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Columbia International Publishing American Journal of Heat and Mass Transfer (2016) Vol. 3 No. 3 pp. 186-224 doi:10.7726/ajhmt.2016.1012

Research Article

______________________________________________________________________________________________________________________________ *Corresponding email: [email protected] (M. M. Rahman), Phone: +968 24141423, Fax: +968 24141490 1 Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, P. O. Box 36, P.C. 123 Al-Khod, Muscat, Sultanate of Oman. 2 On leave from the Department of Mathematics, Jagannath University, Dhaka-1100, Bangladesh

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Finite Element Analysis of Unsteady Natural Convective Heat Transfer and Fluid Flow of Nanofluids inside a Tilted Square

Enclosure in the Presence of Oriented Magnetic Field

K. S. Al Kalbani1, M. S. Alam1, 2, and M. M. Rahman1*

Received: 7 June 2016; Published online: 30 July 2016 © The author(s) 2016. Published with open access at www.uscip.us

Abstract In this paper, the problem of unsteady natural convective heat transfer flow of nanofluids having various sizes of nanoparticles inside an inclined square enclosure in the presence of oriented magnetic field is investigated numerically. The Brownian motion of nanoparticles is taken into consideration in the thermal conductivity model construction. Two opposite walls of the enclosure are insulated and the other two walls are kept at different temperatures. Galerkin weighted residual finite element technique has been employed to solve the governing nonlinear dimensionless equations. In order to ensure the accuracy of the present numerical code, comparisons with previously published works are performed and excellent agreement is obtained. The effects of model parameters such as Rayleigh number, Hartmann number, nanoparticles volume fraction, inclination angle of magnetic field, inclination angle of the geometry, diameter and Brownian motion of the nanoparticles on the fluid flow and heat transfer are investigated. The results indicate that an increment in Rayleigh number and nanoparticle volume fraction increases the heat transfer rate in a significant way, whereas, an increment in Hartmann number decreases the overall heat transfer rate. It is also observed that the heat transfer enhancement strongly depends on the diameter of the nanoparticles as well as the types of the nanofluids. It is observed that the time taken to reach the steady state is controlled by the different model parameters and in particular, it is longer for low Rayleigh number and shorter for high Rayleigh number. A comparison between the two studies of with and without Brownian motion shows that when Brownian motion is considered, the solid volume fraction has more significant effects on the heat transfer rate at all Rayleigh numbers considered in the square cavity. Finally, the distribution of average heat transfer rate for different cavity inclination angle along with various model parameters has been found as almost parabolic shape. Keywords: Finite element method; Nanofluid; Inclined magnetic field; Natural convection; Square enclosure; Brownian motion

K. S. Al Kalbani , M. S. Alam, and M. M. Rahman / American Journal of Heat and Mass Transfer (2016) Vol. 3 No. 3 pp. 186-224

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1. Introduction Natural convection heat transfer in inclined devices has been the subject of many studies in the past since rarely is the earth’s surface aligned with geo-potential lines. The position relative to the vertical direction of the hot active source generating the flow plays an essential role in natural convection problems. To properly adopt the convective exchanges to the application requirements, it is necessary to know the influence of the inclination angle. As an example, electronic devices contained in an airborne cavity change position relative to gravity during take-off, landing and in normal flight. To achieve correct thermal regulation of such devices subject to natural convection, it is thus necessary to control the heat exchanges in all positions. The inclination angle relative to the direction of gravity also plays an important role on temperature and flow fields especially in electronic system, such as laptop computers and crystal growth process in an inclined cylinder etc. Markham and Rosenberger (1984) showed that improved transport rates in the crystal growth process can be achieved by tilting the cylinder. Aminossadati and Ghasemi (2005) numerically investigated the flow and temperature fields in an inclined enclosure simulating an inclined electronic device. They showed that placing the enclosure at different orientations significantly affected the heat transfer rate. Ben-Nakhi and Chamkha (2006) argued that tilting the enclosure considerably affect the flow and temperature fields as well as the heat transfer characteristics of a partitioned enclosure. Jeng et al. (2008) presented an experimental and numerical study of the transient natural convection due to mass transfer in inclined enclosures and showed that the streamlines and fluid concentration vary with the inclination angle. Recently, Tian et al. (2014) studied numerically the problem of unsteady natural convection in an inclined square enclosure with heat-generating porous medium and their result shows that inclination angle plays an important role in the heat transfer characteristics of the walls. When the fluid is electrically conducting and exposed to a magnetic field, the Lorentz force is also active and it interacts with the buoyancy force in governing the flow and temperature fields. Since the Lorentz force suppresses the convection currents by reducing the velocities, employment of an external magnetic field has a wide range of application in different context. In some cases, where the heat is transferred by natural convection mechanism, the electrically conducting fluid may be in the presence of a magnetic field with an arbitrary inclination. Some examples of these could include the fusion reactors, metal casting, geothermal energy extractions, and crystal growth in fluids. Thus, some researchers investigated the effect of magnetic field orientation within the enclosures for two- or three-dimensional heat transfer problems and all of them revealed that the orientation of the magnetic field changed the flow field and consequently the thermal performance of the enclosure (see Ozoe and Okada, 1989; Garandet et al., 1992; Venkatachalappa and Subbaraya, 1993; Alchaar et al., 1995; Krakova and Nikiforovb, 2002; Pirmohammadi and Ghassemi, 2009). Ece and Buyuk (2006) studied natural convection flow under a magnetic field in an inclined rectangular enclosure heated and cooled on adjacent walls. Their numerical results indicate that the magnetic field suppresses the convective flow and the heat transfer rate. They also showed that the orientation and the aspect ratio of the enclosure; the strength and direction of the magnetic field had significant effects on the flow and temperature fields. Grosan et al. (2009) considered the inclination angle of magnetic field on the natural convection within a rectangular enclosure and it was found that the convection mode depends upon both the strength and the orientation of the magnetic field. Their results also indicate that the applied magnetic field in the horizontal direction


