# Study of Natural Convection of Heat Transfer in A Vertical Cone Embedded with Porous Medium

DOI : 10.17577/IJERTV3IS060884

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#### Study of Natural Convection of Heat Transfer in A Vertical Cone Embedded with Porous Medium

Department of Mathematics, Faculty of Science, P.O.Box 741, University of Tabuk,

Zip. 71491, Kingdom of Saudi Arabia

Abstract: In this paper, we concentrate on the study of natural convection of heat transfer confined in a vertical cone embedded with porous medium. Finite Element Method (FEM) has been used to solve the governing partial differential equations. Results are presented in terms of

average Nusselt number ( Nu ), streamlines and Isothermal

lines for various values of Rayleigh number (Ra), Cone angle (CA) & Radius ratio (Rr).

Keywords: Porous medium, Nusselt number ( Nu ), Rayleigh number (Ra), Cone angle (CA) & Radius ratio (Rr).

Hollow portion

Porous media ro

ri Porous

media

z

Fig. (A) : Schematic diagram of Vertical Cone Filled with porous medium

r

Fig. (B) Mesh Pattern of Vertical Cone embedded with porous medium

Nomenclature: List of Symbols:-

= Cone Angle

= Specific heat

= Particle diameter

g = Gravitational acceleration

Ht = Height of the vertical annular cone K = Permeability of porous media

P = Pressure

Greek Symbols:-

= Thermal diffusity

T =Co-efficient of thermal expansion

= Viscous dissipation parameter

T = Temperature difference

= Stephan Boltzman constant

= Density

= Coefficient of Kinematic viscosity

=Coefficient of dynamic viscosity

= Porosity

= Stream function

Nu = Average Nusselt number

= Non-dimensional stream function

= Total heat flux

r, z = Cylindrical co-ordinates

r z = Non-dimensional co-ordinates ri , ro = Inner and outer radius

Ra =Rayleigh number Rr = Radius ratio

Rd = Radiation parameter T= Temperature

T =Non-dimenstional Temperature u = Velocity in r direction

w = Velocity in z direction

Subscripts:-

= Wall

= Conditions at infinity h = Hot

c = Cold

t = Total

1. INTRODUCTION

Natural convective heat transfer in porous media has received considerable attention during

the past few decades. This interest can be attributed to its wide range of applications in ceramic processing, nuclear reactor cooling system, crude oil drilling, chemical reactor design, ground water pollution and filtration processes. The work on free convection about a vertical imperiable flat plate are studied by Cheng and Minkowycz [1], Cheng [2], Na and Pop [3] Gorla and Zinalabedini [4]. The vertical cylinder cases are investigated by Minkowycz and Cheng [5], Kumari et.al [6] and Bassom et.al [7] Cheng et al. [8] use the local non-similarity method to analyze the natural convection of Darcian fluid about a cone. External free convection in a porous medium adjacent to heated bodies was analyzed by Nield and Bejan [9], Merkin [10, 11], Minkowycz and Cheng [12, 13, 14], Pop and Cheng [8, 15], Ingham and Pop [16]. All through these studies, it is assumed that the boundary layer approximations are applicable and the coupled sets of governing equations are solved by numerical methods.

In the present work, we study the problem of natural convection of heat transfer in a inverted cone embedded with porous medium . In the case of full cone similarity solutions exist if the prescribed wall temperature or surface

heat flux is a power function of distance from the vertex of the inverted cone [8, 9, 17, 18]. Bejan and Khair [19] used Darcys law to study the vertical natural convective flows driven by temperature and concentration gradients. Nakayama and Hossain [20] applied the integral method to obtain the heat and mass transfer by free convection from a vertical surface with constant wall temperature and concentration. Yih [21] examined the coupled heat and mass transfer by free convection over a truncated cone in porous media for variable wall temperature and variable heat and mass fluxes, Also he [22] applied the uniform transpiration effect on coupled heat and mass transfer in mixed convection about inclined surfaces in porous media for the entire regime. Cheng [23] used an integral approach to study the heat and mass transfer by natural convection from truncated cones in porous media with variable wall temperature and [24] studies the Soret and Dufour effects on the boundary layer flow due to natural convection heat and mass transfer over a vertical cone in a porous medium, saturated with Newtonian fluids with constant wall temperature. Natural convective mass transfer from upward-pointing vertical cones, embedded in saturated porous media, was studied using the limiting diffusion [25]. The natural convection along with an isothermal wavy cone embedded in a fluid-saturated porous medium were presented in [26, 27]. Singh and Queeny [28] applied the integral method to obtain the heat and mass transfer by

free convection from a vertical surface with constant wall temperature and concentration. In [17, 18] fluid flow and heat transfer of vertical full cone embedded in porous media were solved by Homotopy analysis method.

