In this work, the effect of suction/injection on transient free convective flow in vertical porous (suction/injection on the channel surfaces) channel filled with porous material in the presence of thermal dispersion was studied. The Boussinesq assumption is applied and the nonlinear governing equations of motion and energy are developed. The time dependent problem is solved using implicit finite difference method while steady state problem is solved by perturbation technique method. The solution obtained is graphically represented and the effects of suction/injection, time, Darcy number, thermal dispersion, and Prandtl number on the fluid flow and heat transfer characteristics. During the course of computation, an excellent agreement was found between the well-known steady state solutions sand transient solutions at large value of time. Furthermore, the time required to reach steady state velocity and temperature field strongly dependent on suction/injection parameter, Prandtl number and thermal dispersion parameter. The introduction of suction/injection has distorted the symmetric nature of the flow formation.
Because of its relevance to a variety of applications including solar
receiver devices, building thermal insulation, heat exchangers, energy
storage units, ceramic processing, catalytic reactors and energy
recovery in high temperature furnaces, convection in porous media is a
well-developed field of investigation. Many studies have been
conducted on unsteady and steady free convective flow in vertical
channel filled with porous material under different physical situations
(see refs. [1,2, 3]). A numerical study of natural convection inside a
porous wavy cavity filled with a nanofluid under the effect of thermal
dispersion has been carried out using the Forchheimer-Buongiomo approach
by Sheremet et al., [4]. Umavathi et al., [5]
investigate the combined effect of variable viscosity and thermal
conductivity on mixed convection flow of a viscous fluid in a vertical
channel in the presence of first order chemical reaction. Kasaeipour
et al., [6] studied on free convection heat transfer and
entropy generation analysis of MWCNT-Mgo water nanofluid using Lattice
Boltzmann method in cavity with refrigerant solid body experimental
thermo-physical properties. Rahimi et al., [7] investigate
the free convection heat transfer performance using Lattice Boltzmann
method within the cavities filled with nanofluid in existence of
internal rigid bodies.
The natural convection heat transfer analysis is performed by different
approaches such as headline visualization, total and local entropy
generation, average and local nusselt number using Lattice Boltzmann
method in nanofluid filled cavity with partially heated and cooled walls
included by internal heaters was carried out by Rahimi et al.,
[8]. Salari et al., [9] reported a numerical study of
the natural convection and entropy generation for a layered fluid system
in a cuboid enclosure which is differentially heated from sides and
filled by two immiscrible gas/liquid fluids. Three dimensional natural
convection in a rectangle cuboid filled by two immiscible fluid of water
and air at different liquid height ratio has been studied experimentally
and numerically by Malekshak and Salari [10]. Al-Rashed et
al., [11] evaluate numerically entropy generation inside an inclined
cubical differentially heated cavity wit CNT- water nanofliud using FVM
based on 3D vertically-vector potential formalism.
On the other hand, the thermal dispersion occurs term exists as a result
of both the micro-structure of the porous material as well as the heat
convection effect. One of the most important applications of including
thermal dispersion effect in the mathematical modelling of the heat
equation is the manufacturing processes of polymer composites such as
liquid composite molding. However, all the above literature review
indicates that there is still a scarcity of research on unsteady free
convective flow in vertical channel filled with porous material. Murthy
and Singh [12] have reported in a paper about the effect of viscous
dissipation on a non-Darcy natural convection boundary layer along an
isothermal vertical wall embedded in a fluid saturated porous medium,
that in the case when inertia terms are prevalent, the thermal
dispersion effects will become important. However, Hong and Tien
[1]30} have analyzed the problem of thermal dispersion effects on
natural convection about a heated horizontal cylinder in an enclosed
porous medium. Hsiao et al., [13] discussed the influences of
non-uniform porosity and thermal dispersion on natural convection about
a heated horizontal cylinder in an enclosed porous medium. Moreover,
Hsiao et al., [14] concluded that the effects of variable
porosity and thermal dispersion increases the average Nusselt number and
reduces the error between the experimentl data available and their
solutions. Kuznetsov [15] presented an analytical study of the
effect of the transverse thermal dispersion on fully developed forced
convection in a parallel plate channel filled with an isotropic fluid
saturated porous medium.
