Abstract
This paper intends to numerically study the steady-state free convection heat transfer in the presence of an exothermal chemical reaction governed by Arrhenius kinetics within a right-angled enclosure of triangular shape filled by porous media saturated with magnetized nanofluid. An approximation named as Darcy–Boussinesq approximation along with a nanofluid model mathematically propounded by Buongiorno has been implemented to model physical phenomenon representing fluid flow, heat transfer, and nanoparticle concentration. The mathematical equations in a dimensionless form describing the stream function for circulation of the fluid, the energy equation for heat, and nanoparticle volume fraction for concentration are solved using the finite difference method. The validity of the numerical procedure is established by comparing present results with the formerly available works in both statistical and graphical approaches. Streamlines, isotherms, and isoconcentrations are plotted and discussed for the various parametric regimes. The graphical description depicts that the average Nusselt and Sherwood numbers are the decreasing function of the Rayleigh number. The study revealed the accountable influence of model parameters such as thermophoresis and Brownian diffusion on the local Sherwood number, whereas a minimum impact on the local Nusselt number is observed.
1. Introduction
The flow of fluid and transfer of heat in a cavity filled by a nanofluid saturated with porous media are of vital importance in numerous practical situations. Some of these applications include geophysical applications, chemical reactors, building heating and cooling operations, electronic cooling equipment, and so on. Convective flows are studied extensively in enclosures filled with porous media as accounted in books by Pop and Ingham [1] and Bear [2]. Similarly, the enclosure of various geometric shapes is an essential aspect in engineering applications such as boilers or ovens with porous materials. Baytas and Pop [3] investigated the heat transfer characteristics for a steady-state flow in an inclined cavity filled with a porous medium. Sun and Pop [4] studied free convection in a triangular region filled with a porous medium. In their study, they placed a finite size flush mounted heater on the vertical wall of the cavity. They concluded that the average Nusselt number has a maximum value with a smaller aspect ratio of enclosure dimensions. It also has a lesser value by the lowest position of the heater with the highest magnitude of Rayleigh number and the maximum size of the heater. Izadi et al. [5] performed a numerical investigation of the natural convection heat transfer of a hybrid nanofluid into a porous cavity exposed to a variable magnetic field. Their findings indicate that at higher Rayleigh number, by increasing the Hartmann number, a significant decrease in Nusselt number occurs, which can be attributed to the decreased power of the flow. Arpino et al. [6] studied unsteady free convection in a partially porous annular enclosure and employed an adaptive local time-stepping procedure based on a properly stabilized full matrix-inversion scheme. They concluded that the position of the porous layer in the annulus strongly affected the stability of the free convection. Arpino et al. [7] have also studied porous medium-free fluid interface problems in the presence of high source terms by using a stable, accurate, and efficient artificial compressibility (AC) characteristic-based split (CBS) algorithm. In another paper, Arpindo et al. [8] presented an efficient three-dimensional algorithm based on the fully explicit matrix-inversion free finite element version of the characteristic-based split (CBS) scheme. The scheme was applied for the first time to the simulation of complex thermo-fluid-dynamic problems in domains containing a fluid and a porous layer. Massaroti et al. [9] adopted a dual time-stepping procedure based on the transient artificial compressibility version of the characteristic-based split algorithm to solve the transient equations of the generalized model for heat and fluid flow through porous media. Shermet and Pop [10] also studied steady-state free convection heat transfer in a right-angle triangular cavity filled with a porous medium using the mathematical nanofluid model proposed by Buongiorno [11].