2. MATHEMATICAL FORMULATION

A vertical annular cone of inner radius ri and outer radius r0 as depicted by schematic diagram as shown in figure (A) is considered to investigate the heat transfer behavior. The co-ordinate system is chosen such that the r- axis points towards the width and z-axis towards the height of the cone respectively. Because of the annular nature, two important parameters emerge which are Cone angle (CA) and Radius ratio (Rr) of the annulus. They are defined as

A

r

t

C Ht , R r0 ri , where H is the height of the cone.

r0 ri ri

The inner surface of the cone is maintained at isothermal temperature and outer surface is at ambient temperature

. It may be noted that, due to axisymmetry, a section of the annulus is sufficient for analysis purpose.

We assume that the flow inside the porous medium is assumed to obey Darcy law and there is no phase change of fluid. The properties of the fluid and porous medium are homogeneous, isotropic and constant except for variation of fluid density with temperature. The fluid and porous medium are in thermal equilibrium.

Continuity equation:

(ru)

r

(rw) 0

z

(2.1)

The velocity in r and z directions can be described by Darcy law as

Velocity in horizontal direction

u K p

(2.2)

r

Velocity in vertical direction

K p

w z g

(2.3)

The permeability K of porous medium can be expressed as Bejan (29)

D2 3

K p

180(1 )2

(2.4)

The variation of density with respect to temperature can be described by Boussinesq approximation as

= [1 – T (T – T) ] (2.5)

Momentum Equation :

w u gK T

r z r

(2.6)

Energy equation

T T

1

T

2T

u r

• w z

r r r r z 2

(2.7)

The last term in the right hand side of the equation (2.7) represents radiation effect.

The continuity equation (1) can be satisfied by introducing the stream function as

u 1

(2.8)

r z

w 1

(2.9)

The corresponding dimensional boundary conditions are

r r

at r = ri , T = Tw , = 0 (2.10a)

at r = r0 , T = T , = 0 (2.10b)

(except at z= 0)

The new parameters arising due to cylindrical co-ordinates system are

Non-dimensional Height

r (2.11a)

r

z

L

z (2.11b)

L

Non-dimensional stream function

L

(2.11c)

Non-dimensional Temperature

T (T T )

(Tw T )

(2.11d)

Rayleigh number

Ra gT TKL

(2.12e)

The non-dimensional equations for the heat transfer in vertical cone are

2 1 T

Momentum equation: r rRa

(2.13)

r

z 2

r r

Energy equation :

1 T T 1 T 2 T

r 2

(2.14)

r r z

z r

r r

r

z

The corresponding non-dimensional boundary conditions are

at r = ri , T=1 , = 0 (2.15)

at r = r0 , T=0 , = 0 (2.16)

3. SOLUTION OF GOVERNING EQUATIONS:

Equations (2.13) and (2.14) are coupled partial differential equations to be solved in order to predict the heat transfer behavior. These equations are solvied by using FEM. A simple 3-noded triangular element is considered.

Applying Galerkin method to momentum equation (2.13) yields:

e T 2 1 T

R N r rRa d

(3.1)

A

z 2

r r

r

r

e T 2

1

T

R

N r rRa 2rdA

(3.2)

A

z 2

r r

r

r

Where Re is the residue. Considering the individual terms of equation (3.2)

The differentiation of following term results into

[N T

] [N T

2

]

[N T ]

(3.3)

r r

T 2

r 2

T

r r

2

[N T ]

Thus

N 2

dA

[N

r

] 2 2rdA

r r

(3.4)

A r A r A

The first term on right hand side of equation (3.4) can be transformed into surface integral by the application of Greens theorem and leads to inter-element requirement at boundaries of an element. The boundary conditions are incorporated in the force vector.

Let us consider that the variable to be determined in the triangular area as T.

The polynomial function for T can be expressed as

T = 1 + 2 r + 3z (3.5)

The variable T has the value Ti, Tj & Tk at the nodal position i, j & k of the element. The r and z co-ordinates at these points are ri, rj, rk and zi, zj, zk respectively. Since T = Ni Ti + Nj Tj + Nk Tk (3.6)

Where Ni, Nj & Nk

are shape functions given by

N am bm r cm z

m 2A

(3.7)

1

T 2

N T

N

Making use of (3.7) gives

N r 2

2rdA

r r 2 dA

(3.8)

A A

3

b1

1

Substitution of (3.7) into (3.8) gives

1 b

b b b

2rdA

(2 A)2 2

1 2 3

2

A b

b2

3

b b b b

3

2R 1

1 2 1 3

1

b b b2 b b

(3.9)

4A 1 2 2

2 3 2

b b b b

b2

1 3 2 3

3 3

c2 c c

c c

2

2R 1

1 2 1 3

1

z2 4 A 1 2 2

2 3

2

Similarly, N T 2rdA

A

c c c2 c c

c c c c c2

(3.10)

1 3 2 3

3 3

The third term of equation (3.2) gives

N T A

rRa T 2rdA Ra

r A

N T r T 2rdA

r

(3.11)

Since M1 = N1, M2 = N2, M3 = N3

Where M1, M2 and M3 are the area ratios of the triangle and N1, N2 and N3 are the shape functions.

