Journal of Applied Science and Engineering, Vol. 19, No. 3, pp. 313-319 (2016)
DOI: 10.6180/jase.2016.19.3.09
Convection of Cu-water Nanofluid in a Partially Active
Porous Cavity with Internal Heat Generation
N. Nithyadevi and M. Rajarathinam*
Department of Mathematics, Bharathiar University,
Coimbatore 641 046, Tamil Nadu, India
Abstract
In this study, the effect of internal heat generation for Cu-water nanofluid on natural convection
heat transfer in a fluid saturated porous cavity with partially active walls has been numerically
investigated. The governing non-dimensional Darcy-Brinkman-Forchheimer equations are solved
using the finite volume approach together with SIMPLE algorithm. Benchmark results are compared
with present study which furnish that the present results are to be reliable. The addition of
nanoparticles produces an augmented heat transfer rate for low values of internal heat generation. On
the other hand, the base fluid water induces the maximum heat transfer rate than the nanofluid for high
values of internal heat generation parameter. This means that in the presence of high internal heat
generation, there is no need to add nanoparticles inside the cavity to generate the augmented heat
transfer rate.
Key Words: Heat Generation, Nanofluid, Partially Active Walls, Porous Medium
1. Introduction
Natural convection heat transfer in a porous medium
has greatly stimulated the researchers due to plenty of
applications in the field of thermal insulation, oil recovery, ground water flow modeling, underground coal gasification, solar power collectors, food processing and
migration of sea water etc., Nield and Bejan [1] and Vafai
[2] gave a detailed note on various models such as Darcy,
Darcy-Brinkman, Darcy-Brinkman-Forchheimer, and are
mainly used to compute the fluid flow in the fluid saturated porous medium. The last two models are generally
called as non-Darcy models and are mostly handled by
the authors, because it has rectified the limitations obtained from Darcian model since Darcy law holds only
for low values of Reynolds number. The authors [3,4]
had taken the non-Darcy models to examine the flow
and heat transfer rate in a porous cavity. The effect of
porosity in a three dimensional flow of heat and mass
transfer in a porous medium is numerically investigated
*Corresponding author. E-mail: [email protected]
by Attia [5]. Convection of nanofluids is one of the ongoing research fields due to the excellent thermal properties of nanofluids, where the nanofluid is a new kind
of fluid produced by the dispersion of nanoparticles in
the base fluid such as water, oil and ethylene glycol etc.,
Sheikhzadeh et al. [6] studied the natural convection heat
transfer in a partially active walls using Cu-water nanofluid. They found out that increasing the value of solid
volume fraction leads to increasing amount of heat transfer rate. The effect of magnetic field and heat generation/absorption in a nanofluid filled cavity is analyzed by
Teamah and El-Maghlany [7]. They proposed that for
low values of heat absorption coefficient the heat transfer rate increases with solid volume fraction. On the other
hand, the influence of heat generation/absorption in a
stagnation point flow over a permeable stretching surface is analyzed by Attia [8].
One of the main industrial applications of nanofluid
in porous media is the Enhanced Oil Recovery (EOR).
The EOR is the most growing industry due to demand of
oil worldwide. In EOR, various nanofluids are used in
the porous structured machines to getting enhanced oil.
314
N. Nithyadevi and M. Rajarathinam
Thermal (heat) injection is one of the techniques in EOR.
In this approach, various methods are used to heat the
crude oil, and hence the effect of heat generation in a
nanofluid porous media is a worth of study. Also, the dispersion of nano-sized particles in the traditional fluid
increases the thermal conductivity of the fluid, and the
presence of porous medium enhances the effective thermal conductivity of the base fluid. The recent works [9,
10] concerns nanofluid in porous filled cavity to get a
higher heat transfer rate. The main objective of the present problem is to investigate the effect of heat generation in a Cu-water nanofluid filled porous cavity. The
problem description, numerical procedure, results and
discussion and conclusions of the present study are stated
in the following sections.