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is most effective in suppressing the convection flow for a stronger magnetic field in comparison with the vertical direction. Sathiyammmoorthy and Chamkha (2010) used different thermal boundary conditions to examine the steady laminar two-dimensional natural convection in the presence of inclined magnetic field in a square enclosure filled with a liquid gallium. They found that heat transfer decreases with an increase of the magnetic field strength. They also noted that vertically and horizontally applied magnetic fields affect the heat transfer rate differently. However, most of the above studies on the natural convection in enclosures with the magnetic field effect have considered the electrically conducting fluid having low thermal conductivity. This, in turn, limits the enhancement of heat transfer in the enclosure, particularly, in the presence of a magnetic field. An innovative technique to enhance heat transfer is by using nano-scale particles in the base fluid. Nanotechnology has been widely used in industry since materials with sizes of nanometers possess unique physical and chemical properties. Nano-scale particles added to base fluids are called as nanofluid which is firstly utilized by Choi (1995) in order to develop advanced heat transfer fluids with substantially higher conductivities. The most important characteristics of this new type of fluid are their higher thermal conductivities in comparison with pure fluids. Therefore, one of the most significant issues regarding these substances is the accuracy of proposed models for calculation of effective thermal conductivity. As revealed in the recent comprehensive reviews by Das et al. (2006) and Yu et al. (2008), over the past decade there have been tremendous attempts to identify and model mechanisms of thermal conductivity enhancement of nanofluids, including size and shape of nanoparticles, the hydrodynamic interaction between nanoparticles and base fluid, clustering of particles, temperature or Brownian motion, and so on. Jou and Tzeng (2006) reported a numerical study of the heat transfer performance of nanofluids inside 2D rectangular enclosures. Their results indicated that increasing the volume fraction of nanoparticles produced a significant enhancement of the average rate of heat transfer. Santra et al. (2008) conducted a study of heat transfer augmentation in a differentially heated square cavity using copper-water nanofluid. Their results show that the Bruggeman model predicts higher heat transfer rates than the Maxwell-Garnett model. Abu-Nada (2009) implemented new models for nanofluids properties and examined the heat transfer enhancement under a wide range of temperatures and solid volume fractions. He argued that the heat transfer enhancement depends on the nanofluid viscosity and thermal conductivity models, and the range of Rayleigh numbers and solid volume fractions. But for thermal conductivity, the above- mentioned models do not consider neither the main mechanisms for heat transfer in nanofluids such as Brownian motion nor the nanoparticles size or temperature dependence. Therefore, numerical simulations need more robust model for thermal conductivity that takes into account temperature dependence and nanoparticle size. In order to consider the movement of nanoparticles, some researchers have included the contribution of a dynamic component related to particle Brownian motions in their model development (see Xuan et al., 2003; Koo and Kleinstreuer, 2004; Koo and Kleinstreuer, 2005; Palm et al., 2006; Akbarinia and Behzadmehr, 2007). In a quiescent suspension, nanoparticles move randomly and thereby carry relatively large volumes of surrounding liquid with them. As a result of Brownian motion, the effective thermal conductivity, which is composed of the particles’ conventional static part and the Brownian motion part, increases to result in a lower temperature gradient for a given heat flux. To


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capture these transport phenomena, a new thermal conductivity model (see Koo and Kleinstreuer, 2004) for nanofluids has been considered, which takes into account the effects of particle size, particle volume fraction and temperature dependence as well as properties of base liquid and particle phase by considering surrounding liquid traveling with randomly moving nanoparticles. Ghasemi and Aminossadati (2010) studied the Brownian motion of nanoparticles on natural convective heat transfer flow in a triangular enclosure filled with nanofluid. Their reported results indicate that when the Brownian motion of nanoparticles is taken into account, the resulting average Nusselt numbers are greater than those of not considering the Brownian effects. Seyf and Nikaaein (2012) investigated forced convection heat transfer of ethylene-Glycol based nanofluids in a microchannel heat sink (MCHS) with aluminum-oxide, zinc-oxide and copper-oxide as nanoparticles. Based on their observations, the effect of Brownian motion was more significant for smaller nanoparticles. Wang et al. (2012) studied the heat transfer enhancement of copper-water nanofluids considering Brownian motion of nanoparticles in a singular cavity. Their results showed that when Brownian motion is considered, the solid volume fraction has more significant effects on the heat transfer rate at all Richardson numbers considered in a singular cavity. Very recently, Ehteram et al. (2016) studied the effect of various conductivity and viscosity models considering Brownian motion on nanofluids mixed convection flow and heat transfer. Their results showed that when the Brownian motion of nanoparticles is taken into account, the resulting average Nusselt numbers are greater than those of not considering the Brownian effects. Moreover, the effect of temperature, nanoparticle size, and nanoparticles volume fraction on thermal conductivity was studied by Chon et al. (2005) and they showed that nanofluid thermal conductivity is also affected by temperature, volume fraction of nanoparticles, and nanoparticle size. Thus, such physics cannot be neglected and the dependence of nanofluid thermal conductivity on temperature, volume fraction and diameter of nanoparticles, must be taken into account in order to predict the correct role of nanoparticles on heat transfer enhancement. In a very illuminating paper on nanofluids Khanafer and Vafai (2011) have shown that it is not clear which analytical model should be used to describe the thermal conductivity of nanofluids as there are many models are available in the literature. Additional theoretical and experimental research studies are required to clarify the mechanisms responsible for heat transfer enhancement in nanofluids. These authors have established new correlations for effective thermal conductivity and viscosity are synthesized and developed in this study in terms of pertinent physical parameters based on the reported experimental data. The characteristic feature of nanofluids is thermal conductivity enhancement, a phenomenon observed by Masuda et al. (1993). This phenomenon suggests the possibility of using nanofluids in advanced nuclear systems (Buongiorno and Hu 2005). Therefore, in this paper a finite element simulation is performed in order to investigate the effects of magnetic field strength and its orientation on the thermal performance of an inclined square enclosure filled with different types of nanofluids having various shapes of nanoparticles, where Brownian motion is taken into consideration. We also propose a more appropriate model for the calculation of nanofluid thermal conductivity and study the effect of this model on heat transfer through natural convection in an inclined square enclosure. Besides, this model will be compared to frequent used model in literature namely Maxwell-Garnett model for thermal conductivity. To the best knowledge of the authors, no study which considers this problem in an inclined square cavity has yet been reported in the open literature. Our numerical results provide information that may be


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useful for design optimization as well as for thermal performance enhancement of energy systems such as solar thermal collectors, radiators and advanced cooling of nuclear system.

2. Problem Formulation 2.1 Physical modeling We consider an unsteady, laminar, incompressible two-dimensional natural convection flow in the presence of oriented magnetic field in an inclined square enclosure of length L filled with nanofluids. Dimensional coordinates with the x -axis measuring along the bottom wall and y -axis

being normal to it along the left wall is used. The geometry and coordinate systems are schematically shown in Fig. 1. The angle of inclination of the enclosure from the horizontal axis is denoted by . The cavity is permeated by a uniform magnetic field x yB B B i j of constant

magnitude 2 2

0 x yB B B , where i , j are the unit vectors along the coordinate axis. The direction

of the magnetic field makes an angle with the positive x -axis. The top and bottom walls are

insulated and nanofluids are isothermally heated and cooled by the left and right side walls at uniform temperatures of

HT and ,CT respectively. Under all situations, H CT T is maintained. In the

present study, we have taken water, kerosene and engine oil (EO) as base fluids; Cu , Co , and

3 4Fe O as nanoparticles. It is also assumed that thermal equilibrium exists between the base fluids

and nanoparticles, and no-slip occurs between the two media. The thermophysical properties of the nanofluids are listed in Table 1. The thermophysical properties of the nanofluids are considered to be constant except the density variation in the body force term of the momentum equation, which is estimated by the Boussinesq approximation. The gravitational acceleration acts in the negative y

direction. All solid boundaries are assumed to be rigid no-slip walls. Table 1 Thermo-physical properties of the base fluid and solid nanoparticles.

