Transp Porous Med (2012) 91:697–715
DOI 10.1007/s11242-011-9867-x
Analysis of Heat and Mass Transfer in a Vertical Annular
Porous Cylinder Using FEM
Irfan Anjum Badruddin · N. J. Salman Ahmed ·
Abdullah A. A. A. Al-Rashed · Jeevan Kanesan ·
Sarfaraz Kamangar · H. M. T. Khaleed
Received: 8 April 2011 / Accepted: 27 August 2011 / Published online: 16 September 2011
© Springer Science+Business Media B.V. 2011
Abstract The present study is intended to study heat and mass transfer in a vertical annular
cylinder embedded with saturated porous medium. The inner surface of cylinder is maintained at uniform wall temperature and uniform wall concentration. The governing partial
differential equations are non-dimensionalised and solved by using finite element method
(FEM). The porous medium is discritised using triangular elements with uneven element
size. Large number of smaller-sized elements are placed near the walls of the annulus to
capture the smallest variation in solution parameters. The results are reported for both aiding
and opposing flows. The effects of various non-dimensional numbers such as buoyancy ratio,
Lewis number, Rayleigh number, aspect ratio, etc on heat and mass transfer are discussed.
The temperature and concentration profiles are presented.
Keywords
Heat and mass transfer · Porous medium · Darcy flow · Vertical annulus · FEM
I. A. Badruddin (B) · N. J. Salman Ahmed · S. Kamangar
Department of Mechanical Engineering, Faculty of Engineering, University of Malaya,
50603 Kuala Lumpur, Malaysia
e-mail: [email protected]
A. A. A. A. Al-Rashed
Public Authority for Applied Education and Training, Industrial Training Institute, 13092 Adiliya, Kuwait
J. Kanesan
Department of Electrical Engineering, Faculty of Engineering, University of Malaya,
50603 Kuala Lumpur, Malaysia
H. M. T. Khaleed
Department of Mechanical Engineering, Anjuman Engineering College, Bhatkal 581320, India
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List of symbols
A
Area of an element
Aspect ratio
Ar
Cp
Specific heat
C, C̄
Species concentration (dimensional and non-dimensional, respectively)
D
Mass diffusivity
g
Gravitational acceleration
H
Height of cylinder
k
Thermal conductivity
K
Permeability of porous media
L
= r o − ri
Le
Lewis number
M
Shape function
n
Refractive index
N
Buoyancy ratio
N u z , N̄ u Local and average Nusselt number
p
Pressure
qr
Radiation flux
r, z
Cylindrical coordinates
r̄ , z̄
Non-dimensional coordinates
R̄
Mean radial distance of an element
R
Radius ratio
Ra
Rayleigh number
Rd
Radiation parameter
Sh z , S̄h
Local and average Sherwood number
T, T̄
Dimensional and non-dimensional temperature, respectively
u, w
Velocity in r and z directions
Greek symbols
α
Thermal diffusivity
βC Coefficient of concentration expansion
βT Coefficient of thermal expansion
β R Rosseland extinction coefficient
ρ
Density
v
Coefficient of kinematic viscosity
μ
Coefficient of dynamic viscosity
σ
Stefan Boltzmann’s constant
ψ Stream function
ψ̄ Non-dimensional stream function
Subscripts
w Wall
∞ Conditions at infinity
i
Inner
o
Outer
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Analysis of Heat and Mass Transfer
699
1 Introduction
The study of heat and mass transfer in a saturated porous medium has received a noticeable attention from the researchers during the last few decades. This is supported by the
fact that the heat and mass transfer in porous media involves plenty of applications such as
migration of moisture through the air contained in fibrous insulations, the contamination of
chemicals in the soil, grain storage installations, cryogenic containers, etc. A deep insight
into the various aspects of convection in PM is provided by Nield and Bejan (2006), Ingham
and Pop (1998), Pop and Ingham (2001), Bejan and Kraus (2003) and edited book by Vafai
(2000). The literature reveals an extensive work in heat and mass transfer in porous media.
