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Notes on Steady Natural Convection Heat

Transfer by Double Diffusion From a Heated
Cylinder Buried in a Saturated Porous Media
ARTICLE in JOURNAL OF HEAT TRANSFER FEBRUARY 2015
Impact Factor: 1.45 DOI: 10.1115/1.4029878

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Notes on Steady Natural Convection

Heat Transfer by Double Diffusion
From a Heated Cylinder Buried
in a Saturated Porous Media
Carlos Alberto Chaves1
Department of Mechanical Engineering,
University of Taubate,
Taubate, SP 12060-440, Brazil
e-mail: carlosachaves@yahoo.com.br

Keywords: cylinder buried, heat transfer, natural convection,

steady solution

Wendell de Queiroz Lamas

Department of Basic Sciences and Environment,
School of Engineering at Lorena,
University of Sao Paulo,
Lorena, SP 12602-810, Brazil
e-mail: wendell.lamas@usp.br

Luiz Eduardo Nicolini do Patrocinio Nunes

Department of Mechanical Engineering,
University of Taubate,
Taubate, SP 12060-440, Brazil
e-mail: luiz.nunes@unitau.com.br

Jose Rui Camargo

Department of Mechanical Engineering,
University of Taubate,
Taubate, SP 12060-440, Brazil
e-mail: rui@unitau.br

Francisco Jose Grandinetti

Department of Mechanical Engineering,
University of Taubate,
Taubate, SP 12060-440, Brazil
e-mail: grandi@unitau.br

This paper aims to present numerical solutions for the problem of

steady natural convection heat transfer by double diffusion from a
heated cylinder buried in a saturated porous media exposed to
constant uniform temperature and concentration in the cylinder
and in the media surface. A square finite domain 3  3 and acceptance criterion converged solution with an absolute error under
1  103 were considered to obtain results presented. The Patankars power law for approaching of variables calculated T, C,
and / also was adopted. In order of method validation, an investigation of mesh points number as function of Ra, Le, and N was
done. A finite volume scheme has been used to predict the flow,
temperature, and concentration distributions at any space from a
heat cylinder buried into a fluid-saturated porous medium for a
bipolar coordinates system. Examples presented show that the differences in the flow distribution caused not only when Rayleigh
number range is considered but also when Lewis number range
is considered. Further, increase in the Rayleigh number has a significant influence in the flow distribution when the concentration
distribution is considered. Steady natural convection heat transfer
1
Present address: Rua Daniel Danelli, s/n, Jardim Morumbi, Taubate, SP 12060440, Brazil.
Contributed by the Heat Transfer Division of ASME for publication in the
JOURNAL OF HEAT TRANSFER. Manuscript received May 1, 2014; final manuscript
received February 12, 2015; published online March 24, 2015. Assoc. Editor: Jose L.
Lage.

Journal of Heat Transfer

by double diffusion from a heated cylinder buried in a saturated

porous medium is studied numerically using the finite volume
method. To model fluid flow inside the porous medium, the Darcy
equation is used. Numerical results are obtained in the form of
streamlines, isotherms, and isoconcentrations. The Rayleigh number values range from 0 to 1000, the Lewis number values range
from 0 to 100, and the buoyancy ratio number is equal to zero.
Calculated values of average heat transfer rates agree reasonably
well with values reported in the literature.
[DOI: 10.1115/1.4029878]