The subject of nanofluids has received considerable concentration from the researchers for its role in engineering and Biophysics fields. The presence of nanoscale particles in the fluid enhances heat transfer characteristics and has been used widely in the industry. For example, materials with sizes of nanometers possess unique physical and chemical properties (Oztop and Abu-Nada [12]). They tested Cu (copper), (alumina), and (titania) as the base fluid to study the effect of nanoparticles on natural convection and thermal profiles. Choi [13] first introduced the term nanofluid which rooted from the addition of nanoscale particles to the fluids. The presence of nanoparticles in the fluid enhances the thermal conductivity within it. In a recent catalogue of articles by Kuznetsov and Nield [14] and Nield and Kuznetsov [15], a lot more have implemented Buongiorno’s mathematical model for nanofluids to study problems on Newtonian fluids and porous media filled by nanofluids. The authors reported that the use of surfactant or surface charge technology could cause suspension of nanoparticles in the nanofluid, hence preventing the particles from deposition on the porous matrix. Nanofluids have vital importance in the areas involving solar collectors to absorb solar thermal radiations. A wide set of nanofluid applications is recently reported in the books by Nield and Bejan [16] and YuS et al. [17]. Among several nanofluid mathematical models, Buongiorno’s model [11] received considerable attention since he noted that the effect of the velocity between nanoparticles and base fluids. Also, he has written down the conservation equations considering the two important effects which are the Brownian diffusion and the thermophoresis effects.
The area of fluid mechanics that investigates the characteristics of fluids with electrical conductivity in the presence of an electromagnetic field is known as Magnetohydrodynamics (MHD). MHD convection flow has a wide range of applications in the area of science and engineering. These include geothermal energy extractions, petroleum reservoirs thermal insulation, plasma confinement, crystal growth, and nuclear reactor coolers. Chamkha [18] investigated the hydrodynamic convection flow in a vertical lid-driven cavity with absorption or internal heat generation. Malleswaram and Sivasankaran [19] presented a numerical analysis on magnetohydrodynamic-mixed convection in a lid-driven cavity when both the side-walls are provided with nonuniform heating. AlimM et al. [20] have presented a numerical study for the conjugate effect of Joule heating and MHD-mixed convection in a lid-driven square cavity. Rahman et al. [21] also studied the free convection within a tilted cavity filled with nanofluid in the presence of a porous medium and a sloping magnetic field energized by an exothermic chemical reaction. They concluded that the presence of the magnetic field causes the nanoparticles to separate from the fluid with small velocity. The process of free convection in the presence of an exothermic surface reaction has widely been studied by Merkin and Chaudhary [22], Mahmood and Merkin [23], Ikeda et al. [24], Gray and Merkin [25, 26], and Merkin and Pop [27]. Single first-order kinetic equation with Arrhenius [28] thermal energy dependence models the reaction on the surface and is defined aswhere A and B are the reactants, denotes pre-exponential factor, a represents the reactant concentration, and E and R represent the activation energy and fluid constant, respectively. Rahman [29] investigated a higher order chemical reaction along with internal heat generation on a non-Darcian forced convection for an incompressible viscous fluid flow with thermal conductivity across an elastic surface interlaced in a porous medium. He concluded that with the increase in Darcy number, the dynamic viscosity alongside the thermal conductivity reduces.
It is apparent from the preceding literature study that most of the reviews are performed considering the nanofluids in cavities, but very less attention is paid to study the cavities filled with the porous medium saturated with nanofluids particularly in the presence of MHD and chemical reaction. So, in this paper, we study free convection within a triangular cavity filled with porous media saturated with nanofluid using Buongiorno’s model [11] incorporating Darcy’s law for the flow in the porous medium. We also employed the Boussinesq approximation to model the governing equations for the convective forces. The resulting partial differential equations of the advection-diffusion type are solved numerically using a finite difference method. The description of the scheme is presented in Appendix. To the best of authors’ knowledge, the problem of natural convection within a triangular cavity in the presence of a magnetic field driven by an exothermic chemical reaction using the mathematical nanofluid model has not been reported yet in the literature. As such, the focus of this paper is to study the detailed and attributed analysis on the flow and heat transfer due to the effects of pertinent parameters arising in the problem.