Yih (1999) has studied the coupled heat and mass transfer by natural convection adjacent to a
horizontal cylinder in a saturated porous medium by using Keller box method. Cheng (2000)
has investigated the heat and mass transfer near a wavy surface and shown that the increasing
amplitude–wavelength ratio tends to increase the amplitude of the local Sherwood number
and local Nusselt number. The investigation into heat and mass transfer in an unsaturated
soil was carried out by Thomas and Ferguson (1999). The study by Hossain et al. (1999)
explained that the Nusselt number and Sherwood number were found to decrease with the
increased curvature parameter when the fluid was injected through the permeable surface of
the cylinder. Trevisan and Bejan (1987) have reported the results for heat and mass transfer in
porous medium subjected to high Rayleigh number flow. Yang (2007) investigated the coupled heat and moisture in double-layer hollow cylinder. Singh and Chandarki (2009) studied
the effects of the governing parameters such as buoyancy ratio (N) and Lewis number (Le)
on the local Nusselt and local Sherwood Numbers for the integral treatment of coupled heat
and mass transfer by natural convection from a cylinder in porous media.
El-Amin (2004) studied the double dispersion effects on natural convection heat and
mass transfer in non-Darcy porous medium by employing fourth-order Runge–Kutta method
with shooting technique. Angirasa et al. (1997) investigated the heat and mass transfer with
opposing buoyancy effects. The mixed convention heat and mass transfer was considered
by Yih 1998. The analysis of heat transfer through natural convection in case of the vertical
annular cylinder was reported by Badruddin et al. (2006a, 2006b, 2007) by considering radiation parameter effect using thermal equilibrium model, whereas the thermal non-equilibrium
model was applied for the similar geometry in another study by Badruddin et al. (2006a).
Cheng (2006a) reported the double-diffusive natural convection in a vertical annular porous
medium to demonstrate the effect of modified Darcy number on the volume flow rate. In
another case of his research, Cheng (2006a) investigated the heat and mass transfer rates
of the elliptical cylinder with slender orientation and blunt orientations. Cheng (2009) also
studied the Soret and Dufour effects on the boundary layer flow due to natural convection
heat and mass transfer over a downward pointing vertical cone in a porous medium, whereas
Kairi and Murthy (2011) have investigated the effect of viscous dissipation on natural convection heat and mass transfer from vertical cone. The radiation effect in case of the natural
convection was reported by Salman et al. (2009) using the thermal equilibrium approach,
whereas thermal non-equilibrium approach was adopted to study the mixed convection in
an annular vertical cylinder by the same author Salman et al. (2011). The present study is
aimed to investigate numerically the heat and mass transfer characteristics in a vertical annular cylinder embedded in porous media with constant wall temperature and concentration.
The heat and mass transfer in an annular porous body is an important area of interest as it
involves practical applications such as insulation of pipe lines, high-temperature gas-cooled
reactor vessels, biomass converters, etc. The fluid flow is assumed to be governed by Darcy
law.
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I. A. Badruddin et al.
Fig. 1 Annular cylinder
2 Analysis
Consider the flow of fluid under the influence of natural convection in a saturated porous
medium embedded in a vertical annular cylinder. Figure 1 depicts the geometry of problem
under consideration. Since the geometry is axisymmetric in nature, only two dimensions, i.e.,
r and z can describe the flow behaviour completely in the medium. The cylinder has inner
radiusri and outer radius ro ,. where r and z-axes point towards the width and height of the
medium, respectively. Following assumptions are made:
•
•
•
•
Porous medium is saturated with fluid.
The fluid and medium are in local thermal equilibrium everywhere inside the medium.
The porous medium is isotropic and homogeneous.
Fluid properties are constant except the variation in density.