Introduction
Many transport processes presented in environment occur due
to the flow off with a simultaneous occurrence of temperature and
concentration gradients. Some oceanographic phenomena like
salt sources are explained by coupled presence of thermal gradients and saline. There is also an application dissemination control of pollutants contaminators from chemical industrial waste
and radioactive waste that solves the problem in area of radioactive spreading out in ground, in water contamination, and in other
correlates that still ask for solution [1].
A study on transient natural convection promoted by double
diffusion in saturated porous media was done in Ref. [2]. This
study proposed a numerical solution to variations of Rayleigh
number (0 to 1000), Lewis number (0 to 100), and the buoyancy
ratio number (3 to 3) using the control volume method idealized in Ref. [3]. This method has been widely used and its implementation to a bipolar coordinates system was realized.
Patankars method was applied to the case of transient natural
convection heat transfer by double diffusion from heated cylinders
buried in saturated porous media in Ref. [2].
Numerical solutions for transient natural convection heat transfer by double diffusion from a heated cylinder buried in a saturated porous mediawhere both, cylinder and media surfaces,
were kept at constant uniform temperature and concentration
were presented in Ref. [4], where heat and mass transfer were
studied as a function of Rayleigh, Lewis, and buoyancy ratio
numbers.
Reference [5] turned its attention to processes of combined
(simultaneous) heat and mass transfer that were driven by buoyancy. The density gradients that provide the driving buoyancy
force were induced by the combined effects of temperature
and species concentration nonuniformities present in the fluidsaturated medium. This paper was guided by the review of
Ref. [6]. Review articles on double-diffusive convection such as
those given in Refs. [79] are useful in this work.
This paper is guided by a study made previously in Ref. [4]
applied to the case of the problem of steady natural convection
heat transfer by double diffusion from heated cylinders buried in
saturated porous media presented in Ref. [10].
This paper aims to present numerical solutions for the problem
of steady natural convection heat transfer by double diffusion
from a heated cylinder buried in a saturated porous media exposed
to constant uniform temperature and concentration in the cylinder
and in the media surface.

Problem Statement
An infinite cylinder buried in a saturated porous media is considered. The cylinder has a radius r1 and it is buried in depth d
from the porous media superior surface. The cylinder outside
cover is kept at temperature TW and at concentration CW, constant
while the porous media superior surface is kept at temperature TS
and at concentration CS, according to Fig. 1. The hypothesis of
steady state and impermeable wall is considered as well.

C 2015 by ASME
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Fig. 1 Bipolar coordinates system

To obtain the equations that describe the problem, it is assumed

that [2]:
a) The porous media and the fluid that saturates it are isotropic
and homogeneous; the Boussinesq approximation is valid to
intensities variation due to changes in both temperature and
concentration. It is possible, this way, to express the specific
mass (q) by the following equation [11]:
q q0  1  b  T  T 0  bc  C  C0 

(1)

where To, Co, and qo are references temperature, concentration, and specific mass, respectively, and b and bc are the
coefficients of thermal and chemical expansions, respectively, defined by the following equations:

 
1 @q

q0
@T p

(2)

bc

 
1
@q

q0
@C p

(3)

b) Darcy law is assumed to describe the fluid flow in porous

media, this way, in Ref. [12], it is possible to obtain the
following equation:
l
r  V  rq  g
(4)
K
c) The porous media is rigid, and the thermodynamics properties (except the density in the buoyancy ratio term) are considered constant.
d) There are no chemical reactions, and the viscous dissipation
is negligible.
e) The porous media and the fluid present thermodynamic
equilibrium.

f) In order of comparison with this proposal and validation of

program implemented for bipolar coordinates, it reproduced
the Baus conditions [10]:
cylinder radius r1 0.25 m
impermeable and isotherm surface of the cylinder
silica floor external to the cylinder (media size of the corn
close to 2.54  104 m) of permeability 6  1011 m2
temperature difference between the cylinder and the floor
is 60  C
water as saturating liquid, with its properties calculated
at 40  C
cylinder deepness from the surface floor is 2 m.
It had used Baus problem [10] as reference to validate
this methodology and programing, and then in this reference, u
and v are used to represent bipolar coordinate systems, also in
Ref. [13]. It will be used in the current function formulation in
Eqs. (5)(7).
The natural convection heat transfer by double diffusion in saturated porous medias problem can be written to steady state and
incompressible fluids in bipolar coordinates (u,v) presented in
Fig. 1 in dimensionless terms in the formulation of the stream
function [4] by the following equations:




@T 
@T 
@C
@C
r2 w a  H   G   BN  H   G  
@u
@v
@u
@v
(5)
where
H

1  cos u cosh v
cosh v

Vu

and G

cos u 2

1 @W

hv @v
hu hv

sinh v sin u
cosh v  cos u 2

and V v  

1 @W

hu @u

a
 cos u

cosh v

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@W @T  @W @T 
r T Ra



@v @u @u @v
2 

 

