Abstract
In this study, the free convection of nanofluids between a rectangular enclosure and a sinusoidal cylinder is numerically analyzed using the finite element method (FEM). Two-phase Buongiorno’s formulation was used to model the fluid layer, and Brinkman-Forchheimer equation was used to formulate the porous layer. The enclosure has a low temperature, while the cylinder is maintained at a high temperature. The governing equations are expressed in PDEs and converted into weak formulations (Galerkin FEM). In numerical simulations, the average concentration, the amplitude of undulated cylinder, the number of undulated, and the Rayleigh number are investigated. It is observed that the homogeneous nanofluid model could be valid for low heating intensity with higher waviness frequency and/or higher amplitude. The higher the alumina concentration, the higher the heat transfer rate. The heat transfer rate can be boosted by up to 13% by suspending 1% alumina particles. The heat transfer enhancement decreases with increasing the amplitude and/or increasing the waviness number.
1. Introduction
The phenomena of natural convection induced by a high cylinder temperature inside a low enclosure temperature are analyzed. One of the applications happens in a pipe carrying hot fluid that passes through an enclosed space. The enclosure has saturated porous adiabatic material to decrease the thermal gradient from the cylinder. The flow movement between a pipe and the environment of fluid space induces a pattern of convection cells. In each cell, the fluid circulates in a path and the direction of circulation is contrary with successive vortices. Oosthuizen and Naylor [1] studied the bottom and sides of the enclosure in insulated condition, and the above wall is cold while the cylinder is constantly heated. They found that small change in the Nusselt number with changing the cylinder radius. Misirlioglu [2] investigated the active cylinder put in the middle enclosure and conclude that the rotating cylinder enhances the heat transfer in the square enclosure. Saleh and Hashim [3] filled the enclosure with nanofluids. They concluded that there is an optimum radius, below which adding its size, increases the fluid movements and above which the cylinder size reduces the fluid movement. Alhashash [4] concluded that the heat transfer enhancement for the half heating and high value of permeability is at the denser nanoparticle volume fraction. Tayebi and Chamkha [5] and Tayebi et al. [6] reported that the convective flow circulation of nanofluid in a horizontal elliptical porous annulus is improved as the Darcy number and porosity increase.
Undulated surfaces are utilized in many thermal systems as a means of enhancing the mass and heat transfers, such as solar collectors, microelectronic devices, electric machinery, water storage, and condensers in refrigerators. The published works on the wavy walls have shown that this problem is limited due to the complexness of the mesh generation and parametric analyses of the impacts of several major variables. Murthy et al. [7] investigated the impact of the waviness at the horizontal wall and found that the convection intensity is decreased by using the wavy wall. Kumar [8] showed that the high undulation frequency of the vertical wall enhances the natural convection. Later, Kumar [9] reported that the heating produces a complex oscillatory structure of period equivalent to the wavy wall. Misirlioglu et al. [10] utilized the sinusoidal undulation and compared their findings with those reported in the previous result for a square enclosure with flat walls. They investigated some values of heating intensity, aspect ratio, and waviness factors. Later, Misirlioglu et al. [11] found that the convective flow was found to be sensitive on undulation for inclination less than 45° at high heating intensity. The waviness modified the thermal distributions illustrated by Khanafer et al. [12]. Mushatet [13] considered double undulation of the side walls having differentially heated and concluded that the orientation angle, the number, and the amplitude of corrugation and Rayleigh number have a considerably impact on the convective flow. Mansour et al. [14] looked at the effect of radiation when conditions are not in equilibrium and concluded that the heat transfer decreases by increasing the thermal conductivity ratio. Sompong and Witayangkurn [15] found that convective flows were slightly affected by varying the corrugation number. Sheremet et al. [16] reported the heat transfer improvement by dispersing nanoparticle, heating intensity, undulated number, and thermal deviation quantity. Cheong et al. [17] analyzed the impact of trigonometrical external heating and inner heat combustion to the wavy wall. Hoghoughi et al. [18] studied the effect of corrugation parameters on the free convection of nanoliquid over a circular heater under nonequilibrium state. They concluded that when the circular heater is raised, the heat transfer is less at smaller undulation amplitudes. Alhashash and Saleh [19] considered the effect of undulated surfaces and nanoparticle parameters. Alhashash [20] found that the thermal performance of the corrugated surface is slightly enhanced under specific conditions. Alsabery et al. [21] reported that increasing the length of the heater might enhance heat transfer within the wavy enclosure with the rotating cylinder.
