoparticles volume fraction of (𝜑 = 0.0, 0.06, 0.12, 𝑎𝑛𝑑 0.2) and inclination angles of

(𝜙 =0o, 30o, 30o, and 90o) is shown in Figure (6). It is noticed from this figure that the average Nusselt number

increases with increase in nanoparticles volume fraction due to rises of the thermal conductivity of the nanofluid

with the rise in the nanoparticles volume fraction. At a certain value of 𝜑 and lower Rayleigh numbers and, the

convection strength is weak and increases with increase Rayleigh number for each angles of inclination leading

to increase average Nusselt number. For 𝜑=0, the increase in average Nusselt number is very small as Rayleigh

increases from 104 to 105, and be clearer at 𝜙 = 60𝑜. At Ra=104, there is no effect for changing angle of

inclination on the heat transfer process. Generally, the maximum average Nusselt number is obtained by using

the nanofluids and it is more pronounced at high Rayleigh numbers. Moreover, the average Nusselt number

increases as the angle of inclination increases from 0o to 90o. The decreasing of resistance to the flow is

happened with decrease in the probable area of hot cylinder usual to the buoyancy upward flow. This leads to

increasing the flow velocity adjacent to the hot wall and decreasing the thickness of thermal boundary layers and

hydrodynamic to give high heat transfer rate.

(𝑎) 𝜙 = 0𝑜 (horizontal position) (𝑏) 𝜙 = 30𝑜 (inclined position)


73

(𝑐) 𝜙 = 60𝑜 (inclined position) (𝑑) 𝜙 = 90𝑜 (vertical position)

Figure 6. Average Nusselt number on hot ellipse cylinder versus Rayleigh number at inclination angle ϕ=0o,

30o, 30o, and 90o.

CONCLUSIONS

Numerical investigation for natural convective heat transfer of Al2O3/water nanofluid in an annular enclosure

forming by outer cold circular cylinder and inner hot elliptic cylinder with constant temperatures 𝑇𝑐 𝑎𝑛𝑑 𝑇ℎ;

respectively. From the results of present work, it is concluded that:

1. The streamline function and flow circulation increase with increase in Rayleigh number and as angle of

elliptic inner inclination moves from horizontal to vertical position because of increase the flow mixing.

2. The formation of thermal plume at the upper of the inner circular cylinder wall occurs only at high Rayleigh

number (Ra=106) due to the hot upward currents lengthways the two sides of internal cylinder wall.

3. There is no effect of angle of inclination of elliptic inner cylinder on the isotherms at Ra=105.

4. The maximum value of local Nusselt number of inner hot ellipse cylinder occurs at right and left side points

of horizontal axis of hot cylinder (at angular position, 𝛾 = 90 𝑎𝑛𝑑 270𝑜; respectively) for 𝜙 = 0𝑜

(horizontal position) and at = 270𝑜 (left hand side) only for other angles of inclination.

5. The maximum average Nusselt number is obtained by using nanofluids and it is more pronounced at high

Rayleigh numbers.

6. At low Rayleigh numbers, there is no effect for changing angle of inner cylinder inclination on the heat

transfer process.

7. The heat transfer rates enhance at higher values of Rayleigh number as the angle of inclination increases.

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