Replacing the shape functions in the above equation (3.11) gives

M1

T 1

N T rRa T 2rdA rRa

M (N ) 2

rdA

(3.12)

r

2 r T 2

A A M

T 3

1 2

3

T 1

Ra A 1 2R b b b 2

3 2 A 1 2

3 T

1

T 3

2 b1T 1 b2 T 2 b3 T 3

2R Ra

b1T 1 b2 T 2 b3 T 3

6

b T 1 b T 2 b T 3

(3.13)

1 2 3

Now the momentum equation (3.13) leads to

b2 b b

b b

c2 c c

c c

2

b T 1 b T 2 b T 3

2R 1

1 2 1 3 1

1 2 1

3 1

2R

Ra 1 2

3

b b b2 b b

• c c c2 c c

b T 1 b T 2 b T 3 0

(3.14)

4 A 1 2 2

2 3

1 2 2

2 3 2

6 1 2 3

b b b b

b2

c c c c

c2

b T 1 b T 2 b T 3

1 3 2 3 3

1 3 2 3

3 3

1 2 3

Which is in the form of the stiffness matrix

[Ks] { }= {f}

Similarly application of Galerkin method to Energy equation (2.14) gives

e T 1 T

T

1

T

2 T

R

N r

2rdA

(3.15)

r r z

z r

r r

r

z 2

A

Considering the terms individually of the above equation (3.15) Thus the stiffness matrix of Energy equation (2.14) is given by:

c1 1 c2 2 c3 3

b1 1 b2 2 b3 3

T 1

2

c

• c

b , b , b 2

b

• b

c , c , c

12A c1 1

2 2

3 3 1 2 3

12A b1 1

2 2

3 3 1 2

3 T 2

c

• c

c

b

• b

b

T 3

1 1 2 2

3 3

1 1 2 2

3 3

b2 b b

b b

T 1

c 2 c c

c c

T 1

2R 1

1 2 1 3 1

1 2 1 3

• b b b2 b b T 2 c c

c 2 c c T 2 0

(3.16)

4 A 1 2 2

2 3

1 2 2 2 3

b b b b

b3

T 3

c c c c

c 2

T 3

1 3 2 3

3

1 3 2 3

3

4. RESULTS AND DISCUSSION:

Results are obtained in terms of the average Nusselt number ( ) at hot wall for various parameters such as Rayleigh number(Ra), Cone angle (CA), Rayleigh number (Ra) and Radius ratio (Rr) when heat is supplied to vertical cone embedded with porous medium.

z

T

The average Nusselt number ( ), is given by =

r

0

a)

b)

c)

Fig: 2. Streamlines (left) and Isotherms(Right) for Ra=100, Rr=1 a) CA =15 b) CA =45 c) CA =75

a)

b)

c)

Fig: 2. Streamlines (left) and Isotherms (Right) for Ra=100, CA =15

1. Rr=1 b) Rr=5 c) Rr=10

a)

b)

c)

Fig:3. Streamlines(left) and Isotherms (Right) for Ra=100, CA =75

a) Rr=1 b) Rr=5 c) Rr=10

a)

b)

c)

Fig:4. Streamlines(left) and Isotherms (Right) for CA =75, Rr=1 a) Ra=25 b) Ra=75 c) Ra=100

Fig.1 shows the evaluation of streamlines and isothermal lines inside the porous medium for various values of Cone angle (CA) at Ra = 100, Rr = 1. The magnitude of the streamlines decreases with the increase in Cone angles(CA), this is because the thermal bounded layer thickness decreases with the increase of Cone angles (C). It can be seen from streamlines and isothermal lines that the fluid movements shifts from lower portion of the hot wall to upper portion of the cold wall of the vertical annual cone with the increase of Cone angles (CA). The circulation of the fluid covers almost whole domain at both lower and higher values of Cone angles (CA) at 15o. Where the relation inversely proportion exists between streamlines and Cone angles (CA). This trend is also observed with isothermal lines.