to be constant except the density variation appeared in
the buoyancy term following from Boussinesq approximation. For the above assumptions the equations of continuity, momentum and energy that are govern the present problem written in the form of dimensionless as [10]
(1)
2. Mathematical Formulation
Figure 1 (a and b) demonstrates the physical representation of the present problem of two-dimensional porous square cavity of size H filled with a high thermally
conducting Cu-water nanofluid. The cavity is partially
heated as well as cooled by active parts of constant temperatures Th and Tc (Th > Tc) for left and right active parts
respectively, such that the active parts are placed on the
middle of the vertical walls. The length of the active part
is equal to half of the cavity height (H/2), while the remaining portions of the walls are perfectly insulated. It
is assumed that the solid matrix of the porous medium
does not undergo deformation and it is homogeneous,
isotropic, fluid saturated and incompressible. Also, the
fluid and solid matrix are in Local Thermal Equilibrium
(LTE), which means that the temperature of solid and
fluid phases are equal. In order to analyze the flow and
heat transfer characteristic in a fluid saturated porous
medium, the Darcy-Brinkman-Forchheimer derived by
Nield and Bejan [1] and Vafai [2] is adopted in the present study. The nanofluid in the cavity has the uniform
shape and size of Cu nanoparticles dispersed in the base
fluid. In fact, due to the extreme size and low concentrations of the suspended nanoparticles, the particles are assumed to move with the same velocity as the base fluid.
Also, we consider the nanoparticles and the base fluid
(water) are in thermal equilibrium, and no slip occurs between them. The properties of nanofluid are considered
where Fc =
175
.
is the Forchheimer constant, U and
150 e 3/ 2
V are the velocities along the X and Y coordinates, P is
the fluid pressure, u is the kinematic viscosity, a is the
thermal diffusivity, b is the thermal expansion coefficient, e is the porosity of the porous medium and K is
the permeability of the medium. Here the subscripts nf
and f represents the nanofluid and base fluid, respectively.
In the momentum balance equations, the left hand
side represents the inertial force term whereas the right
hand side represents respectively the pressure gradient
term, Brinkman viscous term, Darcy term and the Forchheimer term. Moreover the last term appeared in the
V-momentum and energy equations are the body force
term caused by buoyancy and the source term due to the
internal heat respectively.
The dimensionless variables in the above equations
are defined as
(2)
and the non-dimensional paramete
Convection of Cu-water Nanofluid in a Partially Active Porous Cavity with Internal Heat Generation
315
of cavity and the initial and boundary conditions for temperature is
t = 0: q = 0, for 0 £ X £ 1, 0 £ Y £ 1
t > 0: An active part q = 1 at X = 0, q = 0 at X = 1 &
¶q
=
¶n
0, elsewhere
where n is normal direction to plane
The average Nusselt number is calculated by
Nu = -
k nf
kf
¶q
dY at X = 0
¶X
H/ 4
3H / 4
ò
(3)
The effective density, the thermal diffusivity, the thermal conductivity, the heat capacitance and the dynamic
viscosity of the nanofluid are calculated using the classical models reported in the literature [6,10].
3. Numerical Analysis
Figure 1. (a) Three dimensional view and (b) Computational
domain for present study.
Da =
s=
uf
gbL3 (Th - Tc )
qL2
K
,
,
Ra
=
,
Q
=
,
Pr
=
af
(rcp ) nf a f
ufa f
L2
e (rcp ) nf + (1 - e )(rcp ) s
(rcp ) nf
are the Darcy number, is the
relative effect of the permeability of the porous medium
versus its cross sectional area, the Rayleigh number, is
the ratio of buoyancy to viscous force, the Prandtl number, is the ratio of momentum diffusivity to thermal diffusivity, the internal heat generation parameter, is the
measurement of heat inside the cavity and the specific
heat ratio, is the relative importance of heat capacity between the nanofluid and saturated porous media respectively.
