Visualization of heat transfer by free convection with radiation in a vertical cone embedded with...

Reprint ISSN 0973-9424

I J OM S

A EA

NTERNATIONAL OURNAL F

ATHEMATICAL CIENCES

ND NGINEERING

PPLICATIONS

(IJMSEA)

PUNE, INDIA

lA

SC

ENT PUBLICA

TIO

Nl

www.ascent-journals.com

@

ASCENT

International J. of Math. Sci. & Engg. Appls. (IJMSEA)ISSN 0973-9424, Vol. 8 No. II (March, 2014), pp.237-255

VISUALIZATION OF HEAT TRANSFER BY FREE

CONVECTION WITH RADIATION IN A VERTICAL CONE

EMBEDDED WITH POROUS MEDIUM

N. AMEER AHMAD1 & M. AYAZ AHMAD2

1 Department of Mathematics,Faculty of Science, P.O.Box 741,

University of Tabuk, Zip. 71491, Kingdom of Saudi Arabia,E-mail: [email protected]

2 Department of Physics,Faculty of Science, University of Tabuk, Tabuk, KSA

Abstract

In this paper, we concentrate on the study of heat transfer by Free convection in-cluding Radiation confined in a vertical cone embedded with porous medium. Inthis study, Finite Element Method (FEM) has been used to solve the governing par-tial 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) and Radiation parameter (Rd).

−−−−−−−−−−−−−−−−−−−−−−−−−−−−

Key Words : Nusselt number (Nu), Rayleigh number (Ra), Cone angle (CA), Radius ratio (Rr)

and Radiation parameter (Rd).

c© http: //www.ascent-journals.com

237

238 N. AMEER AHMAD & M. AYAZ AHMAD

Nomenclature

List of Symbols Greek SymbolsCA = Cone Angle α = Thermal diffusityCp = Specific heat βT =Co-efficient of thermal expansionDp = Particle diameter ε = Viscous dissipation parameterg = Gravitational acceleration ∆T = Temperature differenceHt = Height of the vertical annular cone σ = Stephan Boltzman constantK = Permeability of porous media ρ = DensityP = Pressure γ = Coefficient of Kinematic viscosityNu = Average Nusselt number µ= Coefficient of dynamic viscosityqt = Total heat flux φ = Porosityr, z = Cylindrical co-ordinates ψ = Stream functionrz = Non-dimensional co-ordinates ψ = Non-dimensional stream functionri, ro = Inner and outer radiusRa =Rayleigh number SubscriptsRr = Radius ratio ω = WallRd = Radiation parameter ∞ = Conditions at infinityT= Temperature h = HotT = Non-dimensional Temperature c = Coldu = Velocity in r direction t = Totalw = Velocity in z direction

1. Introduction

Free convection flows under the influence of a gravitational force have been investi-

VISUALIZATION OF HEAT TRANSFER BY... 239

gated most extensively because they occur frequently in nature as well as in science

and engineering applications. When a heated surface is in contact with the fluid, the

result of temperature difference causes buoyancy force, which induces free convection

heat transfer. From a technological point of view, the study of convection heat transfer

from a cone is of special interest and has wide range of practical applications. Study of

buoyancy induced convection flow and heat transfer in fluid saturated porous medium

has recently attracted considerable interest because of a number of important energy

related engineering and geophysical applications such as thermal insulation of build-

ings, geothermal engineering, and enhanced recovery of petroleum resources, filtration

processes, ground water pollution and sensible heat storage beds.

Free convection about a vertical flat plate embedded in a porous medium at high

Rayleigh numbers was analyzed by Cheng and Minkowycz [1]. Na and Pop [2] stud-

ied free convection flow past a vertical flat plate maintained at a non-uniform surface

temperature embedded in a saturated porous medium and presented numerical results

by employing a two-point finite difference method. Gorla and Zinalabedini [3] stud-

ied the free convection from a vertical plate embedded in a saturated porous medium.

Many studies have appeared concerning the interaction of radiative flux with thermal

convection flows. In view of this, Ghosh and Beg [4] have analysed theoretically the

radiative effects on transient free convection heat transfer past a hot vertical surface

in porous media. Cortell [5] have highlighted the effect of the internal heat generation

and radiation on a certain free convection flow. Badruddin et al [6] studied the free

convection and radiation for a vertical wall with varying temperature embedded in a

porous medium. Salleh et al [7] have investigated numerically the free convection over

a permeable horizontal flat plate embedded in a porous medium with radiation effects

and mixed thermal boundary conditions.