2. Problem Formulation
2.1. Physical Model
We consider an isosceles triangular-shaped cavity filled with a porous medium saturated with nanofluid consisting of water and nanoparticles. The -axis is along the bottom wall of width H, while the -axis is along the vertical wall of the cavity with length L and maintained at the constant temperature and the constant nanoparticle volume fraction . A similar assumption is also made for the inclined boundary, while the bottom wall along the -axis is assumed to be adiabatic. A magnetic field with strength is imposed within the fluid along the positive x-axis. An exothermal surface reaction governed by Arrhenius kinetics [28] is also taken into account inside the cavity which is of order one and nonisothermal. Figure 1 represents the schematic diagram of problem geometry and the coordinate system. Water-based nanofluid inside the cavity is chosen for the analysis and is set to be laminar and incompressible. The base fluid (i.e., water) and the nanoparticles are assumed to be in thermal equilibrium. Hence, no slip occurs between them. The suspension of nanoparticles within the nanofluid is expected to be carried out using either surfactant or surface charge technology. Inside the triangular cavity, the fluid flow is considered to be weak, laminar, and incompressible with no heat transfer due to the viscous dissipation as well as the radiation. The nanofluid density depends on the volume fraction of nanoparticles and the temperature, while the other thermophysical properties of the nanofluid are expected to be constant. Compared to the initial temperature , the increase in temperature inside the flow domain is assumed to be smaller. As reported by CardosoT et al. [30], the rise in temperature before a chemical reaction is approximately below , while general consideration of initial temperature is below . Hence, the Darcy–Boussinesq approximation is adapted for the present problem.
2.2. Governing Equations
With the assumption that viscous dissipation effects are negligible, the Oberbeck–Boussinesq approximation is employed, and Darcy’s law is applicable. By following the nanofluid model proposed by Buongiorno [11], the governing equations for total mass, momentum, energy, and nanoparticles, respectively, are written aswhere is given by , is the constant magnetic field strength along axis, and the buoyancy term is approximated by the Boussinesq approximation:
Here, is the velocity vector with and the nanofluid velocity components along and axis, respectively, T is the nanofluid temperature, C is the nanoparticle volume fraction, and other quantities of physical interest are mentioned in Nomenclature. To nondimensionalize the governing equations along with boundary conditions, we make use of the following similarity transformations:
Following the traditional way of defining the dimensionless stream function ψ from velocity components as follows:
Using the dimensionless variables defined in equation (4) and the stream function in equation (5), the governing equations now take the form:
The boundary conditions are as follows:where is the magnetic field parameter also known as the modified Hartmann number, is the Rayleigh number, is the buoyancy-ratio parameter, is the Brownian motion parameter, is the thermophoresis parameter, is the Frank-Kamenetskii number, is the Lewis number, and the Laplacian operator are all defined as
For problems related to heat and mass transfer across the cavity, it is essential to determine rates of heat and mass transfer in terms of local and average Nusselt and Sherwood numbers, respectively, using the following formulas:
3. Numerical Method
We employ the finite difference method in combination with an iterative procedure [31] to solve the governing equations (6)–(8) along with boundary conditions (9). A five-point cell-centered stencil with a central difference formula is applied to obtain the system of linear equations which are then addressed in MATLAB. The step size is chosen to be uniform in both x and y directions. For the convergence criteria, is chosen as a tolerance error for all dependent variables. The iterative procedure is terminated once the following convergence criteria are satisfied:where χ represents the dependent variables ψ, θ or , while M and N are the number of grid points along x and y directions, respectively, and n is the iteration index.
3.1. Test for Grid Independence
For correct usage of the finite difference method, it is essential to perform grid sensitivity analysis on the converged solutions. Thus, eight cases of a uniform grid varying from to with an increment of are tested. The numerical simulations for the grid independence test are shown for selected values of the model parameters , , , , , and . The test is conducted by plotting the average Nusselt number against the number of grid points. As can be seen in Figure 2, the value of the average Nusselt number almost becomes independent of grid size when the grid points are higher than . Hence, a regular grid of points has been selected for the following analysis.