Physical properties

Water (H2O)

Engine oil (EO)

Kerosene Cu Co Fe3O4

-1 -1Jkg Kpc 4179 1880.3 2090 385 420 670

-3kgm 997.1 888.23 780 8933 8900 5180

-1 -1Wm Kk 0.613 0.145 0.149 400 100 80.4

1 1kg sm 0.001003 0.8451 0.00164 - - -

5 -110 K 21 70 99 1.67 1.3 20.6

-1Sm 65.5 10 23.004 106 10 75.96 10 71.602 10 60.112 10

Pr 6.8377 10958.9 23.004 - - -


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2.2 Mathematical modeling Within the framework of afore-mentioned assumptions, the governing equations for the present study are expressed in dimensional form as follows:

0u v

x y

(1)

2 2

2 2

2

0 2

1sin

sin cos sin

nf nf

C

nf nf nf

nf

nf

u u u p u uu v g T T

t x y x x y

Bv u

(2)

2 2

2 2

2

0 2

1cos

sin cos cos

nf nf

C

nf nf nf

nf

nf

v v v p v vu v g T T

t x y y x y

Bu v

(3)

2 2

2 2nf

T T T T Tu v

t x y x y

(4)

where the variables and the related quantities are defined in the nomenclature.

Fig. 1. Schematic view of the square shape enclosure with boundary conditions.

Fig. 2. Mesh generation for a square enclosure.

2.3 Initial and boundary conditions

The initial and boundary conditions for the above stated model are as follows:


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For 0 : 0, 0, 0

For 0 : 0, for 0; 0

0, for ; 0

0, 0 for 0, L; 0

H

C

t u v T p

t u v T T x y L

u v T T x L y L

Tu v y x L

y

(5)

2.4 Thermal and physical properties of nanofluids The effective density, specific heat, thermal expansion coefficient, viscosity and electrical conductivity of nanofluids that appear in equations (1)-(4) are given by the following formulas (see Rahman et al., 2014):

1nf f p (6)

1p p pnf f pc c c

(7)

1nf f p

(8)

2.5

1

f

nf

(9)

2 2

2

p f f p

nf f

p f f p

(10)

where

nf

nf

p nf

k

c

.

The effective thermal conductivity of the nanofluid for spherical nanoparticles is introduced by Maxwell (1873) as follows:

2 2

2

p f f p

nf f

p f f p

k k k kk k

k k k k

(11)

In the above Maxwell model (equation (11)), the Brownian motion of nanoparticles has not considered. But experimentally, it has been proved that the Brownian motion of nanoparticles plays an important role on the heat transfer enhancement of nanofluid (see Chon et al., 2005). Therefore, we propose an appropriate model for the calculation of nanofluid thermal conductivity which is composed of the particle's convectional static part and a Brownian motion part. This two component thermal conductivity model takes into account the effects of particle size, particle volume fraction and temperature dependence as well as types of particle and base fluid combinations:

nf static Browniank k k (12)

where, statick is the static thermal conductivity based on Maxwell classical correlation which is given

in equation (11). But Browniank , the thermal conductivity component enhanced by the irregular

motion of suspended nanoparticles can be proposed as follows:


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, 2

2 3

p p p B ref

Brownian

nf

c K Tk

d

(13)

which represents the important contribution of the present work. Thus, the apparent thermal conductivity of the nanofluid (that consists of the Maxwell static part and Brownian part) which has been used in the present mathematical model is as follows:

,2 2 2

2 32

p f f p p p p B ref

nf f

nfp f f p

k k k k c K Tk k

dk k k k

(14)

2.5 Dimensional analysis Dimensional analysis is one of the most important mathematical tools in the study of fluid mechanics. To describe several transport mechanisms in nanofluids, it is meaningful to make the conservation equations into non-dimensional form. The advantages of non-dimensionalization are listed as follows: (i) non-dimensionalization gives freedom to analyze any system irrespective of their material properties. (ii) one can easily understand the controlling flow parameters of the system, (iii) make a generalization of the size and shape of the geometry, and (iv) before doing experiment one can get insight of the physical problem. These aims can be achieved through the appropriate choice of scales. As a scale of distance, we choose the length of the cavity of the region under consideration measured along the x -axis. Thus, in order to reduce the dimensionless form of the governing equations (1)-(4) with boundary conditions (5), we incorporate the following dimensionless variables:

2

2 2, , , , , = , =

fC

f f f f H C

tT Tx y uL vL pLX Y U V P

L L T T L

(15)

Introducing the relation (15) into equations (1)-(4), the governing dimensional equations can be written in the following dimensionless form:

0U V

X Y

(16)

2 2

2 2

2 2

Pr

Pr sin Pr sin cos sin

f nf

nf f nf

nf f nf

nf f nf f

U U U P U UU V

X Y X X Y

Ra Ha V U

(17)

2 2

2 2

2 2

Pr

Pr cos Pr sin cos cos

f nf

nf f nf

nf f nf

nf f nf f

V V V P V VU V

X Y Y X Y

Ra Ha U V

(18)

2 2

2 2

nf

f

U VX Y X Y

(19)


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where, Prf

f

is the Prandtl number,

3

f H C

f f

g T T LRa

is the Rayleigh number and

0 /f fHa B L is the Hartmann number.

The dimensionless forms of the boundary conditions are as follows:

For 0 : 0, 0, 0

For 0 : 0, 1 for 0; 0 1

0, 0 for 1; 0 1

0, 0 for 0,1; 0 1

U V P

U V X Y

U V X Y

U V Y XY

(20)

3. Calculation of Average Nusselt Number and Average Shear Rate

The most important physical quantities for this model are the local and average Nusselt numbers along the left heated wall of the enclosure. The local Nusselt number is defined as

w

f H C

LqNu

k T T

(21)

where the heat transfer from the left heated wall wq is given by

0

w nf

x

Tq k

x

(22)

The average Nusselt number at the left heated wall of the enclosure can be calculated from the following expression:

1

0

nf

av

f

kNu dY

k X

(23)

Also, the average shear rate at left heated wall of the enclosure can be calculated as 1

0

VSr dY

X

(24)

where, 2 2

U VV

X X

is the average velocity magnitude of the flow.