In the aspect of vertical cylinder, Minkowycz and Cheng [8] were the first authors to

present free convection about vertical cylinder embedded in a porous medium. Yucel

[9] employed an implicit finite difference method to examine the free convection about

a vertical cylinder in a porous medium. Kumari et al. [10] used finite difference method

and improved perturbation solution for free convection on a vertical cylinder embedded

in a saturated porous medium. Merkin [11] investigated the free convection from an

isothermal vertical cylinder in a saturated porous medium. Bassom and Rees [12] ex-


tended the work of Merkin [11] to investigate the variable wall temperature case. The

governing equations are also solved numerically using the keller box method.

The free convection and the existence of the temperature difference between the surface

and the ambient causes the radiation effect may become important. Hossain and pop [13]

investigated the effect of radiation on Darcys buoyancy induced flow along an inclined

surface placed in porous media employing the implicit finite difference method together

with keller box elimination technique. Steady twodimensional natural convection flow

through a porous medium bounded by a vertical infinite porous plate in the presence of

radiation is considered by Raptis [14].

Heat transfer by mixed convection in laminar boundary-layer flow has been analyzed

extensively for flat plate geometry in saturated porous media in vertical, horizontal, and

inclined orientations. Same results can be found, in [15-18]. On the other hand, heat

transfer by simultaneous natural convection and thermal radiation has not received as

much attention. This is unfortunate because thermal radiation will play a significant

role in the overall surface heat transfer in situations where convection heat transfer

coefficients are small, as is the case of natural convection. Viskanta and Grosh [19]

considered the effects of thermal radiation on the temperature distribution and the heat

transfer in an absorbing and emitting media over a wedge by using the Rosseland diffu-

sion approximation. Natural convection radiation over horizontal surfaces was presented

by Ali et al. [20]. Bakier and Gorla [21] considered the effect of thermal radiation on

the mixed convection from horizontal surfaces in saturated porous media. Hassain et al

[22] investigated the natural convection, radiation interaction on boundary layer along

an isothermal plate inclined at a small angle to the horizontal. Recently A. Y. Bakier

[23] investigated the thermal radiation effect on mixed convection from vertical surfaces

in saturated porous media.

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

CA =Ht

r0 − r1, Rr =

r0 − riri

where Ht is the height of the cone.

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

surface is at ambient temperature T∞. 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 temper-

ature. The fluid and porous medium are in thermal equilibrium.

Continuity equation :∂(ru)∂r

+∂(rw)∂z

= 0. (2.1)

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

velocity in horizontal direction

u =−Kµ

∂p

∂r(2.2)

velocity in vertical direction

w =−Kµ

(∂p

∂z+ ρg

)(2.3)

the permeability K of porous medium can be expressed as Bejan (23)

K =D2pφ

3

180(1− φ)2. (2.4)

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

approximation as

ρ = ρ∞[1− βT (T − T∞)]. (2.5)

Momentum Equation :∂w

∂r− ∂u

∂z=gKβ

v

∂T

∂r. (2.6)

Energy equation

u∂T

∂r+ w

∂T

∂z= α

(1r

∂

∂r

(r∂T

∂r

)+∂2T

∂z2

)− 1ρCp

1r

∂

∂r(rqr) (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 ψ as

u = −1r

∂ψ

∂z(2.8)

w =1r

∂ψ

∂r. (2.9)

Rosseland approximation for radiation is (23)

qr =4n2σ

3βR∂T 4

∂r. (2.10)

The corresponding dimensional boundary conditions are

at r = ri, T = Tw, ψ = 0 (2.11a)

at r = r0, T = T∞, ψ = 0 (2.11b)

(except at z = 0).

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

Non-dimensional Radius r = rL (2.12a)

Non-dimensional Height z = zL (2.12b)

Non-dimensional stream function ψ = ψαL (2.12c)

Non-dimensional Temperature T = (T−T∞)(Tw−T∞) (2.12d)

Rayleigh number Ra = gβT ∆TKLvα (1.12e)

Radiation parameter Rd = 4σn2T 3c

βRKs(2.12f)

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

Momentum equation :∂2ψ

∂z2 + r

(1r

∂ψ

∂r

)= rRa

∂T

∂r(2.13)

Energy equation :

1r

[∂ψ

∂r

∂T

∂z− ∂ψ

∂z

∂T

∂r

]=

(1r

∂r

∂r

((1 +

4Rd3

)r∂T

∂r

)+∂2T

∂z2

). (2.14)