With the above assumptions, the governing equations are given by:
Continuity equation
∂(r u) ∂(r w)
+
=0
∂r
∂z
(1)
Using Darcy law for flow in porous media
−K ∂ p
μ ∂r
−K ∂ p
w=
+ ρg
μ
∂z
u=
(2)
(3)
Energy equation
u
∂T
∂T
+w
=α
∂r
∂z
1 ∂
r ∂r
∂
1
1 ∂
∂T
∂2T
r
+ 2 −
(rqr ) +
(qr )
∂r
∂z
ρC p r ∂r
∂z
(4)
Concentration equation
∂C
∂ 2C
r
+ 2
∂r
∂z
(5)
ρ = ρ∞ [1 − βT (T − T∞ ) − βC (C − C∞ )]
(6)
u
123
∂C
∂C
+w
=D
∂r
∂z
1 ∂
r ∂r
Analysis of Heat and Mass Transfer
701
Corresponding boundary conditions are:
at
r = ri, T = Tw , C = Cw , u = 0,
(7a)
at
r = ro , T = T∞, C = C∞ , u = 0,
(7b)
at
z = 0 and z = H
∂T
∂C
=
=0
∂z
∂z
(7c)
where u and w are Darcy velocities in the r and z directions, respectively.
The continuity Eq. (1) can be satisfied by introducing the stream function ψ as:
1 ∂ψ
r ∂r
1 ∂ψ
w=
r ∂r
u=−
(8a)
(8b)
Invoking Rosseland approximation for radiation Yih (1999), Hossain and Alim (1997)
qr = −
4n 2 σ ∂ T 4
3β R ∂r
(9)
Expanding T 4 about T∞ in Taylor series, Raptis (1998)
3
4
− 3T∞
T 4 ≈ 4T T∞
(10)
The following parameters have been used for non-dimensionalising the governing equations.
r
z
ψ
(T − T∞ )
(C − C∞ )
, C=
,
r= , z= , ψ=
, T =
L
L
αL
(Tw − T∞ )
(Cw − C∞ )
3
4n 2 σ T∞
gβT T K L
α
βc (Cw − C∞ )
Rd =
, Ra =
, Le = , N =
βR k
να
D
βT (Tw − T∞ )
(11)
Making use of Eqs. (6–8) and (11) into (2) and (3) and performing mathematical operations
yield
∂T
∂ 2ψ
∂ 1 ∂ ψ̄
∂C
+ r̄
= r̄ Ra
+N
(12)
∂r r̄ ∂ r̄
∂r
∂r
∂z 2
Replacing u and w by ψ in Eq. (4) and substituting terms from Eqs. (9–11) results
1 ∂
1 ∂ψ ∂ T
∂ψ ∂ T
4Rd
∂T
∂2T
−
= 1+
r̄
+ 2
r̄ ∂r ∂z
∂z ∂r
3
r̄ ∂ r̄
∂r
∂z
After substituting the ψ and non-dimensional parameters, Eq. (5) takes the form:
∂ψ ∂C
1 ∂ψ ∂C
1
1 ∂
∂C
∂ 2C
−
=
r̄
+ 2
r̄ ∂r ∂z
∂z ∂r
Le r̄ ∂r
∂r
∂z
(13)
(14)
when Rd = 0 , the above equations reduce to the case of pure natural convection flow over
an isothermal vertical annular cylinder embedded in saturated porous medium.
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702
I. A. Badruddin et al.
Table 1 N̄ u variation with mesh size
Sl. No.
No of elements
N̄ u
S̄h
Time (s)
1
1,800
21.60
3.70
205.84
2
3,200
21.66
3.67
1,298.09
3
5,000
21.69
3.66
5,260.42
The corresponding boundary conditions are
at
r = r̄i , T̄ = 1, C̄ = 1 ψ̄ = 0
(15a)
at
r = r̄o , T̄ = 0, C̄ = 0 ψ̄ = 0
(15b)
at
z̄ = 0 and z̄ = Ar
∂ T̄
∂ C̄
=
=0
∂ z̄
∂ z̄
(15c)
3 Numerical Method
Equations (12–14) are coupled partial differential equations that govern the heat and mass
transfer behaviour. These equations are solved by using finite element method using Galerkin
approach. A simple three-noded triangular element is considered. T̄ , C̄ and ψ̄ vary inside
the element and can be expressed as:
T̄ = T̄1 M1 + T̄2 M2 + T̄3 M3
(16)
ψ = ψ̄1 M1 + ψ̄2 M2 + ψ̄3 M3
(17)
C̄ = C̄1 M1 + C̄2 M2 + C̄3 M3
(18)
where M1, M2, M3 are the shape functions given as:
Mi =
ai + bi x + ci y
, i = 1, 2, 3
2A
(19)
and ai , bi , ci are matrix coefficients
Details of FEM formulations can be obtained from Segerland (1982), Lewis et al. (2004).