Meanwhile, Amiri and Vafai [16] suggested accounting for thermal
dispersion by assuming that the effective thermal conductivity consists
of both stagnant and dispersion conductivity. In their study, they
constructed their correlation based on the experimental findings of
Wakao and Kaguei [17]. Sheremet and Bachok [18] examined the
effect of thermal dispersion on transient natural convection in a
wavy-walled porous cavity filled with a nanofluid.
Adil et al., [19] study focused on the well-established improved
perturbed {NLSE}~with
the Kerr law non-linearity equation model. The model is integrated with
the help of the extended V-expansion method. Faridi et
al., [20] examined the fractional analysis of fusion and fission
process in plasma physics.
Asjad et al., [21] carried out the study on the
fractional comparative study of the non-linear directional couplers in
non-linear optics.}
Recently, Jha and Aina [22] carried out numerical analysis on
transient free convective flow in vertical channel filled with porous
material in presence of thermal dispersion. They reported in their study
that velocity and temperature is enhanced with the increase in thermal
dispersion parameter as well as time. From all the above discussed work,
these effects were studied and they have confirmed the importance of
thermal dispersion effects studied in these papers. {Therefore,
as a follow-up study of Jha and Aina [22], here, we include the
effects of suction/injection in the fluid. Suction or injection of fluid
through the bounding surfaces, as, for example, in mass transfer
cooling, it can significantly change the flow field and, as a
consequence, affect the rate of heat transfer from the bounding
surfaces. In general, suction tends to increase the skin-friction and
heat transfer coefficients whereas injection acts in the opposite
manners. Injection or suction of fluid through porous heated or cooled
surface is of general interest in practical problems involving film
cooling, control of boundary layers, etc. This can lead to enhanced
heating (or cooling) of the system and can help to delay the transition
from laminar flow. The current work is aimed as an extension of the
previous analytical studies to provide some insight into a number of
industrial applications, which use similar configurations.
2. Mathematical Analysis
Consider the unsteady free convective flow in vertical porous channel
formed by two infinite vertical parallel porous plates filled with
porous material in the pressure of thermal dispersion. Figure 1
represents the configuration of studied geometry in this work where
\(x’ -\)axis is taken along one of the channel plates and the
\(y’ -\)axis is taken normal to the channel porous plates. Both the
channel porous plates are assumed to be separated by a width \(2h\).
The
flow is assumed transient and fully developed. At time \(t’ \leq 0,\)
the fluid, and porous plates are assumed to be same temperature
\(T_{0}\), and there is no fluid motion. At time greater than zero, i.e.
\(t’ > 0,\) the temperature of the channel plates at \(y’ = – h\) is
raised to \(T_{w},\) due to temperature gradient between the plates,
natural convection flow occurs in the channel. Furthermore, at the same
time the flow is subjected to suction of the fluid from left vertical
porous plate and at the same rate fluid is being injected through the
right vertical porous plate. Recently, Jha and Aina [22] carried out
numerical analysis on transient free convective flow in vertical channel
filled with porous material in presence of thermal dispersion.
Figure 1. Physical model and coordinate system.
Following
Jha and Aina [22] and considering the effect of suction/injection in
the fluid, the mathematical model for the present physical situation
under Boussinesq’s approximation in dimensional form is:
Eqs. (\ref{e1}) – (\ref{e3}) can be nondimensionalized using the following
variables
\begin{equation}\tag{4}\label{e4}
\begin{cases}t = \frac{t’\nu}{h^{2}},\quad y = \frac{y’}{h},\quad U = \frac{u’}{u_{0}}, \quad \theta = \frac{T’ – T_{0}}{T_{w} – T_{0}},
\quad u_{0} = g\beta{\left( T_{w} – T_{0} \right)h^{2}}{\nu},\\
\Pr = \frac{C_{p}\mu}{k},\quad \gamma = \frac{\nu_{\text{eff}}}{\nu},\quad Da = \frac{k}{h^{2}},\quad kr = \frac{k_{m}}{k_{f}},\quad
S = \frac{V_{0}h}{\nu},\quad Gr = \frac{u_{0}\text{dp}}{\nu}.\end{cases}
\end{equation}
The physical quantities used in the above equations are defined in the
nomenclature.