@W @C @W @C
r2 C RaLe



@v @u
@u @v

(6)

(7)

where
a
sinh v1
r1
d
d cosh v1
r1
T  Ts

T
Tw  Ts
C  Cs
C
Cw  Cs
a

The dimensionless stream function is given by

W

W
aRa

(8)

The buoyancy ratio, Lewis number, and Rayleigh number are

given by the following equations:
Fig. 2 Typical cell of the volume control method

bc  DC
BN
b  DT
a
D

(10)

K  g  b  DT  r 1
  a

(11)

Le
Ra

(9)

Ai;j  wi;j Ai1;j  wi1;j Ai1;j  wi1;j Ai;j1  wi;j1

Ai;j1  wi;j1 Bi;j
with
Ai1;j

Dv
@ue

Ai1;j

Dv
@uw

Ai;j1

Du
@vn

Ai;j1

Du
@vs

where
DT T w  T s
DC Cw  Cs

Methods
To solve the problem numerically, Eqs. (5)(7) were integrated
related to u and v variables, which are already described in dimensionless bipolar coordinates on a generic volume control. Such
dominion is described in Fig. 2, and the integration is done following the volume control method formulation developed in
Ref. [3], where potential law schematic is taken to calculate the
flux term through the limits of each internal control volume.
Integrating Eq. (5) in the control volume VCP in Fig. 2 related
to the variables u and v takes to Eq. (12). After defining these conditions, a computational program is developed in FORTRAN to solve
the problem proposed

@ 2 w
@ 2 w
 du  dv
 du  dv
2
2
VCp @u
VCp @v


@T 
@T 
a 
H
G
 du  dv
 du  dv
@u
@v
VCp
VCp



@C
@C
BN 
H
G
 du  dv
 du  dv
@u
@v
VCp
VCp

(12)
Patankars power law [3] was adopted to calculate the flow
terms through the frontier of each internal control volume, and
takes to the equation in the following form:
Journal of Heat Transfer

(13)

where Du, Dv, (@u)w, (@u)e, (@v)n, and (@v)s are values represented in Fig. 2.
Ai;j Ai1;j Ai1;j Ai;j1 Ai;j1
  
3
2

Dv


H

T

T
i;j
i1;j
i1;j
7
6
2
7
6
  
 
7
6

Du
Dv
7
6


 Ti;j1
Bi;j a  6 Gi;j 
Ti;j1
N  Hi;j 
7
7
6
2
2
7
6 





5
4
Du




 Ci;j1 Ci;j1
 Ci1;j Ci1;j BNGi;j 
2
(14)
To obtain the equations of the chemical constituent and the
energy, in the discrete mode, it proceeds in an analogical way.
For dimensionless terms, the new coordinate system (u,v) is
established following boundary and initial conditions as the
dominion shows in Fig. 1:
For u 0 and v1 < v  0 ) W 0

(15)

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Distinguished equations together with boundary and initial

conditions make a coupled system involving stream function,
temperature, and concentration variables. The numerical solution in this system is treated using simple schematic purposed
in Ref. [3]. To solve these simultaneous mathematical equations
that come from distinguished process, line-to-line iterative
method is used.
In each process interaction, there was a need for updating
w*, T*, and C* values, and w* equation was solved three times
for each interaction. To reach T* and C* values, it was solved
only once by iteration. Such process was widely useful in cases
where Ra and Le are high, which causes stronger convection
streams.
The acceptance standard of a solution as converged is based on
the maximum mistake possible inside the whole calculation range.
The obtained results convergence was accepted when relative
changes in the dependent variables were below 1.0  105.
Fig. 3 Influence of Ra on Nu in the cylinder and in the wall for
BN 5 0 and Le 5 1

u p and v1 < v < 0 ) W 0;

@T
@C

0
@v
@v

(16)

v v1 and 0  u  p ) W 0;
T  C 1 over the buried cylinder

(17)

v 0 and 0  u  p ) W 0;
T  C 0over floor surface

(18)

Boundary conditions presented in Eqs. (15)(18) refer to the

flow off domain covering the heated cylinder. The condition
W* 0 refers to the stagnated fluid. The condition @T  =@v
@C =@v 0 refers to the mass and fluid absence.