Integrating Eqs. (12–14) using Galerkin method yield a set of coupled matrix equations.
These matrix equations for elements are assembled to get the global matrix equation for the
whole domain, which is solved iteratively to get T̄ , C̄ and ψ̄ in the porous medium. For higher
accuracy in the results, the tolerance level of solution for T̄ , C̄ and ψ̄ is set at 10−5 , 10 −5
and 10 −9 , respectively. Element size in the domain varies, having large number of elements
located near the wall where large variations in T̄ , C̄ and ψ̄ are expected. Sufficiently dense
mesh is chosen to make the solution mesh invariant. Table 1 shows the average Nusselt and
Sherwood numbers for different mesh sizes, at hot surface with Ar = 10, R = 1, Rd = 10,
N = 1, Le = 1 and Ra = 100.
The local and average Nusselt and Sherwood numbers at hot and cold surface of the
annulus are evaluated using following relations:
123
Analysis of Heat and Mass Transfer
703
N uz = −
Ar
N̄ u = −
0
1 + 4Rd ∂ T̄ r̄ =r̄i ,r̄o
3
∂ r̄
∂T 1 + 4Rd
3
∂r r̄ =r̄i ,r̄o dz̄
Ar
Sh z = −
∂ C̄ r =ri ,ro
∂ r̄
Ar
S̄h = −
0
∂ C̄ ∂ r̄ r̄ =r̄i ,r̄o
Ar
(20a)
(20b)
(21)
dz̄
(22)
The computations are carried out on high-end computer. It can be seen from Table 1 that the
variation in N̄ u and S̄h is very small when element size is changed from 1,800 to 5,000.
The variation in N̄ u and S̄h is found to be 0.41 and 1.08%, respectively, when mesh size
is increased from 1,800 to 5,000 elements. However, the time required for a mesh of 5,000
elements is considerably higher than that of 1,800 elements. Thus, the mesh size of 1,800
elements is selected for the present study.
4 Results and Discussion
The inner surface of cylinder is maintained at constant temperatures T̄i = 1 and constant
species concentration C̄i = 1. The outer surface remains at T̄o = 0 and C̄o = 0. Aspect ratio
and Radius ratio are defined as Ar = H/L and R = (r0 −ri )/ri , respectively. The parameter
N indicates the relative importance of concentration and thermal buoyancy force. It may be
noted that N is positive for thermally assisting flow and negative for thermally opposing flow.
N = 0 indicates the absence of concentration buoyant force, and flow is driven by thermal
buoyancy only. Results are obtained in terms of Nusselt number and Sherwood number at
hot and cold surface of annulus for various parameters such as aspect ratio, radius ratio,
buoyancy ratio, Lewis number, Rayleigh number and radiation parameter. The temperature
and concentration gradient in Eqs. (21–22) are evaluated using 4-point polynomial along
the nodes near the surface of cylinder. The results are validated for the present method by
comparing N u and Sh with the available literature, i.e., El-Amin (2004), Yih (1998, 1999),
Hossain and Alim (1997). For the comparison, ri is extended to large value, so that the radial
effect becomes negligible and the domain behaves as a vertical plate embedded in porous
medium. Figure 2 explicates the comparison of the present method with that of (El-Amin
2004). It is evident from this figure that the present method follows the similar trend as the
established results by the previous researcher El-Amin (2004). The comparison with that of
Yih (1998) and Nakayama and Hossain (1995) is also shown in the Table 2. It can be inferred
from Table 2 that the present method has good accuracy in predicting the heat and mass
transfer characteristics. Furthermore, the results are compared with convective heat transfer
in axisymmetric bodies (Rajamani et al. 1995) and presented in Table 3. In this case also the
prediction of heat transfer behaviour is found to be similar. Thus, it is clear from, Table 2,
Fig. 2 and Table 3 that the present method provides sufficient accuracy to simulate the heat
and mass transfer behaviour in porous annulus.