4. Finite Element Formulation and Computational Procedure The finite element method (FEM) is such a powerful method for solving both ordinary and partial differential equations that arises in science and engineering problems. The basic idea of this method is dividing the whole domain into smaller elements of finite dimensions called finite elements. This method is such a good numerical method in modern engineering analysis, and it can be applied for solving integral equations including heat transfer, fluid mechanics, chemical processing, electrical systems, and many other fields. Thus, the governing dimensionless equations


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(16)-(19) along with the initial and boundary conditions (20) have been solved numerically by employing Galerkin weighted residual based finite element technique. The method of weighted residual process as described by Zienkiewicz and Taylor (1991) has been applied to (16)-(19) in order to derive the finite element equations as follows:

0A

U VN dA

X Y

(25)

2 2

2 2

2 2

Pr


f nf

nf f nfA A A

nf f nf

nf f nf fA A

U U U P U UN U V dA H dA N dA

X Y X X Y

Ra N dA Ha N V U dA

(26)

2 2

2 2

2 2

Pr


f nf

nf f nfA A A

nf f nf

nf f nf fA A

V V V P V VN U V dA H dA N dA

X Y Y X Y

Ra N dA Ha N U V dA

(27)

2 2

2 2

nf

fA A

N U V dA N dAX Y X Y

(28)

where A is the element area, ( 1,2,...,6)N are the element shape functions or interpolation

functions for the velocity components and temperature, and ( 1,2,3)H are the element shape

functions for the pressure. Applying Gauss’s divergence theorem to the second order derivative terms of the equations (26)-(28) in order to generate the boundary integral terms associated with the surface tractions and heat flux, we obtain the following equations:

0

2 2

0

Pr


Pr

f nf

nf f nfA A A

nf f nf

nf f nf f

nf

x

f nf S

N NU U U P U UN U V dA H dA dA

X Y X X X Y Y

Ra N dA Ha N V U dA

N S dS

(29)


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0

2 2

0

Pr


Pr

f nf

nf f nfA A A

nf f nf

nf f nf fA A

nf

y

f nf S

N NV V V P V VN U V dA H dA dA

X Y Y X X Y Y

Ra N dA Ha N U V dA

N S dS

(30)

w

nf nf

w w

f fA A S

N NN U V dA dA N q dS

X Y X X Y Y

(31)

where the surface tractions , x yS S along the outflow boundary 0S and velocity components and

fluid temperature or heat flux wq that flows into or out from the domain along wall boundary wS .

The basic unknowns for the above differential equations (29)-(31) are the velocity components

,U V , the temperature and the pressure .P The six node triangular elements are used in this

work for the development of the finite element equations. All six nodes are associated with velocities as well as temperature; only corner nodes are associated with pressure. This means that a lower order polynomial is chosen for pressure and which is satisfied through the continuity equation. The velocity component and the temperature distributions, and linear interpolation for the pressure distribution according to their highest derivative orders in the differential equations (16)-(19) as

,U X Y N U (32)

,V X Y N V (33)

,X Y N (34)

,P X Y H P (35)

where 1,2,...,6 ; =1,2,3

Now substituting the element velocity component distributions, the temperature distribution, and the pressure distribution from equations (32)-(35) into equations (25) and (29)-(31), the finite element equations can be written in the following form,

0x yK U K V (36)

2 2

Pr


x y x xx yy

u

f nf

nf f nf

nf f nf

nf f nf f

K U K U U K V U R P K K U

Ra K Ha K V U Q

(37)

2 2

Pr


x y y xx yy

v

f nf

nf f nf

nf f nf

nf f nf f

K V K U V K V V R P K K V

Ra K Ha K U V Q

(38)


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x y xx yy

nf

f

K K U K V K K Q

(39)

where superposed dot denotes partial differentiation with respect to and the coefficients in element matrices are in the form of the integrals over the element area and along the element edges

0S and wS as,

,K N N dA ,x

NK N dA

X

,y

NK N dA

Y

,xx

N NK dA

X X

,yy

N NK dA

Y Y

,x

NK N N dA

X

,y

NK N N dA

Y

,x

NR N dA

X

,y

NR N dA

Y

0

Pru

nf

x

f nf

Q N S dS

, 0

Prv

nf

y

f nf

Q N S dS

, w

nf

w

f

Q N q dS

.

These element matrices are evaluated in closed-form ready for numerical simulation. Details of the derivation for these element matrices are omitted herein for brevity. The derived finite element equations (36)-(39) are nonlinear. The nonlinear algebraic equations so obtained are modified by imposition of boundary conditions. To solve the set of the global nonlinear algebraic equations in the form of matrix, the Newton-Raphson iteration technique has been adapted through partial differential equation solver with MATLAB interface. The convergence criterion of the numerical

solution along with error estimation has been set to 1 510m m , where is the general

dependent variable ( , , )U V and m is the number of iteration.

The main advantages of finite element method (FEM) over finite difference method (FDM) are that it has ability to deal with complex 2D or 3D domains, higher accuracy and rapid convergence. Other benefit of the finite element method is that of the specific mode to deduce the equations for each element which are then assembled. Therefore, the addition of new elements by refinement of the existing ones is not a major problem. The computational domains with irregular geometries by a collection of finite elements make the method a valuable practical tool for the solution of boundary value problems arising in various fields of engineering. For the other methods, the mesh refinement is a major task and could involve the rewriting of the program. But for all the methods used for the discrete analogue of the initial equation, the obtained system of simultaneous equations must be solved. That is why, the present work emphasizes the use of finite element technique to solve flow and heat transfer problems. 4.1 Mesh generation In the finite element method, the mesh generation is the technique to subdivide a domain into a set of sub-domains, called finite element, control volume, etc. The discrete locations are defined by the numerical grid, at which the variables are to be calculated. It is basically a discrete representation of the geometric domain on which the problem is to be solved. Meshing the complicated geometry make the finite element method a powerful technique to solve the boundary value problems occurring in a range of engineering applications. Fig. 2 displays mesh configuration of the present physical domain with triangular finite elements.


198

4.2 Grid independency test

A grid refinement study has been performed for 6.8377Pr , 510Ra , 20Ha , 30 , 0.05 ,

15 and 10 nmd in a square enclosure. Four different non-uniform grid systems with the

following number of elements within the resolution field: 2540, 6578, 17038, and 26300 are examined. The numerical design is carried out for highly precise key in the average Nusselt number

avNu and average shear rate Sr for the aforesaid elements to develop an understanding of the

grid fineness as shown in Fig. 3 and Table 2. The scale of avNu for 17038 elements show a very little

difference with the results obtained for the elements 26300. Hence the grid size of 17038 and 26300 elements can be used to get the accurate results. In the presence study, 17038 triangular elements have been considered to get the results. Fig. 3. Convergence of the average Nusselt number (left) and average shear rate (right) with grid

refinement for 56.8377, 10 , 20Pr Ra Ha , 30 , 0.05 , 15 and 10 nmd .

Table 2 Grid test using the values 6.8377Pr , 510Ra , 20Ha , 030 , 0.04 , 015 ,

10 nm.d

Wall

Elements (Nodes)

1504 (927)

2540 (1487)

6578 (3752)

17038 (9420)

26300 (14051)

Left

avNu

6.73852 1324.90283

6.73999 1369.90663

6.74342 1391.79755

6.74385 1392.27714

4.3 Code validation In order to verify the accuracy of the present numerical code, we have compared our result for steady state case with Ghasemi et al. (2011). The physical problem studied by Ghasemi et al. (2011)

Sr


199

was a steady two-dimensional natural convection flow in a square enclosure filled with

2 3water-Al O nanofluid which is under the influence of a horizontally applied magnetic field. Using

our code the present numerical predictions have been obtained for Rayleigh number between 310

and 710 . The comparison of the results obtained by our numerical code with those of Ghasemi et al. (2011) with respect to average Nusselt number (at the hot wall) are shown in Table 3 which shows an excellent agreement. This validation boosts the confidence in the numerical outcome of the present study.