3.2. Corroboration of the Numerical Code
We performed graphical as well as statistical tests for the corroboration of a numerical code. One of the two tests compares the average Nusselt number for a regular fluid with different values of Rayleigh number . The results are reported in Table 1 for the parameters , and , which reduce the problem to one mentioned in references [3, 10, 32–37].
The second test is performed to compare the obtained results for streamlines, isotherms, and isoconcentrations with the numerical data of [10]. They have studied the steady free convection flow and heat transfer in a triangular enclosure filled with a porous medium which is of the same geometry as our problem. The numerical simulations are performed by setting and , which reduce the present study to the reference problem. Figures 3 and 4 compare the present results with previous results and are approximately the same compared to those of Sheremet and Pop [10]. These tests for the comparison of results obtained for the present study and their agreement boost the confidence of the current numerical scheme.
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4. Results and Discussion
This section discusses the numerical simulation results and presents the graphical results of streamlines, isotherms, isoconcentrations, and average Nusselt number and average Sherwood number for various model parameters in a triangular enclosure. The effects of different values of Rayleigh number , modified Hartmann number , Brownian motion parameter , thermophoresis parameter , buoyancy-ratio parameter , Lewis number , and Frank-Kamenetskii number on the flow and heat transfer characteristics are analyzed.
4.1. Explanation of Physical Parameters
It is useful to present the physical reasons for variation in each vital parameter. In fluid mechanics, Rayleigh number is the property of a fluid that determines how heat is transferred throughout the fluid. The Rayleigh number is closely related to Grashof number , and both numbers are used to describe natural convection and heat transfer by natural convection. The Rayleigh number is simply defined as the product of the Grashof number, which represents the relationship between buoyancy and viscosity within a fluid, and the Prandtl number, which describes the relationship between momentum diffusivity and thermal diffusivity. A smaller value of Rayleigh number represents the laminar flow, while a higher range indicates a turbulent flow. The buoyancy-ratio parameter describes an effect of the buoyancy force due to the concentration difference in comparison with the buoyancy force due to temperature difference. , the Brownian motion parameter describes the movement of nanoparticles and enlightens the thermal effects of the nanofluid and nanoparticle diameter, whereas the thermophoresis parameter explains the temperature gradient effect on nanoparticle diffusion depending on the thermal conductivity of the fluid and particle materials. The Lewis number is a dimensionless number, named after Warren K. Lewis, defined as the ratio of thermal diffusivity and mass diffusivity. It is used to characterize fluid flows where there is simultaneous heat and mass transfer. The Lewis number is, therefore, a measure of the relative thermal and concentration boundary layer thicknesses. is a dimensionless number known as Hartmann number, which gives a measure of the relative importance of drag forces resulting from magnetic induction and viscous forces in Hartmann flow and determines the velocity profile for such flow.
4.2. Effects of Frank-Kamenetskii and Rayleigh Numbers
The impact of the Frank-Kamenetskii number and the Rayleigh number on the evolution of streamlines, isotherms, and isoconcentrations is described in Figure 5 when and and in Figure 6 when and , respectively. From these figures, it is observed that in the case of streamlines, two opposite rotating vortices appear which are independent of the values of and . The values for other model parameters are , , , , and . A decrease in thermal and concentration boundary layer thicknesses is observed with the increase of and . The value of decreases with the rise in , while it increases with the rise of . The exothermic reaction produces more heat with growing value for , hence accelerating the convection phenomenon within the cavity. For the smaller value of , the isoconcentrations become concentrated towards the vertical wall where the heat is low. This concentration is due to the thermophoresis phenomenon since the nanoparticles flee towards cool regions from the hot regions. However, a vortex appears near a bottom wall for a higher value of , which shifts towards the center of the cavity as the value increases. Increase in and also intensifies the convective flow in the cavity accounting by the maximum .