The corresponding non-dimensional boundary conditions are

at r = ri, T = 1, ψ = 0 (2.15)

at r = r0, T = 0, ψ = 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:

{Re} = −∫ANT

(∂2ψ

∂z2 + r∂

∂r

(1r

∂ψ

∂r

)− rRa∂T

∂r

)dv (3.1)

{Re} = −∫ANT

(∂2ψ

∂z2 + r∂

∂r

(1r

∂ψ

∂r

)− rRa∂T

∂r

)2

∏rdA (3.2)

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

The differentiation of following term results into

∂

∂r

([NT ]

∂ψ

∂r

)= [NT ]

∂2ψ

∂r2+∂[NT ]∂r

∂ψ

∂r. (3.3)

Thus ∫ANT ∂

2ψ

∂r2dA =

∫A

∂

∂r

([NT ]

∂2ψ

∂r2

)2

∏rdA−

∫A

∂[NT ]∂r

∂ψ

∂r. (3.4)

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 = α1 + α2r + α3z. (3.5)

The variable T has the value Ti, Tj and Tk at the nodal position i, j and k of the element.

The r and z co-ordinates at these points are ri, rj , rk and zi, zj , zk respectively.Since

T = NiTi +NjTj +NkTk (3.6)


where Ni, Nj and Nk are shape functions given by

Nm =am + bmr + cmz

2A. (3.7)

Making use of (3.7) gives

∫ANT ∂

2ψ

∂r22

∏rdA = −

∫A

∂NT

∂r

∂N

∂r

ψ1

ψ2

ψ3

dA (3.8)

Substitution of (3.7) into (3.8) gives

=1

(2A)2

∫A

b1

b2

b3

[b1b2b3]

ψ1

ψ2

ψ3

2∏

rdA

= −2∏R

4A

b21 b1b2 b1b3

b1b2 b22 b2b3

b1b3 b2b3 b23

ψ1

ψ2

ψ3

(3.9)

Similarly,

∫ANT ∂

2ψ

∂z2 2∏

rdA = −2∏R

4A

c21 c1c2 c1c3

c1c2 c22 c2c3

c1c3 c2c3 c23

ψ1

ψ2

ψ3

(3.10)

The third term of equation (3.2) gives∫ANT rRa

∂T

∂r2

∏rdA = Ra

∫ANT r

∂T

∂r2

∏rdA. (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

∫AnT rRa

∂T

∂r2

∏rdA = Ra

∫A

M1

M2

M3

∂(N)∂r

T 1

T 2

T 3

2∏

rdA (3.12)


= RaA

3

1

1

1

2∏R

2

2A[b1 + b2 + b3]

T 1

T 2

T 3

=2

∏R

2Ra

6

b1T 1 + b2T 2 + b3T 3

b1T 1 + b2T 2 + b3T 3

b1T 1 + b2T 2 + b3T 3

.

(3.13)

Now the momentum equation (3.13) leads to

∏R

4A

b21 b1b2 b1b3

b1b2 b22 b2b3

b1b3 b2b3 b23

+

c21 c1c2 c1c3

c1c2 c22 c2c3

c1c3 c2c3 c23

ψ1

ψ2

ψ3

+2

∏R

2Ra

6

b1T 1 + b2T 2 + b3T 3

b1T 1 + b2T 2 + b3T 3

b1T 1 + b2T 2 + b3T 3

= 0. (3.14)

which is in the form of the stiffness matrix

[Ks]{ψ} = {f}

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

{Re} = −∫ANT

[1r

(∂ψ

∂r

∂T

∂z− ∂ψ

∂z

∂T

∂r

)−

(1r

∂

∂r

((1 +

4Rd3

)r∂T

∂r+∂2T

∂z2

))]2

∏rdA

(3.15)

Considering the terms individually of the above equation (3.15).

Thus the stiffness matrix of Energy equation (2.14) is given by: 2∏

12A

c1ψ1 + c2ψ2 + c3ψ3

c1ψ1 + c2ψ2 + c3ψ3

c1ψ1 + c2ψ2 + c3ψ3

[b1, b2, b3]−2

∏12A

b1ψ1 + b2ψ2 + b3ψ3

b1ψ1 + b2ψ2 + b3ψ3

b1ψ1 + b2ψ2 + b3ψ3

[c1, c2, c3]

T 1

T 2

T 3

+2

∏R

4A

{

1 +43Rd

} b21 b1b2 b1b3

b1b2 b22 b2b3

b1b3 b2b3 b33

T 1

T 2

T 3

c21 c1c2 c1c3

c1c2 c22 c2c3

c1c3 c2c3 c33

T 1

T 2

T 3

= 0.