Fig.2 predicts the streamlines and isothermal lines inside the porous medium for various values of Radius ratio (Rr) at Ra = 100 and CA = 15. It can be observed that be horizontal scale changes for various values of Radius Ratio(Rr). The magnitude of the streamlines decrease with the increase in Radius ratio (Rr). The thermal boundary layer thickness decreases with the increase in Radius ratio (Rr). It can be seen from the streamlines and isothermal lines that the fluid movement shifts from lower portion of the hot wall to the upper portion of the cold of the vertical annular cone with the increase in Radius ratio (Rr). The circulation of fluid covers almost whole domain at both lower and higher values of Radius ratio (Rr).

Fig.3 analysis the streamlines and isothermal lines inside the porous medium for various values of Radius ratio (Rr) at Ra = 100 and CA = 75. It is seen that the streamlines and the isothermal lines tends to move away from the cold wall and reaches nearer to the hot wall of the vertical annular cone. The thermal boundary layer becomes thinner with the increasing Radius ratio (Rr).

Fig.4 indicates the streamlines and isothermal lines inside the porous medium for various values of Rayleigh number (Ra) at CA = 75 and Rr = 1. The thermal boundary layer becomes thinner with the increase Rayleigh number (Ra). So, the streamlines and isothermal lines tends to move away from the cold wall and reaches nearer to the hot wall of the vertical annular cone.

Fig.5 illustrates the effect of Rayleigh number (Ra) on the average Nusselt number ( ). This Figure is obtained for value of Rr = 1. When cone angle is increased from 15 to 75, at the hot wall of the vertical annular cone, it is found that the average Nusselt number ( )at Ra = 10 is increased by 23.3%. The corresponding increase in average Nusselt number ( ) at Ra =100 is found to be 26.3%. The difference between the average Nusselt number ( ) at two different values of Cone angle (CA) increases with increase in Cone angle (CA). This is due to the reason that high cone angle produces high buoyancy force, which leads to increased fluid movements and thus increased the average Nusselt number ( )with Rayleigh number (Ra) as expected. This increase is almost linear for Cone angles (CA) 15 & 45 degrees.

Rr=1

Fig.5 variation with Ra at hot surface for different values of CA at

Fig.6 shows the variation of average Nusselt number ( )at hot wall with respect to Rayleigh number (Ra). This Figure is obtained for the value of CA = 75. When Radius ratio (Rr) is increased from 1 to 10 at the hot wall of the vertical annular cone, it is found that the average Nusselt number ( ) at Ra = 10 is increased by 20%. The corresponding increases in average Nusselt number ( ) at Ra = 100 is found to be 21%. The difference between the average Nusselt number ( ) at two difference values of Radius ratio (Rr) increases with increase in Radius ratio (Rr). High Radius ratio (Rr) produces high buoyancy force, which leads to faster fluid movements and thus increased the average Nusselt number ( ). i.e., for a given Rayleigh number (Ra) Nusselt number ( ) increases with Radius ratio (Rr).

Fig.7 illustrates the effect of Radius ratio (Rr) on the average Nusselt number ( ).This Figure corresponds to the value Ra = 100. It is seen that the average Nusselt number ( ) at hot wall of the vertical annular cone increases with increase in Radius ratio (Rr). It is found that the average Nusselt number ( ) at Rr = 1 increased by 9.2% when Cone angle (CA) increased from 15 to 45. The corresponding increase in average Nusselt number ( ) at Rr = 10 is found to be 9.8%. This difference between the average Nusselt number ( ) at two different value of Cone angle (CA) increases with increase Cone angle (CA). This difference becomes more prominent with the increase in Radius ratio (Rr) for higher values of Cone angle (CA).

Fig.8 shows the variation of average Nusselt number ( ) at hot wall with respect to Radius ratio (Rr). This Figure is obtained for the value of CA = 75. The average Nusselt number ( ) at hot wall of the vertical annular cone increases with increase in Radius ratio (Rr). It is found that the average Nusselt number ( ) at Rr = 1 increases by 5.8% when Rayleigh number (Ra) is increased from 25 to 100. The corresponding increase in average Nusselt number ( ) at Rr = 10 is found to be 10.8%. This difference between the average Nusselt number ( ) at Rayleigh number (Ra) increase with increase in Rayleigh number (Ra). This difference increases with increase in Radius ratio (Rr) for higher value of Rayleigh number (Ra). The nonlinearity of the curves increases as Rayleigh number (Ra) increases indicating higher curvature activity in the cavity.

Fig.6 variation with Ra at hot surface for different values of Rr at CA = 75

r A

Fig.7 variation with R at hot surface for different values of C

at Ra =

Fig.8 variations with Rr at hot surface for different values of Ra at CA =

100 75

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