Prior research almost exclusively concentrated on convection in porous enclosures with smooth walls, as opposed to flow and heat transfer in walls with undulations. This work analyzes the effect of undulations on the flow and convective heat transfer properties of water alumina in an enclosure with a sinusoidal hot cylinder. The investigated geometry can be utilized to passively regulate heat transmission in an evacuated-tube solar absorber. Accordingly, the main novelty of this study is the investigation of the coeffect of waviness and nanoparticles on fluid circulation within a porous cavity. In addition, the simultaneous influence of the cylinder surface and the Brinkman-Forchheimer on heat transfer characteristics is studied using the two-phase mixture model. The two-phase nanofluid model was previously adopted by Alsabery et al. [22], Alsabery et al. [23], Ghasemian et al. [24], and Hoseininejad et al. [25]. Arrangements of corrugated cylinder were considered by Nabavizadeh et al. [26], Sheikholeslami et al. [27], Hatami and Safari [28], Hashim et al. [29], Jabbaripour et al. [30], and Tayebi and Chamkha [31] for nonporous case and found the streamlines and isotherm pattern sensitive on the waviness surface.
2. Mathematical Formulation
A 2D schematic model of an enclosure having a sinusoidal hot cylinder is presented in Figure 1. The fluids within the annulus are water-based nanofluids having alumina nanoparticles. The outer walls were maintained at a low temperature. When the alumina is suspended in the water, Brownian motion happens from the relative velocity at the interface as illustrated in the schematic representation. When there are no turbulence effects, it follows Buongiorno’s model. Buongiorno evaluated the impacts of various slip mechanisms between the base fluid and nanoparticles that are gravity, thermophoresis, Brownian diffusion, inertia, the Magnus impact, fluid drainage, and diffusiophoresis. The most influential mechanisms on nanofluid flow and heat transmission, which can affect nanoparticle concentration fluctuations, were proven to be thermophoresis and Brownian diffusion. The cylinder having radius is put in the middle of the enclosure. The sine wavy profile of the cylinder surface follows the relation: where is the base radius, parameters and are amplitude and number of corrugations, respectively, and is the rotation coordinate. The thermophysical properties of the applied materials are tabulated in Table 1 [32].
Under the effect of the buoyancy force, the cylinder surface and enclosure at different levels of temperature bring a free convection problem. The Brinkman-Forchheimer model is assumed valid for the porous layer, and the Boussinesq approximation is applied for density variation due to differentially heated. A nanofluid with Newtonian properties fills the pores of the porous material. Using a thermal equilibrium model and accounting for Brownian nanoparticle kinematics and thermophoresis factors, energy equations for the fluid phase of a porous media are derived. Based on these considerations, the continuity, the momentum, and the energy equations are stated as follows [33, 34]: where velocities are zero on the enclosure and cylinder surface. The thermal boundary conditions are where symbol is the gravity acceleration, is the concentration of nanoparticles, and is the solid nanoparticle mass flux. Based on two phase, Buongiorno’s model nanoparticle mass flux can be stated as follows:
The nanofluid heat capacitance is given as
The thermal diffusivity of the nanofluids is evaluated as
The density of the nanofluids can be evaluated:
The nanofluid thermal coefficient is modelled as
The dynamic viscosity ratio of water alumina follows Corcione [35] model:
The conductivity ratio of water alumina is calculated by Corcione [35] as follows: with parameter that has a function where constant is given by Boltzmann. Constant nm is the mean path of the base fluid. is the molecular size of the host fluid given as Corcione [35]. where is the molecular mass of the water, is the Avogadro number, and is the density of the base liquid at our environment temperature. Value of is evaluated as
Now, we introduce the following nondimensional variables:
This, then, yields the dimensionless continuity, momentum, energy, and nanoparticle equations:
Parameters are the reference Brownian diffusion coefficient, is the Schmidt number, is the reference thermophoretic diffusion coefficient, is the diffusivity ratio parameter (Brownian diffusivity/thermophoretic diffusivity), is the Lewis number, is the Rayleigh number, is the Darcy number, and Prandtl number, , is equal to , Forchheimer coefficient, , is equal to , is the permeability of the porous medium, is spherical bead diameter [33], and is the matrix porous conductivity. The dimensionless boundary conditions are as follows:
The dimensionless streamfunction is obtained using the relation and . The local Nusselt number evaluated at the sinusoidal surface is where is the rotation coordinate. The averaged Nusselt number on the hot sinusoidal cylinder is computed by where is a perimeter of the sinusoidal cylinder. This perimeter is in the function of the amplitude and number of undulation, and here, the radius is kept at .