(3.16)


4. Results and Discussion

Results are obtained in terms of the average Nusselt number (Nu) at hot wall for various

parameters such as Rayleigh number (Ra), Cone angle (CA) Radiation parameter (Rd),

Rayleigh number (Ra) and Radius ratio (Rr) when heat is supplied to vertical cone

embedded with porous medium.

The average Nusselt number (Nu), is given by

Fig. 1 : Streamlines(left) and Isotherms(Right) for Ra = 100, Rr = 1, Rd = 1

a) CA = 15 b) CA = 45 c) CA = 75


Fig. 2 : Streamlines(left) and Isotherms (Right) for Ra = 100, CA = 75, Rd = 1

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


Fig. 3 : Streamlines (left) and Isotherms (Right) for Ra = 100, CA = 75, Rr = 1

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


b)

c)

Fig. 4 : Streamlines (left) and Isotherms (Right) for Rd = 1, CA = 75, Rr = 1

a) Ra = 25 b) Ra = 75 c) Ra = 100


Fig.1 (a-c) shows the streamlines and isothermal lines inside porous medium for various

values of Cone angle (CA) at Ra = 100, Rr = 1 and Rd = 1. The fluid gets heated up

near hot wall and moves up towards the cold wall due to high buoyancy force and then

returns to hot wall of the vertical annular cone. The boundary layer thickness decreases

with the increase of the Cone angle (CA).

Fig.2 (a-c) represents the streamlines and isothermal lines for various values of Radius

ratio (Rr) at Ra = 100, CA = 75 and Rd = 1. It is clear from the streamlines and

isothermal lines that the thermal boundary layer thickness decreases as the Radius

ratio (Rr) increases. The magnitude of the streamlines increases as Radius ratio (Rr)

increases and tends to move towards the cold wall of the vertical annular cone. At low

Radius ratio (Rr) the streamlines tend to occupy the half domain of the vertical annular

cone as compared to the higher value, of Radius ratio (Rr).

It is clearly seen that more convection heat transfer take place as the upper portion

of the vertical annular cone. The streamlines and isothermal lines shifts from the left

upper portion of the hot wall to the upper portion of the cold wall of vertical annular

cone as the Radius ratio (Rr) increases.

Fig.3 (a-c) illustrates the streamlines and isothermal lines distribution inside the porous


medium for various values of Radiation parameter (Rd). This figure corresponds to

Ra = 100, CA = 75 and Rr = 1. As the value of Radiation parameter (Rd) increases,

the formation of the streamlines and isothermal lines is also increases. The streamlines

and isothermal lines tend to occupy the major part of the domain from hot wall to cold

wall.

Fig.4 (a-c) shows the streamlines and isothermal lines inside the porous medium for

various values of Rayleigh number (Ra) at Rd = 1, CA = 75 and Rr = 1. The stream-

lines and isothermal lines tends to move towards the hot wall and away from cold wall

of the vertical annular cone as Rayleigh number (Ra) increases. This increases the

thermal gradient at hot wall and decreases the same at cold wall of the vertical annular

cone. Thus the heat transfer rate increases at the hot wall and decreases at cold wall

with increasing Rayleigh Number (Ra). The magnitude of streamline increases as the

Rayleigh number (Ra) increases and the thermal boundary layer becomes thinner with

the increase in Rayleigh number (Ra).

Fig. 5 shows the variation of average Nusselt number (Nu) at hot wall, with respect to

Radius ratio (Rr) of the vertical annular cone for various values of Radiation parameter

(Rd). This figure corresponds to the values Ra = 100, CA = 75. It can be observed that

with the increase of Ra from 50 to 100 there is a negligible effect on the Nusselt number

(Nu) due to the presence of the radiation for values of Rd = 1, 5, 10.

Fig. 6 illustrates the variation of average Nusselt number (Nu) at hot wall, with respect

to Radius ratio (Rr) of the vertical annular cone for various values of Rayleigh number

(Ra) at values Rd = 1, CA = 75. It is found that the average Nusselt number (Nu) at

Rr = 1 is increased by 1.4%. The corresponding increase in average Nusselt number

(Nu) at Rr = 10 is found to be 2.7%. This is due to the reason that high Rayleigh

number (Ra) produces high buoyancy force, which leads to increases fluid movements

and thus increased the average Nusselt number (Nu). For a given Radius ratio (Rr)

increase in Ra variation increase in average Nusselt number (Nu) with very little effect

from Ra = 25 to 50 and the effect increases marginally for Ra = 75 to 100.