No slip velocity conditions is applied for all the walls
The non-dimensional Darcy-Brinkman-Forchheimer
equations are discretised by the finite volume method
reported by Patankar [11]. Due to the complexity between the pressure and velocity terms, SIMPLE algorithm is incorporated to the momentum equations. The
convective and diffusion terms are approximated using
the central and power law schemes respectively. The
resulting set of algebraic equations are solved using the
Thomas algorithm with line by line iterative process, and
this procedure is repeated until a convergent solution in
obtained. The convergence criteria chosen in the present problem as the sum of the residuals of U, V and q
have been met less than the value of 10-5. Figure 2 displays the comparison of streamlines and isotherms for
present and previous work [10]. In that work, the cavity
is filled with a single clockwise cell with high circulation rate due to the high Rayleigh number. Also it is
seen that, the streamlines are highly packed near the top
right and bottom left corners of the cavity. For the case of
isotherms, the lines are perpendicular to the active walls
which mean that the transfer of heat is attained by the
convective mode of heat transfer. In addition to that, a
thermal boundary layer is formed at the isothermal walls,
in particular at the top right and bottom left sides of the
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N. Nithyadevi and M. Rajarathinam
Figure 2. Streamlines and isotherms for present work compared with Nguyen et al. [10] using Q = 0, f =
0.025, e = 0.4, Ra = 107 and Da = 10-4.
cavity. The same result is obtained for the present study
also. Henceforth, the comparison study shows that a
good agreement is found between the two numerical
methods.
4. Result and Discussions
The natural convection in a porous cavity filled with
heat generating Cu-water nanofluid is studied numerically. The numerical simulations are carried out to find
the effect of heat generation parameter (Q) varies from 0
to 10, Darcy number (Da) from 10-5 to 10-1, solid volume
fraction from 0.0 to 0.1 and the Rayleigh number (Ra),
porosity (e) and specific heat ratio (s) are fixed at 106,
0.4 and 1.0 respectively.
Figure 3 displays the streamlines and isotherms of
nanofluid (f = 0.1) and pure water (f = 0.0) for various
internal heat generation parameter Q with fixed e = 0.4
and Da = 10-3. For Q = 0, the streamline nature is characterized by a single clockwise cell occupied in the whole
cavity. The isotherms are crowded near the active walls,
and also the lines are perpendicular to the isothermal
walls at the core of the cavity. This manifests that the
convection is prominent at the interior of the cavity where
as conduction takes place near the active walls. Moreover, the fluid temperature is maintained at the surface
temperature. Increasing the value of Q induces the opposing buoyancy effect to the fluid at the vicinity of the
hot wall. For this reason behind, streamlines are significantly suppressed and secondary cell is created at the left
corner of the top wall for increasing value of Q. Thus for
high Q, the whole cavity occupied a two circulating cells
of different strength and oppositely moving direction of
motion. This result can be found the literature study of
[7]. Also it can be seen that the flow rate increase remarkably when the value of internal heat generation is
augmented. In case of isotherms, the fluid temperature
increases slowly and exceeds the surface temperature as
the value of Q increases. Also it can be viewed that the
Figure 3. Steady state streamlines and isotherms for nanofluid with f = 0.1 (solid line) and pure water (dotted line) for different Q
with fixed Da = 10-3.
Convection of Cu-water Nanofluid in a Partially Active Porous Cavity with Internal Heat Generation
heating effect is gradually getting diminished, due to the
increasing amount of heat within the cavity.
In order to understand the effect of porous medium,
the non-dimensional parameter Da is varied for a particular range which is presented in Figure 4. For Da = 10-5,
a single clockwise cell is observed which is occupied in
the whole cavity. The isotherms are nested near the cooling active part, and are perpendicular to the vertical walls
for remaining places. This shows that conduction mode
of heat transfer is dominant when the value of Da is very
low. Increasing value of Da physically means that the
permeability of the porous medium increases and this
317
leads to increasing rate of fluid motion in the cavity. Also
the streamlines are internally suppressed due to the dominance of permeability. In case of isotherms, thermal
boundary layers are formed at the vicinity of the active
walls which means that the steep temperature gradient
occurs in those regions. This exhibits that for increasing
value of Darcy number, the convection state is more enhanced. Also the thermal boundary layer of cold active
part is thinner than the active hot part due to the fixed
heat generation parameter.
Figure 5(a) and (b) shows the average Nusselt number for various Q and Da over different solid volume frac-
Figure 4. Steady state streamlines and isotherms for nanofluid with f = 0.1 (solid line) and pure water (dotted line) for different
Da with fixed Q = 2.