3. Computational Methodology
The Galerkin FEM method is used to solve numerically the mathematical equations in the form of PDEs to determine the fluid flows and thermal distribution inside the annulus. First, a strong formulation of the governing equation is established by discarding the pressure element using a penalty parameter and the continuity as the following
Strong formulation of the momentum is stated as
Strong formulation of the momentum is stated as
If the multielement technique is used, the strong formulation must take the second derivative, which is a drawback.
Second, by multiplying the equations by a test function, weak formulations of the momentum equations are obtained. The adjusted momentum equations are then integrated over the domain. The weak form provides advantages in terms of computation flexibility. Third, the unknown function is estimated. Finally, expand the , , , and using basis function as
The nonlinear residual equations for the momentum equation were generated from the Galerkin-weighted residual FEM at nodes of internal domain that is
The nonlinear residual equations for the momentum equation is
For both the test and basic functions, the Galerkin FEM employs the same function.
The residual integrations of the momentum and heat transfer, nanoparticle equations in and directions are employed by using Gauss’s method. A Newton-Raphson method is employed to solve the system nonlinear algebraic equations expressed in a matrix. The iteration of the current work is assumed to be a convergence solution when the corresponding error of each variable is equals or less than . In heterogeneous distribution, triangular mesh element is selected to mesh all domains.
To get the optimal grid size, the mesh independency test is performed for , , , and as tabulated in Table 2. The mesh sizes, i.e., M1, M2, M3, M4, and M5 are labelled for identification. The numbers inside the table are observed to be consistent by increasing the number of elements. Considering both time computation and the error, the M4 grid size was chosen for all the computations done in this paper. As a comparison, streamline results are validated with that reported by Kumar [8] for the sinusoidal undulated surface at , , , , and as presented in Figure 2.
4. Results and Discussion
The analysis in this simulation is performed in the following domain of the related dimensionless quantities: the average alumina volume fraction, ; the amplitude of corrugated surface, ; the number of corrugated surface, ; and the Rayleigh number, . The base radius is 0.2, the porosity is 0.7, the Darcy number is 0.02, the Prandtl number is 4.62, the Lewis number is 3.5E5, the normalized temperature is 155, the Schmidt number is 3.5E4, and the ratio of Brownian to thermophoretic diffusivity is 1.1. The alumina concentrations are varied in to avoid clogging in the system.
Figures 3 and 4 present the effects of various values of Rayleigh numbers and corrugated surface on streamlines and isotherms for and . The heat transfer is majority by the conduction at the small Rayleigh number. Hence, as seen in Figure 4, the thermal structures lay over each other circularly with minimum distortions. This figure exhibits almost a conduction mode of temperature from the hot sinusoidal cylinder to the cold enclosure. The plot of the streamlines exhibits a pair of rotating cells, which follows the pattern and location of the isotherm magnitude. By increasing the Rayleigh number, the convective flow becomes stronger where the hot fluid from the sinusoidal surface moves upward and keeps reaching the top wall and then moves along the left cold and right surfaces. At the upper part, the fluid losses its heat to the cold wall and then circulates down next to the left and right wall. Due to the fluid movement transporting mass transfer, the isotherms have deflected along with the fluid flow, and the isotherm magnitude has shifted toward the top. Increasing the heating intensity leads to a higher flow circulation. The movement of the fluid follows how the heating changes and meets up with how its enthalpy changes. The streamlines show a pair circulation surrounding the undulated circle with the vortices moving upward. When the flows rotated as cells in the clockwise direction, the maximum value of the flow circulation is attributed by . The streamline magnitude is stagnant by adjusting the undulated amplitude. This indicates that the undulated wall seems unsensitive to the velocities. At and , single vortices developed on the left and right sides of the annulus. Later, at larger amplitudes, double vortices formed at occupy the annulus’s left and right halves. The vortices have the form of a vertical ellipse. This is due to the inner heat circulation in the free space between the cylinder and the left and right walls being modified by the amplitude especially at moderate heating intensity. The local decreasing of temperature gradient brings to the upwelling plume.