Table 3 Comparison of the present data (Nuav) with those of Ghasemi et al. (2011) for different values of Ra and .

= 0 = 0.02

Ra Present study Ghasemi et al. (2011) Present study Ghasemi et al. (2011) 103 1.002 1.002 1.060 1.060 104 1.182 1.183 1.212 1.212 105 3.138 3.150 3.127 3.138 106 7.901 7.907 7.957 7.979 107 16.912 16.929 17.179 17.197

5. Results and Discussion In this section, simulated numerical results are analyzed to investigate the effects of magnetic field strength and its orientation as well as the inclination angle of the square enclosure filled with various nanofluids having different diameters of the nanoparticles (1nm 100nmd ). Calculations

are made for various values of volume fraction of nanoparticles 0 0.1 , Rayleigh number

3 610 10Ra , Hartmann number 0 60Ha and magnetic field orientation 0 00 90 . In

the numerical simulations, we have considered three different types of base fluids namely water

2H O , engine oil (EO) and kerosene with three different kinds of nanoparticles such as Cu , Co and

3 4Fe O . Streamlines and isotherms evolution with dimensionless time as well as the average

heat flux and the local Nusselt number on the heated left wall are calculated for different model parameters. The results are taken for

2Cu-H O nanofluid and then compare the average Nusselt

number with different nanofluids for different volume fractions and different nanoparticle diameters.

Figure 4 displays the streamline evolutions with dimensionless time for 510 .Ra For a

shorter time, an oval-shaped circulation is formed at the center of the cavity and the streamlines intensify towards the hot wall. As increases, the circulation zone changes to an elliptical pattern and the streamlines intensify at both the hot wall as well as the cold wall. This indicates a high downward velocity of the flow. As increases, the pattern of the streamlines show no significant change until it reaches the steady state.


200

Figure 5 shows the temperature distribution inside the cavity against for 510 .Ra For 0.05 the flow is unsteady and the contour lines are condensing at the bottom of the hot wall which indicates a high temperature gradient due to buoyancy effect. Over some time, the temperature distribution changes in the enclosure while the contour patterns show a marginal variation until it reaches the steady state. 5.1 Effects of Rayleigh number Figure 6 depicts the average Nusselt number on the left heated wall for different Rayleigh numbers

3 4 5 610 ,10 ,10 ,10Ra with dimensionless time , 0.05 4.5 . In all four cases the average

Nusselt number decreases initially. As time progress, it starts to increase and reached its maximum

value, then to the steady state. It is obvious that in the steady part, when 310Ra the heat transfer

is the lowest and as Ra increases the heat transfer increases due to the increase in the buoyancy effect. To find at which time the flow reaches the steady state, the average Nusselt number is calculated numerically and displayed graphically in Fig. 7 for different Rayleigh number. From this

figure we observe that the time taken to reach the solution in steady state is 1.55, 0.9, 0.8 and

0.55 for 3 4 5 610 ,10 ,10 ,10Ra respectively. It means that as Ra increases, the dimensionless time

the flow takes to reach the steady state decreases. Thus, strong buoyancy helps the flow to

reach steady state faster. Figure 8 displays the local Nusselt number along the hot wall of the enclosure. The results are taken at 2 (steady state) and are presented for four values of the Rayleigh numbers

3 4 5 610 ,10 ,10 ,and 10Ra with solid volume fraction 0.05, Hartmann number 20Ha and

the nanoparticle diameter 10d nm. Except in the vicinity of the top left corner of the enclosure, the results show that due to the strengthened buoyant flow the local Nusselt number increases as

the Rayleigh number increases. For 310Ra the local Nusselt number shows a horizontal line

since the conduction regime dominated. For 410Ra , the local Nusselt number shows a slight increase and then a slight decrease . This is due to the convection- dominated regime. The increase

of the local Nusselt number is noticeable for 510Ra and more significant for 610Ra at which the local Nusselt number increases sharply until it reaches a peak in the vicinity of the lower left corner and then decreases quite significantly.

Figures 9-10 display the effect of three different values of Rayleigh number 410Ra , 510 , and 610

on streamlines and isotherms respectively for both unsteady case ( 0.05) and the steady case

( 2). In these figures we have considered solid volume fraction 0.05, the nanoparticle

diameter 10 ,d nm Hartmann number 20Ha and 2Cu-H O nanofluid. The buoyancy driven

circulating flows within the enclosure are evident for all values of the Rayleigh number .Ra For 410 ,Ra a central circulation cell is observed within the enclosure as a dominant characteristic of

the flow field. However, as Ra increases, the circulation pattern gets larger and the streamlines


201

intensifies in the vicinity of the hot and cold wall which is evident of high velocity gradient and indicates a strengthen in the natural convection. The isotherms also indicate a regime where the contribution of the convective flow field in the heat transfer becomes evident. It is obvious that increasing Rayleigh number is associated with the variation of isotherms pattern. As Rayleigh number increases, the contour lines are condensing at the hot and cold walls which indicate higher temperature gradient. Moreover, the thermal boundary layer near the walls becomes thinner, indicating a higher heat transfer rate. 5.2 Effects of solid volume fraction Figure 11 presents the average Nusselt number on the left heated wall for different nanoparticles

volume fractions 0,0.05,0.025,0.1 with dimensionless time , 0.05 4.5, when

Rayleigh number is 510 , Hartmann number is 20Ha and nanoparticle diameter is 10d nm.

For small amount of nanoparticles, the average Nusselt number is high at some short time then it dropped to reach a minimum value and then rises up until it shows a constant value as the flow

reaches the steady state. For large volume fraction, 0.1 , this fluctuation in the average Nusselt

number is not significant. This is due to the conductivity dominant of the nanoparticles. At the steady state of the flow, as the solid volume fraction increases, the average Nusselt number on the left heated wall increases. This is due to the increase of the nanofluid thermal conductivity as the

fraction of the nanoparticles is increased. Figure 12 shows the dimensionless time taken to

reach the flow in steady state. From this figure we observe that the time taken to reach the

steady state of the flow is 1.05, 0.85, 0.8 and 0.65 for 0.0,0.025,0.05,0.1 respectively. Thus,

addition of nanoparticles to the base fluid helps the unsteady problem to reach it steady state. Figure 13 illustrates the local Nusselt number on the left heated wall of the enclosure for four

values of solid volume fractions 0,0.025,0.05 and 0.1 , when 510 ,Ra 20Ha and

10d nm . The results are taken for the steady state, i.e. when 2 . Along the heated wall, as the volume fraction increases, the heat transfer rate also increases. This is due to the high thermal conductivity of the nanoparticles. The pattern of the local Nusselt number lines is similar for all

values of . At vicinity of the heated wall 0 0.05Y the local Nusselt number increases to its

maximum value, then decreases along the other part of the wall. The peaks of the maximum for different occur at 0.05.Y