Substituting Eq. (\ref{e4}) into Eqs. (\ref{e1}) – (\ref{e3}), the dimensionless
momentum and energy equations are:
The governing Eqs. (\ref{e5}) and (\ref{e6}) presented in the previous section
are highly nonlinear and exhibit no analytical solutions. The importance
of analytical solutions which refers to steady natural convection flow
in vertical porous channel filled with porous material in the presence
of thermal dispersion relies on the chance to obtain nontrivial
benchmarks to test the reliability of numerical codes developed for more
complex physical situations. Analytical solutions are often an
opportunity to inspect the internal consistency of mathematical models
and of the approximations adopted, as well as to develop new theoretical
results. The mathematical model representing the steady state natural
convection flow in vertical porous channel filled with porous material
in the presence of thermal dispersion can be obtained by setting
\(\frac{\partial U}{\partial t} = 0\) and
\(\frac{\partial\theta}{\partial t} = 0\) in Eqs. (\ref{e5}) and (\ref{e6}) to
get:
\begin{equation}\tag{10}\label{e10}
\begin{cases}
U = 0,\quad \theta = 1,\quad \text{at} \quad y = – 1,\\
U = 0,\quad \theta = 0,\quad \text{at}\quad y = 1.\end{cases}
\end{equation}
The above equations under given boundary conditions are solved by
perturbation method. In order to construct an approximate solutions of
Eqs. (\ref{e8}) and (\ref{e9}), subject to the boundary conditions in Eq.
(\ref{e10}), we employed a regular perturbation method by taking a power series
expansion in the thermal dispersion parameter \(C\) such as:
\begin{align}\tag{11}\label{e11}
&U\left( y \right) = U_{0}\left( y \right) + \varepsilon U_{1}\left( y \right) + \varepsilon^{2}U_{2}\left( y \right) + … = \sum_{i = 0}^{\infty}{\varepsilon^{i}U_{i}\left( y \right)},\\
&\tag{12}\label{e12}\theta\left( y \right) = \theta_{0}\left( y \right) + \varepsilon\theta_{1}\left( y \right) + \varepsilon^{2}\theta_{2}\left( y \right) + … = \sum_{i = 0}^{\infty}{\varepsilon^{i}\theta_{i}\left( y \right)}.
\end{align}
A two term expansion has been used here and terms smaller than
\(O\left( \varepsilon^{2} \right)\), where \(\varepsilon = Gr*{\Pr*}C\)
are neglected. Here \(\varepsilon\)is chosen as our perturbation
parameter. Obviously, the results are only valid for
\(\varepsilon < < 1\), as was the case in experiments reported in
[23] and it will be shown that for most cases of practical interest,
for which experiments are conducted, this requirement is met. One can
now obtain the zeroth and first order equations as
With no suction/injection parameter, i. e. \(S \rightarrow 0\), the
solution tends to the one given by Eqs. (\ref{e13}) — (\ref{e23}) of Jha and
Aina [22] in the absence of suction/injection parameter.
In the following section, Eqs. (\ref{e5}) – (\ref{e7}) are solved numerically for
velocity and temperature and the skin friction together with the rate of
heat transfer are computed.
4. Numerical solution
The momentum and energy equations given by Eqs. (\ref{e5}) and (\ref{e6}) are
solved numerically by using implicit finite difference method. The
procedure involves discretization of the transport Eqs. (\ref{e5}) and (\ref{e6})
into the finite difference equations at the grid point \((i , j).\) They
are, in order as follows:
The time derivative is replaced by the backward difference formula,
while spatial derivative is replaced by the central difference formula.