Fig. 4 Influence of Le and Ra on Sherwood in the wall (BN 5 0):

(a) Sh up to 6 and (b) Sh up to 35

Results and Discussion

A flow class basically dominated by thrust due to the cylinder
heating was considered. In this case, BN 1 and the distribution
of chemical constituent have a small influence on fluid movement.
The value of BN was fixed on zero for simulation purposes. In
this boundary, temperature field is linked to flow field but is independent of concentration distribution. Both fields are dependent
on Rayleigh number but are not dependent on Lewis number.
However, chemical constituents distribution depends on both the
fields. So, the concentration field is influenced by both Rayleigh
and Lewis numbers, and the difference between the temperature
and concentration fields is dictated by Lewis number. It can be
shown from Eqs. (6) and (7) that both fields are identical for
Lewis equal to 1 and only one of these equations is sufficient to
describe the problem of this singular case.

Fig. 5 Influence of Ra on the distribution of local Nu (BN 5 0):

(a) in the cylinder and (b) in the wall

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For Ra 0.1, the problem is reduced to virtually a cylinder buried in a purely conductive media. In this case, most of the heat is
transported by conduction. When Ra increases, the convective
effects become important.
In case Ra 10, a plume forms between the wall and the heated
cylinder. The very close lines in temperature fields in the vicinity
of the cylinder surface indicate a greater temperature gradient in
this region. The Ra 10 well represents the transition from purely
conductive flow to the highly convective one.
Note that the increase in Rayleigh number intensifies the flow
more and more near to the top of the cylinder buried and the line
of symmetry of the plume that is formed on the cylinder.
In Fig. 3, the influence of Rayleigh on the Nusselt numbers in
the cylinder and the wall is observed. Heat exchange with the
cylinder increases monotonically with the Rayleigh number. This
behavior also occurs for Nusselt in the wall.
Figure 4 shows the influence of Rayleigh and Lewis parameters
on the average Sherwood number of the flow for BN 0, where it

can see the same Sherwood behavior as a function of Rayleigh

and Lewis.
Figure 4(a) shows that for low Rayleigh, the conductive effect
is predominant for every value of Lewis. Similar effect can be
observed in Fig. 4(b), where a little influence of Rayleigh number
is shown for low values range of Lewis number. Figure 4(a)
shows the monotonically behavior of the average Sherwood
curves. The results obtained with Le 0.1 already approximate
closely to the pure conduction effect, that is, Sh 1, where it can
also be noted that curves for Le 50 and Le 100 are quite close
and parallel, indicating a fully convective flow.
Figure 4(b) confirms previous observations. This figure also
shows that for Ra 100, the predominant effect on the convection
flow is indicated by a linear curve with a variation in Lewis number.
Figure 4(b) shows the monotonically behavior of Sherwood
curves in the wall due to Rayleigh. Such behavior is very similar
to that shown in Fig. 4(a) for Sherwood in the cylinder due to
Rayleigh.

Fig. 6 Influence of Ra on local Sherwood distribution in the

cylinder for: (a) Le 5 0.1 and BN 5 0; (b) Le 5 10 and BN 5 0; and
(c) Le 5 100 and BN 5 0

Fig. 7 Influence of Le on local Sherwood distribution in the

cylinder for: (a) Ra 5 1 and BN 5 0; (b) Ra 5 100 and BN 5 0; and
(c) Ra 5 1000 and BN 5 0