Figures 14-15 present the streamlines and isotherms respectively for different volume fractions

0,0.025 and 0.05 at 0.05 (unsteady case) and 2 (steady case) 5when 10 ,Ra 10d nm , 20. Ha For all values of , the flow is rotating clockwise with a circulation zone at

the center of the enclosure. At an unsteady case 0.05 , as the volume fraction increases, the

circulation at the center of the cavity becomes more uniform. At the steady case, as increases,


202

the elliptical eye of the circulation at the center of the enclosure gets smaller. The volume fraction plays an insignificant role on the shape of the isotherms as shown in Fig. 15. Nevertheless, the temperature distribution within the flow domain intensifies is varying due the increased thermal conduction ability of the nanoparticles. 5.3 Effects of nanoparticle diameter

Figure 16 displays the variation of the average Nusselt number with dimensionless time for

different nanoparticle diameters 1 ,10 ,50 and 100 .d nm nm nm nm The results are calculated on the

left heated wall of the enclosure for 0.05 0.4, 0.05 , 20Ha , 510Ra and for 2Cu-H O

nanofluid. For a short time the average heat transfer rate dropped from a high value to its minimum and then increases until it reaches to a constant value at steady state. For 100d nm , 50nm , and

10nm the pattern is similar but it shows different behavior for 1d nm at which the average heat transfer rate increases then shows a slight decrease before taking a constant value. As the nanoparticle diameter decreases, the average Nusselt number increases. This is because as the nanoparticle diameter decreases, the specific area increases which enhance the thermal

conductivity of the nanofluid to increase. Figure 17 presents the dimensionless time taken to

reach the flow in steady state for the same parameter values as in Fig. 16. For 1nm particle diameter, it takes 0.55 which is the shortest time when compared to the larger diameters. Figure 18 illustrates the effect of the nanoparticle diameter on the local Nusselt number on the left

heated wall of the enclosure. The Rayleigh number is 510Ra , Hartmann number 20Ha and the volume fraction 0.05 are kept constant. The results are taken for

2Cu-H O nanofluid at the

steady case 2 . The local Nusselt number becomes higher as the nanoparticle diameter gets

smaller. In all cases, the local Nusselt number increases near the bottom left corner of the heated wall and dropped down along the rest of the wall. Figures 19-20 show the effect of different nanoparticle diameters on the streamlines and isotherms

respectively for 2Cu-H O nanofluid. The results are calculated for the unsteady state 0.05 as

well as for the steady state 2 under the same parameters conditions as in Fig. 18. At an

unsteady case, for 1d nm , the streamlines are elongated along the diagonal and condensed at the hot left and cold right walls with an elliptical circulation at the center of the cavity. As the nanoparticle diameter increases, the circulation changes to an oval shape and the streamlines show

more condensation at the hot wall than the cold one. For the steady state 2 , the streamlines

distribution becomes more uniform and more condense for 1d nm which clearly indicates that as the particle size gets smaller, the mixture becomes more homogeneous. No significant differences in the pattern of the isotherms are noticed with the variation of d as can be seen Fig. 20. A slight deformation in contours can be observed as the nanoparticle diameter increases. Nevertheless, the hot region near the hot wall gets larger with decreasing nanoparticle diameter.


203

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Fig. 4. Streamlines evolution with dimensionless time when 015 , 030 , 0.05 ,

Pr 6.8377 , 20Ha , 10d nm , 510Ra .


204

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Fig. 5. Isotherms evolution with dimensionless time ( ) , 0.05 , 0 0 31 , ,5 0

5 Pr 6.8377, 20 , 10 , 10 .Ha d nm Ra


205

Fig. 7. Dimensionless time to reach the solution in steady state for different .Ra

Fig. 6. Average Nusselt number on the left heated wall for different Rayleigh numbers

when 015 , 030 , 0.05 , Pr 6.8377 , 20Ha , 10d nm .


206

Fig. 8. Local Nusselt number on the heated wall in the steady state for different .Ra 410Ra 510Ra 610Ra

0.05

2

Fig. 9. Streamlines for different Rayleigh numbers Ra at 0.05 (unsteady state) and 2 ( steady state ) , 0.05 , Pr 6.8377 , 20Ha , 10d nm .


207

410Ra 510Ra 610Ra

0.05

2

Fig. 10. Isotherms for different Rayleigh numbers Ra at 0.05 (unsteady state) and 2 ( steady state ) , 0.05 , Pr 6.8377 , 20Ha , 10d nm .

Fig. 11. Average Nusselt number on the left heated wall for different solid volume fractions

when 510Ra , 0.05 , Pr 6.8377 , 20Ha , 10d nm .


208

Fig. 12. Dimensionless time to reach the solution in the steady state for different .

Fig. 13. Local Nusselt number on the heated wall at steady state for different .


209

0 0.025 0.05

0.05

2

Fig. 14. Streamlines for different volume fraction at 0.05 (unsteady state) and

2 ( steady state ) , when 510Ra , Pr 6.8377 , 20Ha , 10d nm .


210

0 0.025 0.05

0.05

2

Fig. 15. Isotherms for different volume fraction at 0.05 (unsteady state) and

2 ( steady state ) , when 510Ra , Pr 6.8377 , 20Ha , 10d nm .

Fig. 16. Average Nusselt number on the left heated wall for different nanoparticle diameter

d when 510 , 6.8377, 20 , 0.05.Ra Pr Ha


211

Fig. 17. Dimensionless time to reach the solution in steady state for different .d

Fig. 18. Local Nusselt number on the heated wall at steady state for different .d


212

1d nm 10d nm 100d nm

0.05

2

Fig. 19. Streamlines for different particle diameter d at 0.05 (unsteady state) and

2 ( steady state ) , when 510Ra , Pr 6.8377 , 20Ha , 0.05 .


213

1d nm 10d nm 100d nm

0.05

2

Fig. 20. Isotherms for different particle diameter d at 0.05 (unsteady state) and

2 ( steady state ) , when 510Ra , Pr 6.8377 , 20Ha , 0.05 .

Fig. 21. Average Nusselt number on the left heated wall for different Hartmann number ( )Ha

when 510Ra , Pr 6.8377 , 10d nm , 0.05 .


214

(a)

(b)

Fig. 22. (a) Local Nusselt number on the heated wall at the steady state for different Hartmann number (left) , and (b) different magnetic field inclination angle (right).

(a)

(b)

Fig. 23. (a) Average Nusselt number with τ , and (b) local Nusselt number on the heated wall at steady state for two cases (with Brownian effect) and (without Brownian effect).


215

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 24. Average Nusselt number on the heated wall for different geometry inclination angles for 2Cu-H O nanofluid.