The above equations are solved by the Thomas algorithm by manipulating
into a system of linear algebraic equations in the tridiagonal form. In
each time step, the process of numerical integration for every dependent
variable starts from the first neighbouring grid point of the plate
at \(y = – 1\), using the tridiagonal form of the finite difference
Eqs. (\ref{e5}) and (\ref{e6}) until it reaches at immediate grid point of the
plate at \(y = 1\). In each time step the temperature field has been
solved and then the evaluated values are used to obtain the velocity
field. The process of computation is advanced until a steady state is
approached by satisfying the following convergence criterion:
with respect to the temperature and velocity fields, here,
\(A_{i,j}\) stands for the velocity and temperature fields, \(M\) is the
number of interior grid points and \(|A|_{\max}\) is the maximum
absolute value of\(\text{ A}_{i,j}\).
In the numerical computation, there is a need to specify \(\Delta t\) to
get a steady solution as rapidly as possible, yet small enough to avoid
instabilities. It is set, which is suitable for the present computation,
as
\[\Delta t = stabr \times \left( \Delta y \right)^{2}\,.\]
The parameter `\(\text{stabr}\)’ is determined by numerical experiment
in order to achieve convergence and stability of the solution procedure.
Numerical experiments show that the value 2 is suitable for numerical
computations. In order to confirm the validity of this numerical model,
the numerical scheme are compared with analytical solution derived for
steady state problem using perturbation technique. At large value of
time, the obtained numerical values using implicit finite difference
method for velocity, temperature, skin-friction and rate of heat
transfer are in excellent agreement with the obtained steady state
values using the perturbation method.
5. Results and discussion
Effect of suction/injection on transient free convective flow in
vertical porous channel formed by two infinite vertical porous plates
filled with porous material in presence of thermal dispersion has been
considered. The basic parameters that governed this flow are the
suction/injection parameter \(\left( S \right)\) which were
simultaneously applied each to opposite porous plates, Prandtl number
\(\left( \Pr() \right)\), which is inversely proportional to the thermal
diffusivity of the working fluid, the non-dimensional time
\(\left( t \right)\), Darcy number \(\left( \text{Da} \right)\), and
thermal dispersion parameter \(\left( C \right)\). For the purpose of
discussions, some numerical calculations are carried out for
dimensionless velocity, temperature, skin friction and rate of heat
transfer. The effects of various parameters on the flow are presented in
graphical form in Figures 2–10 so as to clearly reveal the influence
of each governing parameter on the flow formation and temperature
distribution. In addition, in experimental analysis of air flow through
aluminium foam heat exchangers, Calmidi and Mhajan [23] reported a
value of \(0.06\) for their \(C\) while their \(\text{Gr}\) (here
\(\text{Gr}\) is pore-Reynolds, i.e. square root of permeability is
defined as the length scale) was limited to \(135\) with their maximum
\(\Pr\)value being \(0.0075\). Note that this \(\Pr\) value is based on
effective thermal conductivity rather that of air. This combination
leads to a maximum of \(0.061\) for \(C\). Hunt and Tien [24], on
the other hand, reported theoretical and experimental analysis of
dispersion effects in foams with water as the working fluid leading to
\(0.6\) as their maximum \(C\) value. This study has been performed over
the reasonable ranges of \(0.01 < Da < 0.1,\) \(0.0 < t < 0.6,\) and
\(0.0 < C < 0.5\). The selected reference values of \(Da,t,\) and \(C\)
for the present analysis are \(0.1,\) \(0.2,\) and \(0.2,\) respectively
as given in Jha and Aina [22]. Furthermore, the value of suction/
injection has been performed over the reasonable ranges of
\(- 5 < S < 5\) and the selected reference values of \(S\) for the
present analysis is \(0.5\).
The effect of time on velocity profile is depicted in Figure 2. It is
clearly seen in Figure 2 that, by increasing the non-dimensional time,
the fluid velocity increases and finally attains its steady state. This
is consequence of temperature increase that results from increase in
time, since convection current becomes stronger and hence velocity
increases with time.
Figure 2. Velocity profile for different values of time \(\left( S = 0.5,Da = 0.1,C = 0.2,\Pr = 0.71 \right)\).
Figure 3 depicts the role of non-dimensional time \(\left( t \right)\)
on the transient temperature profiles. It is reveals that, temperature
increases with increase in time and finally attains its steady state.