Figure 3 shows the S̄h variation with aspect ratio of annular cylinder for different values
of Le. Lewis number Le has pronounced effect on Sherwood number. The S̄h increases initially when aspect ratio is increased, reaches a maximum value and then starts decreasing.
123
704
I. A. Badruddin et al.
1/ 2
Shz / Raz
Shz / Raz
1/ 2
Nuz / Raz
1/2
[10]
Nuz / Raz
1/2
Le
Fig. 2 Comparison of results with the previously published work
Table 2 Comparison of the present method with available literature
N
Le
1/2
1/2
N u z /Raz
Sh z /Raz
Yih (1998)
Nakayama
and Hossain
(1995)
Present
Yih (1998)
Nakayama
and Hossain
(1995)
Present
0
1
0.4437
0.444
0.4325
0.4437
0.444
0.4325
0
2
–
0.444
0.4325
–
0.693
0.6912
0
4
–
0.444
0.4325
–
1.053
1.1093
0
6
–
0.444
0.4325
–
1.332
1.4075
0
8
–
0.444
0.4325
–
1.568
1.6357
0
10
0.4437
0.444
0.4325
1.6803
1.776
1.8268
0
100
0.4437
0.444
0.4325
5.5445
6.061
6.3780
1
1
0.6276
0.628
0.6551
0.6276
0.628
0.6551
1
2
–
0.593
0.6307
–
0.937
1.0420
1
4
–
0.559
0.5944
–
1.383
1.5808
1
6
–
0.541
0.5687
–
1.728
1.9535
1
8
–
0.529
0.5496
–
2.019
2.2471
1
10
0.5214
0.521
0.5347
2.2020
2.276
2.4967
1
100
0.4700
0.470
0.4386
7.1389
7.539
7.7022
The maxima for S̄h at Le = 1 occur at an aspect ratio around 1.25, but it keeps shifting
towards lower aspect ratio when Le is increased. The effect of Lewis number on average
Nusselt number N̄ u variation is comparatively small, due to which it has not been explicated
123
Analysis of Heat and Mass Transfer
705
Table 3 Comparison of results for annular body
R
N̄ u (Rajamani et al. 1995)
N̄ u present
0.25
1.791
1.7551
1
2.344
2.2534
4
4.016
3.8485
8
5.466
5.5502
Fig. 3
S̄h versus aspect ratio Rd = 4, Ra = 100, R = 1, N = 0.5
2.9x10-13
K = 2.9x10-14
2.9x10-15
Fig. 4
N̄ u and S̄h versus aspect ratio Rd = 1, R = 1, N = 1, Le = 5
in the figure. S̄h increases with increase in Le. The concentration boundary layer becomes
thinner as Le increases, which in turn increases the concentration gradient and thus the S̄h.
Figure 4 shows the effect of permeability on the heat and mass transfer characteristics of
porous medium. The permeability range corresponds to the well-packed soil. It is obvious
from Fig. 4 that the permeability has greater effect on mass transfer than the heat transfer. The Nusselt and Sherwood numbers increase with increase in the permeability of the
123
706
I. A. Badruddin et al.
1
0.9
0.8
0.7
(a)
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
1
0.9
0.8
0.7
(b)
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
1
0.9
0.8
0.7
(c)
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
Fig. 5 a Isotherms, b Isoconcentration, c Streamlines Left Ar = 0.5, Right Ar = 1 at Ra = 100, R = 1,
N = 0.5, Le = 5 and Rd = 1
123
Analysis of Heat and Mass Transfer
Fig. 6
707
S̄h versus aspect ratio at Rd = 10, Ra = 25, R = 1, Le = 2
Fig. 7 S̄h variations with radius ratio at hot surface for Rd = 2, Ra = 50, Ar = 5, N = 1
medium. This could be attributed to decrease in the resistance for fluid flow with increase
in permeability which in turn transports the heat and mass from inner surface to the porous
medium. It is observed that the Nusselt number is not much affected at low values of permeability (2.9×10−15 − 2.9×10−14 ) which is shown as single line in Fig. 4. The effect of
aspect ratio variation is negligible at low permeability.