Fig. 7 shows the variation of average Nusselt number (Nu) at hot wall, with respect

to Radiation parameter (Rd) of the vertical annular cone for various values of Cone

angle (CA). This figure corresponds to the values Rr = 1, Ra = 100. It is found that

the average Nusselt number (Nu) increases with increase in Radiation parameter (Rd).


It can be seen that the average Nusselt number (Nu) increases with increase in Cone

angle (CA). For a given Radiation parameter (Rd), the difference between the average

Nusselt number at two different values of Cone angle (CA) increases with increase in

Cone angle (CA). The increase in Nu for increase of Cone angle (CA) from 45o to 75o

is substantial. It is interesting to note that the variation of (Nu) with Rd is linear for

the three different angles of the cone.

Fig. 8 illustrates the variation of average Nusselt number (Nu) at hot wall, with respect

to Radiation parameter (Rd) of the vertical annular cone for various values of Radius

ratio (Rr) at values Ra = 50, CA = 75. It is found that the average Nusselt number

(Nu) increases with Radiation parameter (Rd) linearly. It can be seen that the average

Nusselt number (Nu) increases with increase in Radius ratio (Rr). For a given Radiation

parameter (Rd), the difference between the average Nusselt number at two different

values of Radius ratio (Rr) increases with increase in Radius ratio (Rr). The average

Nusselt number (Nu) increases with the increase in Radius ratio (Rr).


to Rayleigh number (Ra) of the vertical annular cone for various values of Cone angle

(CA) at Rr = 1, Rd = 1. It is found that the average Nusselt number (Nu) increases

with increase in Rayleigh number (Ra). It can be seen that the average Nusselt number

(Nu) increases with increase in Cone angle (CA) for a given Rayleigh number (Ra). The

difference between the average Nusselt number (Nu) at two different values of Cone

angle (CA) increases with Cone angle (CA) for instance, the average Nusselt number

(Nu) increased by 11.4% when Cone angle (CA) is increased from 15 to 45 Ra = 10.

However the average Nusselt number (Nu) increased by 13.8%, when Cone angle (CA)

is increased from 15 to 45 at Ra = 100. This difference becomes more prominent as the

Rayleigh number (Ra) increases for particular value of Cone angle (CA).


to Rayleigh number (Ra) of the vertical annular cone for various values of cone angle

(CA) at Ra = 100, Rd = 1. It is found that the average Nusselt number (Nu) increases

with increase in Radius ratio (Rr) and also in Cone angle (CA). For a given Radius

ratio (Rr) the difference between the average Nusselt number (Nu) at two different val-

ues of Cone angles (CA) increases with increase in Cone angle (CA). For instance, the

average Nusselt number (Nu) increases 11.8%, when Cone angle (CA) increased 15 to


45 at Rr = 1. However the average Nusselt number (Nu) increased by 6.2% when Cone

angle (CA) is increased from 15 to 45 degrees at Rr = 10. This difference becomes more

prominent as the Radius ratio (Rr) increases. So, the average Nusselt number (Nu)

increases substantially when the Cone angle (CA) and Radius ratio (Rr) increases.

References

[1] Cheng P. and Minkowycz W. J.,,Free Convection about a vertical flat plateembedded in a porous medium with application to heat transfer from a dike,Journal of Geophysical Research, 82 (1977), 2040-2044.

[2] Na T. Y. and Pop I., Free Convection flow past a vertical flat plate embeddedin a saturated porous medium, International journal of Engineering Sciences,21(5) (1983), 517-526.

[3] Gorla R. S. R. and Zinalabedini A.H., Free convection from a vertical plate withnonuniform surface temperature and embedded in a porous medium, ASME,Journal of Energy Resources Technology, 109(1) (1987), 26-30.

[4] Ghosh S. K. and Anwar Beg O., Theoretical analysis of radiative effects ontransient free convection heat transfer past a hot vertical surface in porousmedia, Nonlinear Analysis : Modelling and Control, 13 (2008), 419-432.

[5] Cortell R., Internal heat generation and radiation effects on a certain free con-vection flow, Int. Journal of Nonlinear Science, 9 (2010), 468-479.