Figure 3. Temperature profile for different values of time \(\left( S = 0.5,Da = 0.1,C = 0.2,\Pr = 0.71 \right)\).
Figures 4 and 5 describe the influence of Prandtl number on the velocity
as well as temperature profile, respectively. It is clearly shows in
these Figures that, the fluid velocity and temperature decreases as
Prandtl number increases. The physical explanation to this is that,
fluids with high Prandtl number have a lower thermal diffusivity and
high viscosity, which causes low heat penetration and reduce thermal
boundary layer.
Figure 4. Velocity profile for different values of Prandth number \(\left( S = 0.5,Da = 0.1,C = 0.2,t = 0.2 \right)\).Figure 5. Temperature profile for different values of Prandth number \(\left( S = 0.5,Da = 0.1,C = 0.2,t = 0.2 \right)\).
Figure 6 illustrate the impact of Darcy number on the velocity profile.
The velocity is observed to increases as Darcy number increase.
Figure 6. Velocity profile for different values of Darcy number \(\left( S = 0.5,C = 0.2,\Pr = 0.71,t = 0.2 \right)\).
Figures 7 and 8 show the effect of suction/injection parameter
\(\left( S \right)\) on the dimensionless temperature and velocity
profiles, respectively. In this study, when suction takes place at the
plate \(\left( Y = 1 \right)\) with a corresponding injection at the
plate\(\left( Y = – 1 \right)\), \(S\)is positive
\(\left( S > 0 \right)\). For the reverse phenomenon, \(S\) is negative
\(\left( S 0 \right)\) while the
reverse case is observed for \(\left( S 0 \right)\). This increases the temperature
thus strengthening the convection current in the channel, which in turn
increases the velocity. On the other hand, cold fluid particles are
introduced into the channel by injection through the plate
\(\left( Y = 1 \right)\) and hot fluid particles are removed from the
channel by suction through the plate \(\left( Y = – 1 \right)\) in the
case of \(\left( S < 0 \right)\), thus the reverse phenomenon is
observed.
Figures 9 and 10 shown the effect of thermal dispersion
parameter\(\left( C \right)\) on the velocity and temperature profiles,
respectively.. It is evident from Figures 9 and 10 that, the fluid
velocity and temperature increases with the increase in the thermal
dispersion parameter. This is physically true because, an increase in
thermal dispersion adds more heat to the fluid leading to an increase in
temperature, which causes velocity increase. Furthermore, it is
interesting to note from these figures that, the impact of thermal
dispersion parameter on the fluid velocity and temperature is more
pronounced for higher values of Darcy number.
Figure 7. Velocity profile for different values of C \(\left( Da = 0.1,S = 0.5,\Pr = 0.71,t = 0.2 \right)\).Figure 8. Temperature profile for different values of suction/injection \(\left( C = 0.2,Da = 0.1,\Pr = 0.71,t = 0.2 \right)\).Figure 9. Velocity profile for different values of C \(\left( Da = 0.1,S = 0.5,\Pr = 0.71,t = 0.2 \right)\).Figure 10. Temperature profile for different values of C \(\left( Da = 0.1,S = 0.5,\Pr = 0.71,t = 0.2 \right)\).
Figures 11 and 12 presents variation of skin-friction at
\(\left( Y = – 1 \right)\) and \(\left( Y = 1 \right)\), respectively,
for different values of suction/injection parameter \(\left( S \right)\)
and time \(\left( t \right)\). These figures reveal that the
skin-friction decreases with increase in suction/injection parameter on
the plate \(\left( Y = – 1 \right)\) while the result is reversed on the
plate \(\left( Y = 1 \right)\). In addition, the skin-friction is
observed to increase with increase in time on both walls.
Figure 11. Effects of suction/injection on the skin friction at \(y = – 1\).Figure 12. Effects of suction/injection on the skin friction at \(y = 1\).