Figure 5 depicts the isotherms, isoconcentration lines and streamlines for different values
of aspect ratio. It is evident from the figure that when aspect ratio is increased from 0.5 to 1,
the isothermal lines tend to move towards lower corner of inner surface of cylinder, which
indicates that the convective heat transfer has increased. However, at the same time, isoconcentration lines also move towards the lower and upper corner of inner and outer surfaces of
annular cylinder, respectively. The fluid activity increases at lower and upper corner of inner
and outer surfaces, respectively, as indicated by streamlines.
123
708
I. A. Badruddin et al.
Fig. 8 S̄h variations with radius ratio at cold surface for different values on Le at Rd = 2, Ra = 50,
Ar = 5, N = 1
Fig. 9 N̄ u and S̄h variations with radius ratio at hot surface for different values on N at Rd = 2, Ra = 50,
Ar = 5, Le = 2
Figure 6 explains the S̄h as a function of aspect ratio for both aiding and opposing
flows. N̄ u is not much affected by buoyancy ratio; thus, it has not been shown in the figure.
S̄h increases with increase of N . The maximum S̄h shifts to lower aspect ratio when N is
increased. Figure 7 illustrates the effect of radius ratio on the Sherwood numbers S̄h, with
respect to Lewis number at hot surface of the annulus. It can be noticed that the Sherwood
123
Analysis of Heat and Mass Transfer
709
Fig. 10 N̄ u and S̄h variations with radius ratio at cold surface for different values on N at Rd = 2, Ra = 50,
Ar = 5, Le = 2
number at hot surface increases with increase in radius ratio. The increase of Sherwood
number is found to be almost linear with respect to radius ratio. Figure 8 demonstrates the
effect of radius ratio on the Sherwood number with respect to Lewis number at cold surface
of the annulus. At cold surface of annulus, the S̄h decreases with increase in radius ratio. It
is found that the rate of decrease in S̄h is higher at lower values of radius ratio.
Figure 9 reveals the effect of radius ratio on Nusselt and Sherwood numbers with respect
to buoyancy ratio at hot surface. Sherwood number increased by 2.44 times when radius ratio
is varied from 0.5 to 10 at N = −0.5. The corresponding increase in Sherwood number at
N = 2 is found to be 1.28 times. The Nusselt number increased by 2.47 times and 1.85 times
at N = −0.5 and N = 2, respectively, when radius ratio is changed from 0.5 to 10. The
effect of radius ratio on cold surface for various values of buoyancy ratio has been shown in
Fig. 10. The trend of N̄ u and S̄h variation is found to be similar to that of Fig. 8. The Nusselt
and Sherwood numbers are smaller for opposing flow as compared to that of aiding flow.
The Sherwood number at R = 0.5 increased by 2.65 times when N is changed from −0.5
to 2, and corresponding increase in Nusselt number is found to be 0.67 times. It is noticed
that the difference in Nusselt and Sherwood numbers at hot and cold surface decreases with
increase in buoyancy ratio at R = 0.5. However, this difference increases at R = 10 when
buoyancy ratio is increased.
Figure 11 shows the isothermal lines, isoconcentration lines and streamlines for two different values of radius ratio. It can be seen that the isothermal lines move towards the hot
surface and away from cold surface when radius ratio is increased. This leads Nusselt number
at hot surface to increase, whereas it decreases at cold surface. Similar behaviour for isoconcentration lines is observed as shown in Fig. 11b. It is interesting to note that the fluid flow
regime gets divided into two distinct regions at R = 10 as shown by streamlines in Fig. 11c.
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I. A. Badruddin et al.
(a)1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
(b) 1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
0.1
0.2
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1
1.1
0
0
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
(c) 1
1
0.9
0.9
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0.8
0.7
0.7
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0.2
0.2
0.1
0.1
0
0
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
Fig. 11 a Isotherms, b Isoconcentration, c Streamlines Left R = 0.5, Right R = 2 at Ra = 50, Ar = 1,
N = −0.5, Le = 10 and Rd = 2
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Analysis of Heat and Mass Transfer
711
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
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0.2
0.2
0.1
0.1
(a)
0
0
1
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
2
(b) 1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
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0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
1
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
2
1
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
2
1
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
2
0
1
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
2
(c) 1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
1
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
2
Fig. 12 a Isotherms, b Isoconcentration, c Streamlines, Left N = −0.5, Right N = 2 at Ra = 25, Ar = 1,