216

Table 4 Average Nusselt number on the heated wall for different nanofluids and different volume

fractions when 015 , 030 , 20,Ha 10 ,d nm

510 .Ra

Nanofluids

I

II (

)

III

(

)

IV

(

)

0 0.025 0.05 0.1

Water – Cu 4.2392 5.72042 34.94 6.74342 59.07 7.9734 88.09

Water – Co

4.2392 5.82343 37.37 6.90079 62.79 8.17087 92.75

Water-Fe3O4

4.2392 5.972 40.88 7.34097 73.17 9.41399 122.07

Engine oil- Cu

4.24062 4.56775 7.71 4.84118 14.16 5.25535 23.93

Engine oil- Co

4.24062 4.59174 8.28 4.88508 15.2 5.32949 25.68

Engine oil – Fe3O4

4.24062 4.63467 9.29 4.98612 17.58 5.58431 31.69

Kerosene - Cu

4.24041 8.21477 93.73 10.5253 148.21 13.23239 212.05

Kerosene- Co

4.24041 8.49111 100.24 10.93415 157.86 13.8396 226.37

Kerosene – Fe3O4

4.24041 8.34948 96.9 10.86271 156.17 14.04478 231.21


217

Table 5 Average Nusselt number on the heated wall for different nanofluids and different

nanoparticle diameters d when 015 , 030 , 20,Ha 0.05, 510 .Ra

Nanofluids

d

I

II (

)

III (

)

IV (

)

100nm 50nm 10nm 1nm

Water – Cu 5.211 5.53093 6.14 6.74342 29.41 10.27207 97.12

Water – Co

5.26535 5.60855 6.52 6.90079 31.06 10.61396 101.58

Water-Fe3O4

5.64103 5.99511 6.28 7.34097 30.14 11.29177 100.17

Engine oil- Cu 4.56453 4.61847 1.18 4.84118 6.06 5.6474 23.72

Engine oil- Co 4.58465 4.64331 1.28 4.88508 6.55 5.7555 25.54

Engine oil –

Fe3O4 4.69748 4.75375 1.2 4.98612 6.14 5.82747 24.06

Kerosene - Cu 6.88556 7.70864 11.95 10.5253 52.86 18.15617 163.68

Kerosene- Co 7.08314 7.95889 12.36387 10.93415 54.37 19.09461 169.58

Kerosene –

Fe3O4 7.08014 7.93486 12.07208 10.86271 53.43 18.68528 163.91


218

Table 6 Average Nusselt number on the heated wall for different volume fractions and different

Rayleigh number Ra for two cases (with Brownian effect) and (without Brownian effect).

5.4 Effects of magnetic field and its orientation Figure 21 depicts the effect of the magnetic field intensity on the average Nusselt number which is calculated on the left heated wall of the enclosure for 0.05 0.5, taking Hartmann number

0Ha , 20, 40, and 60, 510 ,Ra 10d nm and 0.05. As Hartmann number ( )Ha increases, the

average Nusselt number in steady state decreases. This is because the magnetic field suppresses the convective flows as the intensity of the magnetic field increases, which in turn slow down the heat transfer rate. Figure 22 displays the effect of the magnetic field intensity and its orientation on the local Nusselt number on the left heated wall. The results are calculated when the solution becomes steady

2 , for the same parameters as in Fig. 21. Figure 22(a) shows the significant effect of

Hartmann number on the local Nusselt number. In the case when no magnetic field affects the flow, the heat flux is higher on the wall and as its intensity increased, the heat flux decreases. The magnetic field inclination angle influences the heat flux as shown in figure 22(b). The local Nusselt number rises in the vicinity of the heated wall and then declines away from the wall. The effect of the magnetic field inclination angle on the top half of the heated wall is minimal.

Ra avNu

(

)

(

)

(

)

(

)

(

)

0 0.01 0.02 0.03 0.04 0.05 I II III IV V VI

310 With Brownian effect

1.009

1.2959

28.45 1.577

56.39 1.8544

83.8 2.1255

110.68 2.3909

136.98

Without Brownian effect

1.009

1.0383

2.9 1.068

5.88 1.099

8.92 1.1305

12.04 1.1626

15.22


1.599

1.8191

13.7 2.027

26.73 2.2362

39.77 2.4487

53.05 2.6652

66.59


1.599

1.6023

0.15 1.605

0.33 1.6085

0.54 1.6125

0.79 1.6175

1.1


4.233

4.9039

15.68 5.4685

29 5.9547

40.47 6.3764

50.42 6.7434

59.07


4.233

4.2675

0.67 4.295

1.3 4.3203

1.91 4.3448

2.5 4.368

3.04


9.234

10.889

17.92 12.34

33.64 13.628

47.58 14.779

60.05 15.812

71.24


9.234 9.336 1.1 9.435 2.17 9.532 3.22 9.627 4.25 9.72 5.26


219

5.5 Average heat transfer rate for different nanofluids So far we have discussed the results for water-Cu nanofluid. Here, we considered various base fluids and nanoparticles to see how the results depend on them. Table 4 illustrates the average

Nusselt number on the heated wall of the enclosure for the steady state 2 for three different

types of base fluids namely water (H2O), engine oil (EG) and Kerosene with three different kinds of

nanoparticles Cu, Co and Fe3O4 . Rayleigh number is considered to be 510 ,Ra Hartmann number

20Ha and the nanoparticle diameter 10d nm . For all nine types of nanofluids, as the volume fraction increases, the average Nusselt number increases. Kerosene-based nanofluids show the highest heat transfer when compared to the water-based and engine oil-based nanofluids. Although, Cu nanoparticle has higher thermal conductivity than Co and Fe3O4, kerosene-Co and kerosene-Fe3O4 show higher heat transfer rate. This is due to the Brownian effect on the thermal conductivity of the nanofluids. Engine oil-based nanofluids show the lowest heat transfer due to the high dynamic viscosity and low thermal conductivity of the base fluid. The effect of different nanoparticle diameters on the heat transfer rates is listed in Table 5. As the nanoparticle diameter increases, the average heat transfer rate decreases. Kerosene based nanofluids show a significant enhancement in heat transfer rate as the nanoparticle diameter decreases. For 1 nm nanoparticles diameter, it reaches 163.68%, 169.58% and 163.91% enhancement for Cu, Co and Fe3O4 when compared with 100 nm nanoparticle diameter. For water-based nanofluids, the heat transfer rate intensified by around 100% for 1nm nanoparticle diameter compared with 100 nm particle diameter. Engine oil-based nanofluids show around 24 % increased heat transfer for 1 nm particle diameter compared with 100 nm particle diameter. This is due to the high dynamic viscosity of engine oil which suppresses the Brownian motion of the nanoparticles. 5.6 Effect of Brownian motion All the results obtained are by taking into account the Brownian motion effect on the effective thermal conductivity of the nanofluid as shown in equation (14). In order to examine the impact of the Brownian motion on heat transfer rates, we have calculated the average Nusselt number and local Nusselt number on the heated wall for two cases (with Brownian motion and without

Brownian motion) considering Rayleigh number 510Ra , Hartmann number 20Ha and the

nanoparticle diameter 10d nm for water-Cu nanofluid. The results are illustrated in Fig. 23. It is obvious that the Brownian motion plays a significant role in enhancement of the heat transfer rate. The contribution of Brownian motion of nanoparticles to the heat transfer is due to the movement of the nanoparticles which can transfer heat and micro-convection of the fluid around individual nanoparticles. Table 6 presents a comparison study for the increase in the average Nusselt number

for the nanofluid with respect to pure water at various Rayleigh numbers 3 610 10Ra and solid

volume fractions 0 0.05 The influence of on the average Nusselt number avNu is more

significant at low Rayleigh numbers for both models with and without Brownian motion. For

example, at 310Ra , for the nanofluid with 0.05 , avNu increased by 136.98% when Brownian

motion is taken into account and by 15.22% when neglected it.