Figure 5. Average Nusselt number versus f (a) for different Q with fixed Da = 10-3 and (b) for different Da with fixed Q = 2 at f =
0.05.
318
N. Nithyadevi and M. Rajarathinam
tions. From Figure 3, the temperature gradient near the
hot active part is decreased when the value of Q is increased, and this due to the enhancement of heat within
the cavity. This concept leads to, the decreasing rate of
heat transfer inside the cavity. The above explanations
are derived from Figure 5(a). Also it can be seen that, the
addition of nanoparticles generate the augmented heat
transfer rate for very low values of internal generation, in
particular Q = 0 and 2. The increasing rate of heat transfer for f is slowly transmitted into decreasing rate when
Q is augmented, and this is because of overwhelmed production of heat within the cavity. The similar results
were noticed in the earlier work [7] also. On the other
hand, the heat transfer rate increases with increase in
both Da and f. In addition, for low values of solid volume
fractions (f = 0.0, 0.05), the heat transfer rate shows no
significant variation, and this is true for all Da.
Figure 6 shows the time history of average Nusselt
number for different Q. Initially, the heat transfer falls
down and increases slightly as time increases and finally
reaches the steady state. Furthermore, the required running time increases by increasing the value of Q.
Darcy number (10-5 £ Da £ 10-1) and solid volume fractions (0.0 £ f £ 0.1). The following conclusions can be
drawn in the present analysis.
The flow rate and fluid temperature increases significantly with increasing value of Q.
The heat transfer rate increases with Da but decreases
with Q.
The addition of nanoparticles inside the cavity, produce the enhancement of heat transfer especially at low
values of solid volume fraction. But in the case of high
internal heat generation, the opposite results were obtained. That is, the heat transfer rate decreases as the value
of solid volume fraction increases and this is due to the
extreme amount of heat generated inside the cavity.
5. Conclusions
[1] Nield, D. A. and Bejan, A., Convection in Porous Media, 4th ed., Springer-Verlag, New York (2013). doi:
10.1007/978-1-4614-5541-7
[2] Vafai, K., Handbook of Porous Media, 2nd ed., Taylor
and Francis, Boca Raton (2005). doi: 10.1201/9780
415876384.pt3
[3] Nithiarasu, P., Seetharamu, K. N. and Sundararajan,
T., “Natural Convection Heat Transfer in a Fluid Saturated Variable Porosity Medium,” Int. J. Heat Mass
Transfer, Vol. 40, No. 16, pp. 3955-3967 (1997). doi:
10.1016/ S0017-9310(97)00008-2
[4] D. Santhosh Kumar, D., Dass, A. K. and Dewan, A.,
“Analysis of Non-Darcy Models for Mixed Convection in a Porous Cavity Using Multigrid Approach,”
Numer. Heat Transfer Part A, Vol. 56, No. 8, pp.
685-708 (2009). doi: 10.1080/10407780903424674
[5] Attia, H. A., “On the Effectiveness of Porosity on
Unsteady Mixed Convection Flow along an Infinite
Vertical Porous Plate with Heat and Mass Transfer,”
Tamkang. J. Sci. Engg., Vol. 14, pp. 285-291 (2011).
doi: 10.6180/jase.2011.14.4.01
[6] Sheikhzadeh, G. A., Arefmanesh, A., Kheirkhah, M.
The present study deals the effect of internal heat
generation in a nanofluid saturated porous cavity with
partially active walls. The numerical simulations are presented in the form of streamlines, isotherms and average
Nusselt number graphs for various non-dimensional parameters such as internal heat generation (0 £ Q £ 10),
Figure 6. Time history of average Nusselt number for different Q with fixed Da = 10-3, f = 0.05.
Acknowledgements
The author M. Rajarathinam would like to acknowledge the Rajiv Gandhi National Fellowship (RGNF) for
their financial supports.
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Convection of Cu-water Nanofluid in a Partially Active Porous Cavity with Internal Heat Generation
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Manuscript Received: Jan. 18, 2016
Accepted: Apr. 24, 2016