[6] Badruddin I. A. et al, Free convection and radiation for a vertical wall withvarying temperature embedded in a porous medium, Int. J. Thermal Sci, 45(2006), 487-493.

[7] Salleh M. Z. et al, Free Convection over a permeable horizontal flat plate em-bedded in a porous medium with radiation effects and mixed thermal boundaryconditions, Journal of Mathematics and Statistics, 8 (2012), 122-128.

[8] Minkowycz W. J. and Cheng P., Free convection about vertical cylinder embed-ded in a porous medium, International journal of Heat and Mass Transfer, 19(1976), 805-813.

[9] Yucel A., The influence of injection or withdrawal of fluid on free convectionabout a vertical cylinder in a porous medium, Numer. Heat Transfer, 7(4)(1984), 483-493.

[10] Kumari M., Pop I. and Nath G., Finite-difference and improved perturbationsolutions for free convection on a vertical cylinder embedded in a saturatedporous medium, Int. J. Heat and Mass Transfer, 28(11) (1985), 2171-2174.

[11] Merkin J. H., Free Convection from a Vertical cylinder embedded in a saturatedporous medium, Acta Mechanica, 62 (1986), 19-28.

[12] Bassom A. P. and Rees D. A. S., Free Convection from a Heated VerticalCylinder Embedded in a Fluid Saturated Porous Medium, Acta Mechanica, 116(1996), 139-151.


[13] Hossain M. A. and Pop I., Radaiation effect on Darcy free convection along aninclined surface plate in a porous media, Heat and Mass Transfer, 32 (1997),223-227.

[14] Raptis A., Radiation and free convection flow through a porous medium, Int.comm. Heat Mass Transfer, 25 (1998), 289-295.

[15] Cheng P., Free convection about a flat plate embedded in a saturated porousmedium, Int. J. Heat Mass transfer, 20 (1977), 893-898.

[16] Kazmierczak M., Poulikakos D. and Pop I., Melting from a flat plate embeddedin a porous medium in the presence of steady natural convection, NumericalHeat Transfer, 10 (1986), 571-581.

[17] Cheng P., Adv. Heat Transfer, 14 (1979), ii-xi, 1-363.[18] Hassanien I. A., Bakeir A. Y. and Gorla R. S. R., Effects of thermal dispersion

and stratification on non-Darcy mixed convection from a vertical plate in aporous medium, Heat and mass transfer, 34(2-3) (1998), 209-212.

[19] Viskanta R. and Grosh R. J., Boundary layer in thermal radiation absorbingand emitting media, Int, J. Heat and Mass Transfer, 5 (1962), 795-806.

[20] Ali M. A., Chen T. S. and Armaly B. F., Natural Convection-Radiation inter-action in Boundary-Layer Flow over Horizontal Surfaces, AIAA Journal, 22(12)(1984), 1797-1803.

[21] Bakier A. Y. and Gorla R. S. R., Thermal radiation effect on mixed convectionfrom horizontal surfaces in saturated Porous media, Transport in porous media,23 (1996), 357- 363.

[22] Hassain M. A., and Ress D. A. S. and Pop I., Free Convection-Radiation in-teraction from an isothermal plate inclined at an small angle to the horizontal,Acta Mechanica, 127 (1998), 63- 73.

[23] Baker A. Y., Thermal radiation effect on mixed convection from a verticalsurfaces in saturated porous media, Int. comm. Heat mass Transfer, 28(1)(2001), 119-126.

[24] Bejan A., convective Heat Transfer, 2nd edition, New York, John Wiley & Sons,(1995).

[25] Lewis R. W., Nithiarasu P. and Seetharamu K. N., Fundamentals of the finiteelement method for heat and fluid flow, John Wiley and sons, Chichester (2004).

[26] Segerland L. T., Applied Finite Element Analysis, John Wiley and Sons, NewYork (1982).

[27] Modest M. F., Radiative heat transfer, New York, Mc.Graw-Hill (1993).[28] Kays W. H., Convective Heat and Mass Transfer, TMH, Edition, (1975).[29] Ingham D. B., Pop I., Transport Phenomena in Porous Media, Elsevier,Oxford,

(2005).[30] Nield D.A., Bejan A., Convection in Porous Media, Third ed., Springer, New

York, (2006).

Visualization of heat transfer by free convection with radiation in a vertical cone embedded with...

Documents

Transcript of Visualization of heat transfer by free convection with radiation in a vertical cone embedded with...