220

5.7 Effect of geometric inclination angle on optimum heat transfer

An average Nusselt number is calculated on the heated wall at the steady state 2 for different

geometry inclination angles 0 90 . The other model parameters are kept fixed such as510 , 0.05, 10 , 20Ra d nm Ha , and =30 for

2Cu-H O nanofluid. It is found that as

increases, the average heat transfer rate increases until it reaches a peak at an angle of 41 at which maximum heat transfer occurs. The heat transfer then decreases until it reaches a minimum value

at an angle of 90 where the hot wall becomes horizontal. The critical angle is influenced by the buoyancy force and the Lorenz force. As the Rayleigh number varies as displayed in Fig. 24 (a), the

critical angle becomes 18 and 26 for 4 510 ,10Ra respectively. For

310 ,Ra the heat transfer

rate shows a slight decreasing linear behavior. The critical angle becomes 35 ,37 ,38 ,39 ,40

when the volume fraction varied to 0, 0.01, 0.02, 0.03 and 0.04 respectively. Which means as the

volume fraction increased by 1%, the critical angle also increased by 1 as presented in Fig. 24 (b).

The critical angle remains 39 when the nanoparticle diameter d varied from 50nm to 100nm and it remains unchanged when 1 .d nm This result is depicted in Fig. 24 (c). It is noted from Fig. 24

(d) that, in the absence of any applied magnetic field 0Ha to the cavity, the critical angle

becomes 29 whereas it becomes 37 and 27 for 40Ha and 60 respectively. When the

magnetic field is inclined with different angles 0 ,30 ,45 ,60 ,90 as in Fig. 24 (e), a

significant result is observed. The critical angle is slightly changed, while some lines of different

intersect with each other’s which indicates that the heat flux is equivalent for different values of

at a specific value of . For instance, the lines of 0 and 90 intersect at 80 which

means that at a geometry inclination angle of 80 , when the magnetic field is applied horizontally

or vertically, the heat flux for both cases are equal. The above-results are calculated when the Brownian motion of the nanoparticles is taken into account. When the effect of the Brownian

motion is not considered, the critical angle shifted to 37 as shown in Fig. 24 (f). From these figures we also observe that, the distribution of the average heat transfer rate for different cavity inclination angle along with various model parameters has been found almost parabolic shape which is investigated in the present research for the first time.

6. Conclusions In this paper, the problem of unsteady natural convection flow and heat transfer enhancement under the influence of an orientated magnetic field in an inclined square enclosure filled with various types of nanofluids is studied numerically. The flow and heat transport structure are presented in terms of streamlines and isotherm, respectively. Results for different situations such

as the Rayleigh number ,Ra the nanoparticles volume fraction , the nanoparticle diameter

,d Hartmann number ( )Ha , the magnetic inclination angle , geometric angle ( ) as well as


221

the Brownian motion of nanoparticles are presented and discussed. Heat transfer enhancement for nine different nanofluids is obtained and explained for different model parameters. In view of the acquired outcomes, the following findings are listed: (1) The time taken to reach the steady state of the flow is controlled by different parameters

namely, the Rayleigh number ,Ra the volume fraction , and the nanoparticle diameter

d .

(2) As the Rayleigh number increases, the flow and thermal field varied and the heat transfer increases significantly.

(3) As the volume fraction of nanoparticle increases, the flow and thermal fields modify slightly. However, the heat transfer rate increases remarkably.

(4) The flow and thermal fields change slightly as the nanoparticle diameter decreases while the heat transfer increases rapidly due to the increase in the specific surface area of the nanoparticles.

(5) The magnetic field intensity has a noticeable influence on the heat flux but its inclination angle shows a marginal effect on it.

(6) Kerosene-based nanofluid has a remarkable heat transfer rate when compared to water or engine oil based nanofluids.

(7) Due to the random movement of the nanoparticles, the micro-convection around individual nanoparticles intensifies. Thus, Brownian motion plays a significant role in enhancement of heat transfer rate.

(8) When the cavity inclination angle increases from 0 to 90 , the average heat flux increases until

it reaches an optimum value then decreases making almost a downward parabolic shape. (9) The critical point for the maximum heat transfer rate is highly influenced by the Rayleigh

number, nanoparticle volume fraction and Hartmann number. However, it is slightly affected due to the change in nanoparticle diameter and magnetic field inclination angle.

(10) At some cavity inclination angles, the heat flux is equivalent for different magnetic field inclination angles.

(11) The Brownian motion of nanoparticles, which is a key mechanism in the thermal conductivity enhancement of nanofluid, has significant effect on the heat transfer enhancement.

Nomenclature

0B magnetic field strength [kg·s−2·A−1] Greek symbols

pc specific heat [J.kg-1K-1] geometry inclination angle [deg]

d particle diameter [m] thermal expansion coefficient [K-1] g gravitational acceleration [m.s-2] magnetic field inclination angle [deg] Ha Hartmann number dynamic viscosity [kg.m-1.s-1] k thermal conductivity [W.m-1.K-1] kinematic viscosity [m2.s-1]

BK Boltzmann constant [J.K-1] density [kg.m-3]

L enclosure length [m] electrical conductivity [S.m-1] Nu Nusselt number dimensionless temperature p dimensional fluid pressure [Pa] nano-particle volume fraction

P dimensionless fluid pressure dimensionless time Pr Prandtl number subscripts


222

Ra Rayleigh number av average T temperature [K] C cold wall

,u v dimensional velocity components [m.s-1] f base fluid

,U V dimensionless velocities [m.s-1] H hot wall ,x y dimensional coordinates [m] nf nanofluid

,X Y dimensionless coordinates p solid particle

Acknowledgments M. M. Rahman would like to thank The Research Council (TRC) of Oman for funding under the Open Research Grant Program ORG/SQU/CBS/14/007. K. S. Al Kalbani is grateful to TRC for a Doctoral Fellowship. M. S. Alam is also grateful to TRC for a Post-Doctoral Fellowship.

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finite element analysis of unsteady natural …paper.uscip.us/ajhmt/ajhmt.2016.1012.pdffinite...

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