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The monotonic effects of the Rayleigh number and the Frank-Kamenetskii number on heat transfer in terms of local Nusselt are depicted in Figure 7. An accretion of with leads to an increase in due to an interaction of the thermal boundary layers close to the vertical walls. Similar results are obtained for local Sherwood number and plotted in Figure 8. For the higher value of , the local Sherwood number decreases with , and particles become concentrated towards the vertical wall with a less significant effect of increasing Frank number , resulting in a decrease in the nanoparticle volume fraction (Figure 9(b)). Figure 9 depicts that average Sherwood and Nusselt numbers decrease with the increasing value of . This shows that the rate of heat and mass transfer decreases along the vertical wall, and the heat transfer is dominated by the conduction. Figure 10 is the mesh diagram for stream function ψ depicting the formation of two vortices. This pattern can also be viewed in Figures 5 and 6, where we noticed that the nanofluid flows inside the triangular cavity which occurs in two major vortices. One vortex has a clockwise rotation in the left, while the second vortex has an anticlockwise rotation in the right half of the cavity. So, the Lorentz force increases the motion in the left half, and it decelerates the motion in the other half because the flow and the field act in the opposite direction. Since the applied magnetic field prevents the convection current, as a result, the average Nusselt number and Sherwood number decreases.
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4.3. Effects of Hartmann and Rayleigh Numbers
In Figures 11 and 12, the effects of the Hartmann number on local Nusselt and Sherwood numbers with the variation of the Rayleigh number are plotted against the vertical wall of the cavity. For a weak flow (Ra = 10), the effect of Hartmann number on local Nusselt number is negligible, as shown in Figure 11(a). This is because when the Rayleigh number is small, the flow convection is insignificant and the heat transfer in the cavity is dominated by conduction. However, in the case of higher Rayleigh number, the increases but depicts a small variation with , as observed in Figure 11(b). Due to high Rayleigh number, the flow convection becomes stronger which in turn increases the heat transfer in the cavity. Similar to Figure 12(a), the effect of the Hartmann number on Sherwood number is minimal, and it decreases for both the lower and higher values of Rayleigh number, as shown in Figure 12(b).
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It is also noted that a magnetic field with high intensity prevents the convection and may cause separation among nanoparticles. They are hence breaking the linearity of the gradient of nanoparticle concentration. To further check this aspect, we have plotted the concentration gradients along the center line () of the enclosure for both cases of Rayleigh number, i.e., and , for different values of Hartmann number, as displayed in Figure 13. For higher Rayleigh number, , the nanoparticles gather and start concentrating along the boundaries of the cavity away from the center, as can also be seen from Figure 6. Also, the deviations in gradients for nanoparticle concentration are nonlinear for a strong flow (), while for a weak flow (), is completely linear. Figure 14 shows the effects of the Hartmann number and Rayleigh number on local Nusselt and local Sherwood numbers. As the Hartmann number increases, the Nusselt number decreases. Similar is the trend observed for the Rayleigh number. The existence of the maximum and minimum points in the local Nusselt number profile corresponds to the presence of thermal plumes.
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4.4. Effects of Brownian Diffusion and Thermophoresis Parameter
This section elaborates the effects of thermophoresis and Brownian diffusion parameters on the rate of heat and mass transfer which are observed along the vertical wall (). Figure 15 illustrates the variation of local Nusselt and local Sherwood numbers for Brownian diffusion parameter. It is noticed from Figure 15(a) that the impact of is quite less on the Nusselt number which distinctively agrees with the conclusions of Saghir and Mohamed [38]. However, the effect of these parameters on the mass transfer rate (Sherwood number) is quite noticeable. From Figure 15(b), it can be observed that as the particle size is smaller, the mass transfer is quite increased, while an increase in particle size diminishes the Brownian diffusion effect resulting in no change in the rate of mass transfer.