Figures 13 and 14 displays the variation of rate of heat transfer for
different values of suction/injection parameter \(\left( S \right)\) and
time \(\left( t \right)\) at \(\left( Y = – 1 \right)\) and
\(\left( Y = 1 \right)\), respectively. It is observed that the rate of
heat transfer decreases with increase in suction/injection parameter and
time at \(\left( Y = – 1 \right)\) and converse is the case at
\(\left( Y = 1 \right)\).
Figure 13. Effects of thermal dispersion on the Nusselt number at \(y = 0\).Figure 14. Effects of thermal dispersion on the Nusselt number at \(y = 1\).
Finally, in order to see the accuracy of numerical solutions, numerical
values of velocity and temperature are presented in Tables 1 and 2,
respectively, for steady state operating conditions using perturbation
technique and implicit finite difference method for transient
mathematical model using large value of time \(\left( t = 2.6 \right)\). An excellent agreement is found between them.
Table 1. Comparison of numerical value of the transient state
velocity obtained using the implicit finite difference and the steady
state velocity obtained analytically.
Table 2. Comparison of numerical value of the transient state
temperature obtained using the implicit finite difference and the steady
state Temperature obtained analytically.
The effect of suction/injection on transient and steady free convective
flow in vertical porous channel formed by two infinite vertical parallel
porous plates filled with porous material in the presence of
thermal dispersion has been investigated. The temperature field and
velocity field are obtained analytically by perturbation method for the
steady situation and numerically by implicit finite difference technique
for transient situation. The skin-friction and rate of heat
transfer expressed as Nusselt number are derived from the velocity and
temperature, respectively. Graphical results for the temperature,
velocity, skin friction and rate of heat transfer are presented and
discussed for various physical parameter values. The main findings are
as follows:
The time required to reach steady state velocity and temperature field
strongly dependent on Prandtl number, suction/injection parameter and
thermal dispersion parameter.
The impact of thermal dispersion parameter on the fluid velocity and
temperature is more pronounced for higher values of Darcy number.
The heat transfer is higher at left porous plate
\(\left( Y = – 1 \right)\) where injection takes place in comparison
to right porous plate where suction takes place
\(\left( Y = 1 \right)\).
The introduction of suction/injection has distorted the symmetric
nature of the flow formation.
An increase in thermal dispersion parameter and time enhanced
the skin friction and rate of heat transfer.
Nomenclature
\(C =\) Dispersion coefficient
\(C_{\rho} =\) specific heat of the fluid at constant pressure
\(Da =\) Darcy number
\(g =\)acceleration due to gravity
\(Gr =\) Grashof number
\(h =\) gap between the plates
\(Nu_{0} =\) Nusselt number at \(\left( Y = – 1 \right)\)
\(Nu_{1} =\) Nusselt number at \(\left( Y = 1 \right)\)
\(\Pr =\)Prandtl number
\(S =\)suction/injection parameter
\(t’ =\) dimensional time
\(t =\) dimensionless time
\(T =\)temperature of the fluid
\(T_{0} =\) temperature of the fluid and plates in reference state
\(\left( t’ \leq 0 \right)\)
\(u’ =\) dimensional velocity of the fluid
\(U =\) dimensionless velocity of the fluid
\(V_{0} =\)constant suction/injection velocity
\(y’ =\)dimensional coordinate perpendicular to the channel walls
\(y =\) dimensionless coordinate perpendicular to the channel walls
Greek Letters
\(\beta =\) coefficient of thermal expansion
\(\gamma =\) ratio of kinematics viscosity
\(\theta =\) dimensionless temperature
\(\rho =\) density
\(\tau_{0} =\)dimensionless skin friction at \(\left( Y = – 1 \right)\)
\(\tau_{1} =\) dimensionless skin friction at \(\left( Y = 1 \right)\)
\(v =\)fluid kinematic viscosity
\(v_{\text{eff}} =\)effective kinematic viscosity
\(k =\) thermal conductivity
\(k_{m} =\) thermal conductivity of the solid phase
\(k_{f} =\) thermal conductivity of the fluid phase
Author Contributions:
All authors contributed equally in this paper. All authors read and approved the final version of this paper.
Conflicts of Interest:
The authors declare no conflict of interest.
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