Journal of Heat Transfer

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Figure 5 shows the distribution of the local Nusselt around the

cylinder (Fig. 5(a)) and along the wall (Fig. 5(b)) for a broad
range of Rayleigh numbers.
Figure 5(a) shows that the local Nusselt numbers are higher in
the cylinder region opposite to the wall (h 0). Although the fluid
movement in this region is smaller, the fluid is completely cold to
come into contact with it, causing a high gradient. For all
values of Rayleigh considered, the local Nusselt number decays
continuously as the fluid moves toward the plume region.
Figure 5(a) also shows that the transfer of cylinder heat is predominantly conductive when the Ra value is 10.
Figure 5(b) shows that for high values of Rayleigh, there is a
tendency for a constant heat exchange in the wall; in this case, it
is important to note that this is a wall with constant temperature
imposed.
Figure 6 shows the distribution of the local Sherwood
along the cylinder for Le 0.1, 10, and 100, respectively, in
Figs. 6(a)6(c).
It can be observed that for Ra of approximately 10 with Le 0.1,
the diffusive characteristic is modified as shown in Fig. 6(a). With
increasing in Lewis, this limit drops for Ra 0.1 with Le 10, as
shown in Fig. 6(b). For Le 100, diffusive characteristics are not
observed for any values of Ra, as shown in Fig. 6(c).
Figure 6(a) shows that for low Lewis even with high Rayleigh,
it is difficult to achieve purely convective effects across the surface of the cylinder. The change is noted in Fig. 6(b) for Le 10.
Figure 6(c) with Le 100 shows that from Ra 100 has an
influence that is convective dominant as can be noted by the
appearance of uniform distribution of Sherwood.
Figure 7 shows the influence of the Rayleigh distribution on the
local Sherwood distribution in the cylinder, which is similar to
that observed in Fig. 6 for varying of the Lewis number. For the
case BN 0, the great similarity of Rayleigh and Lewis influences
on the distribution of local Sherwood in the cylinder.

Conclusions
The study of doubly diffusive natural convection for a cylinder
buried in a homogeneous porous and saturated media was investigated numerically.
The equations describing the problem were expressed in bipolar
coordinates according to stream function formulation and numerically solved by the method of control volume.
The program implemented allowed to achieve satisfactory
results and enabled a better understanding of the influence of
Rayleigh and Lewis numbers on flows driven by heat.
The Nusselt and Sherwood numbers as a function of Rayleigh,
Lewis, and the ratio of thrust N were widely analyzed and their
effects relatively explained within the range of values specified in
this work.
The results for the heat driven flow directed allowed basically
demonstrating purely conductive, conductive, and convective, and
strongly convective regions for different values of Rayleigh and
Lewis considered.
A finite volume scheme has been used to predict the flow,
temperature, and concentration distributions at any space from a
heat cylinder buried into a fluid-saturated porous medium for a
bipolar coordinates system.
Examples presented show that the differences in the flow
distribution caused not only when Rayleigh number range is
considered but also when Lewis number range is considered.
Further, increase in the Rayleigh number has a significant influence in the flow distribution when the concentration distribution is
considered.
Steady natural convection heat transfer by double diffusion
from a heated cylinder buried in a saturated porous medium is
studied numerically using the finite volume method. To model
fluid flow inside the porous medium, the Darcy equation is used.
Numerical results are obtained in the form of streamlines, isotherms, and isoconcentrations. The Rayleigh number values range

from 0 to 1000, the Lewis number values range from 0 to 100,

and the buoyancy ratio number is equal to zero. Calculated values
of average heat transfer rates agree reasonably well with values
reported in the literature.

Acknowledgment
Dr. Wendell de Queiroz Lamas thanks his "Productivity Scholarship in Research," supported by the Brazilian National Council
for Scientific and Technological Development (CNPq), Grant
No. 301260/2013-3.
Dr. Jose Rui Camargo and Dr. Francisco Jose Grandinetti thank
their "Productivity Scholarship in Technological Development
and Innovative Extension," supported by the Brazilian National
Council for Scientific and Technological Development (CNPq),
Grant No. 311795/2012-9.