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Figure 16 depicts the effects of thermophoresis parameter () on heat and mass transfer. Physically, the nanoparticles move from the hotter region towards the colder region due to the thermophoresis effect. From Figure 16(a), we found that the effect of is almost negligible on the rate of heat transfer for the nanofluid flow inside the cavity filled with the porous medium. Once again the results reconfirm the conclusions of Saghir and Mohamed [38]. Figure 16(b) shows that the rate of mass transfer decreases along the vertical wall. But, the rate of the mass transfer becomes prominent with the growing effect which leads to the homogeneous distribution of particles within a cavity.
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5. Conclusions
A numerical analysis has been carried out to study the free convection within the triangular cavity filled with a porous medium saturated with nanofluids in the presence of magnetic field initiated by a chemical reaction influenced by Arrhenius kinetics. The finite difference method has been applied to solve the governing equations. The numerical results have been obtained and discussed for a wide range of physical parameters appearing in the problem. It is found that increasing the value of Rayleigh number decreases both the average Nusselt number and average Sherwood number. Moreover, the local Nusselt number increases with the variation of Frank-Kamenetskii number, Hartmann number, Brownian diffusion, and thermophoresis parameter for the increasing value of Rayleigh number. But, local Sherwood number decreases with the variation of Frank-Kamenetskii number, Hartmann number, Brownian diffusion, and thermophoresis parameter for the increasing value of Rayleigh number.
Appendix
Finite Difference Discretization
Based on the similar approach adopted in [31], equation (6) is discretized by using five-point stencil:where
Similarly, equation (7) is discretized as follows:wherewhere
The Neumann boundary condition for temperature equation (11) is discretized as
Also, equation (8) is discretized as follows:where
The discretized form of the aforementioned equation iswhere
Nomenclature
| a: | Concentration of the reactant A |
| B: | Magnetic field strength T |
| C: | Nanoparticle volume fraction |
| C0: | Reference nanoparticle volume fraction |
| DB: | Brownian diffusion coefficient m2·s−1 |
| DT: | Thermopheric diffusion coefficient m2·s−1 |
| E: | Activation energy |
| Fk: | Frank-Kamenetskii number |
| : | Gravitational acceleration m2·s−1 |
| H: | Height of the triangular cavity m |
| Ha: | Magnetic field parameter |
| K: | Permeability of the porous medium md−1 |
| k0: | Pre-exponential factor |
| km: | Effective thermal conductivity of the porous medium |
| Le: | Lewis number |
| Nb: | Brownian motion parameter |
| Nr: | Buoyancy-ratio parameter |
| Nt: | Thermophoresis parameter |
| Nu: | Local Nusselt number |
| : | Average Nusselt number |
| P: | Pressure |
| P: | Dimensionless pressure |
| R: | Fluid constant |
| Ra: | Rayleigh number |
| Q: | Exothermicity factor |
| T: | Nanofluid temperature |
| T0: | Reference value for temperature |
| , : | Velocity components in the , directions m·s−1 |
| V: | Nanofluid velocity |
| , : | Dimensional coordinates in the direction of bottom and vertical wall m (B,01) αm thermal diffusivity of the porous medium m2·s−1 |
| β: | Thermal expansion coefficient K−1 |
| ε: | Porosity |
| μ: | Viscosity of the fluid kg·m−1·s−1 |
| ψ: | Dimensionless stream function |
| ψ: | Dimensionless nanoparticle volume fraction |
| ρ: | Fluid density kg·m−3 |
| : | Reference density of the nanofluid |
| ρp: | Nanoparticle mass density |
| : | Heat capacity of the nanofluid J·K−1·m−3 |
| : | Effective heat capacity of the porous medium |
| : | Heat capacity of the nanoparticle material |
| θ: | Dimensionless temperature. |
Data Availability
All the data are included in the article. If there is a further demand for data, the author can provide depending on their availability.
Conflicts of Interest
The author declares that there are no conflicts of interest.