R = 1, Le = 2 and Rd = 1
123
712
I. A. Badruddin et al.
(a) 1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
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0.3
0.2
0.2
0.1
0.1
0
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1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
(b) 1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
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0.3
0.3
0.2
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0.1
0.1
0
1
1.1
1.2
1.3
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1.5
1.6
1.7
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1.9
2
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
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2
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
0
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
(c) 1
1
0.9
0.9
0.8
0.8
0.7
0.7
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0.6
0.5
0.5
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0.4
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0.2
0.2
0.1
0.1
0
0
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
Fig. 13 a Isotherms, b Isoconcentration, c Streamlines, Left Rd = 0, Right Rd = 10 at Ra = 100, Ar = 1,
R = 1, Le = 0.1 and N = 0.5
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713
Fig. 14 Temperature and concentration profile for various values of N at Le = 2, Ar = 10, R = 1, Rd = 10,
and Ra = 25
Fig. 15 Temperature and concentration profile for various values of Ra at Le = 2, Ar = 10, R = 1, Rd = 2
and N = −0.5
Figure 12 shows the isothermal lines, isoconcentration lines and streamlines for different values of N . The isothermal lines are almost parallel to vertical surface of cylinder at
N = −0.5, which indicates that the heat transfer is dominated by conduction. When N is
123
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I. A. Badruddin et al.
increased from −0.5 to 2, then the isothermal lines and isoconcentration lines concentrate at
the lower and upper corner of inner and outer surfaces, respectively. The streamlines are distorted and crowded near the surface of cylinder when N is changed from −0.5 to 2 (Fig. 12c),
indicating the increased fluid velocity near the surface of cylinder. This increased velocity
helps in carrying more amount of heat and mass from the surface to the porous medium, thus
increasing the Nusselt and Sherwood numbers.
Figure 13 depicts the isothermal lines, isoconcentration lines and streamlines for two
different values of Rd, i.e., Rd = 0 (pure natural convection) and Rd = 10 (convection +
radiation). The value of Rd corresponds to the ambient temperature of 30◦ C for water as fluid
in the medium. It is obvious from this figure that the conduction mode of heat transfer dominates when radiation effect increases. There is slight change in isoconcentration lines that
move towards the hot wall at bottom portion of the annulus with increase in Rd. This indicates
that the mass transfer rate increases slightly at bottom portion of annulus with increase in
radiation parameter. The increase in mass transfer at bottom portion is counterbalanced by
decrease at upper portion; thus, there is negligible effect of radiation parameter on average
Sherwood number (figure not shown to conserve the space).
Figure 14 shows the temperature and concentration profile for different values of N . For
a given value of Ra, Rd, Ar , Le and R, temperature and concentration gradient increases with
increase in N . The concentration profile is affected to a greater degree due to changes in
N , as compared to that of temperature profile. At little distance from the inner surface of
cylinder, the variation in concentration profile with respect to N ceases for aiding flow.
The effect of Rayleigh number on temperature and concentration profile is shown in
Fig. 15. The thermal and concentration boundary layer becomes thinner with increase in
Rayleigh number as thermal energy and species concentration diffuses quickly leading to
increased heat and mass transfer rate. When Ra is varied, then the temperature profile is
affected to a greater extent as compared to concentration profile.
5 Conclusion
Heat and mass transfer in a vertical annular cylinder embedded with saturated porous medium
is studied. The governing equations are solved using finite element method. Effects of various
parameters such as buoyancy ratio, Lewis number, Rayleigh number, aspect ratio, and radius
ratio on N̄ u and S̄h are investigated. It is found that S̄h increases initially when aspect ratio
is increased, reaches a maximum value and then starts decreasing. The maxima for S̄h keep
shifting towards lower aspect ratio when Le is increased. Similar effect on S̄h with respect
to aspect ratio is observed due to changes in N . In general, it is found that S̄h increases with
increase in Le. Temperature and concentration gradient increases with increase in buoyancy
ratio or Rayleigh number.
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