Nomenclature
a
A
B
BN
C
d
D
F
g
G
g
h
H
K
Le
Nu
Qx
Qy
r1
Ra
Sh
T
u, v
V
V
v1
x, y

scale factor to bipolar coordinates, dimensionless

coefficient of volume control method dimensionless
coefficient of volume control method dimensionless
buoyancy ratio number, dimensionless
chemical concentration, kg m3
cylinder depth at the superior surface, m
chemical diffusivity, m s2
function, dimensionless
gravity acceleration, m s2
function, dimensionless
gravity acceleration vector, m s2
scale factor, dimensionless
function, dimensionless
porous media permeability, m2
Lewis number, dimensionless
Nusselt number, dimensionless
dimension on x axis, m
dimension on y axis, m
buried cylinder radius, m
Rayleigh number, dimensionless
Sherwood number, dimensionless
temperature, K or  C
bipolar coordinates, m
average velocity in u or v direction, m s1
average velocity vector, m s1
v coordinates values at the cylinder, m
Cartesian coordinates, m

Greek Symbols
a
b
bc
D
r
h
l

q
W
@

thermal diffusivity, m2 s1

coefficient of thermal expansion,  C1
coefficient of chemical expansion, m3 kg1
variation in a variable, dimensionless
differential operator, dimensionless
value of variable u over cylinder (v v1), dimensionless
dynamic viscosity, N s m2
kinetic viscosity, m2 s1
specific mass, kg m3
stream function, dimensionless
variation defined in typical cell of control volume method
(Fig. 2)

Subscripts
e
i
j
N

index defined in control volume method (Fig. 2)

index
index
point defined in a typical cell at control volume method
(Fig. 2)
o reference value

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S point defined in a typical cell at control volume method

(Fig. 2)
u u direction
v v direction
VCp control volume defined in point P (Fig. 2)
W point defined in a typical cell at control volume method
(Fig. 2)

Superscript
* dimensionless parameters

References
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Around a Cylinder Buried in a Saturated Porous Media, IV Brazilian Congress
of Thermal Sciences and Engineering, Rio de Janeiro, pp. 110 (in Portuguese).
[2] Chaves, C. A., 1990, Natural Convection Double Diffusion Around a Cylinder
Buried in a Saturated Porous Media, Ph.D. thesis, State University of Campinas, Campinas, SP, Brazil (in Portuguese).
[3] Patankar, S., 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere,
New York.

Journal of Heat Transfer

[4] Chaves, C. A., Camargo, J. R., Cardoso, S., and Macedo, A. G., 2005,
Transient Natural Convection Heat Transfer by Double Diffusion From a
Heated Cylinder Buried in a Saturated Porous Media, Int. J. Therm. Sci.,
44(8), pp. 720725.
[5] Nield, D. A., and Bejan, A., 2013, Double-Diffusive Convection, Convection
in Porous Media, Springer, New York.
[6] Trevisan, O. V., and Bejan, A., 1990, Combined Heat and Mass Transfer by
Natural Convection in a Porous Medium, Adv. Heat Transfer, 20,
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[7] Diersch, H.-J. G., and Kolditz, O., 2002, Variable-Density Flow and Transport
in Porous Media: Approaches and Challenges, Adv. Water Resour., 25(812),
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[8] Mamou, M., 2002, Stability Analysis of Double-Diffusive Convection in
Porous Enclosures, Transport Phenomena in Porous Media II, D. B. Ingham
and I. Pop, eds., Elsevier, Oxford, UK.
[9] Mojtabi, A., and Charrier-Mojtabi, M. C., 2005, Double-Diffusive Convection
in Porous Media, Handbook of Porous Media, 2nd ed., K. Vafai, ed., Taylor
and Francis, New York.
[10] Bau, H. H., 1984, Convective Heat Losses From a Pipe Buried in a SemiInfinite Porous Media, Int. J. Heat Mass Transfer, 27(11), pp. 20472056.
[11] Bejan, A., 1984, Convection Heat Transfer, Wiley, New York.
[12] Bird, R. B., Stewart, W. E., and Lightfoot, E. N., 1976, Transport Phenomena,
Wiley, New York.
[13] Moon, P., and Spenser, D. E., 1971, Field Theory Handbook, Springer, Berlin.

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