Figures 5 and 6 show the influence of several values of Rayleigh numbers and corrugated surface on streamlines and isotherms for and . Single vortices appeared at the left and right part of the annulus at . Later at higher undulation number, double vortices generated by gravitational acceleration occupy the left and right part of the annulus. The shape of the vortices is a vertical ellipse. At , both flow circulations shift to the top wall and change to an almost like elliptical shape which denser lines with high intensity at the upper part of the annulus. The value of the flow circulation decreases by the rise of undulated number, and the same circulations tend to grow vertically because of modifying the hot zone. The hot nanofluids were transferred via conduction at and the small Rayleigh number. At , the isotherms very near to cylinder surface tend to take a circular-parallel shape which is at a small Rayleigh number. Taking higher, the thermal distributions shift to the top wall with raising the density at the upper part. At , the generating force for starting the heat transfer is more strong and apparent, the isotherm pattern transfers to the curved shape. Very dense isotherm distributes above the wavy cylinder. This indicates the large temperature gradient near the top centerline. The mushroom shapes were obtained. The isotherm expands due to the great circulation that hits the top wall. This circulation was boosted by rising maximum heat transport because of increasing the hot surface from adding an undulated number. The streamlines and isotherms are symmetric along the geometric centerline.
Figure 7 shows the influence of combination values of Rayleigh numbers and varying the corrugation amplitude, , and number, , at . It observes that the alumina concentration near the isothermal surfaces is higher than the concentration in the middle. The number of lines with high concentration below the corrugated cylinder increases when the amplitude or number of the corrugation takes higher. This figure shows that the aluminas are distributed uniformly at small Rayleigh number for various combinations of the amplitude and number undulations. Low heating intensity distributes the alumina in a large part of the annulus and homogenizes the solution. Here, the homogeneous nanofluid model could be valid since the insignificant effect of thermophoretic. When the cylinder heat is intensified, the thermal agitation and the circulation are great power at the lower part of the annulus. Furthermore, at a higher Rayleigh number, there is a small concentration gradient below the corrugated cylinder. Thus, the thermophoretic action that causes the dispersing of alumina has a low impact, and the agitation is not great in this area. It also observed denser nanoparticle distribution at this region. When the amplitude is higher at , the agitation of the alumina is improved. The enhancement of agitation causes a greater uniform alumina dispersion in the whole annulus. The improved agitation is a consequence of the increase in the length of heated surface. In the domains of convective cells, there are extensive uniform regions, whereas the nonuniform regions become more limited as they approach the corrugated surface.
Figure 8 shows a relationship between the mean Nusselt number, , and the Rayleigh number, , at various when and . Heat transfer rate increases with increasing the alumina volume fraction. The increasing more pronounced at a high and small Rayleigh number. This fact due to alumina thermal conductivity has a significant effect on increasing the thermal gradient at moderate buoyancy force, but the viscosity and agitation suppress the fluid movement. At the high Rayleigh number, intensity of the upper convective cells enhances considerably, and the sizes of regions with dense concentration decrease. The viscosity action and agitation are to suppress the flow circulation. The cumulative impact is to increase the Nusselt number considerably.
Figure 9 summarizes the variations of the average Nusselt number versus the Rayleigh number for different undulation amplitude at and . The heat transfer rate decreases by increasing the amplitude, and it is being significant at a high heating intensity. The temperature gradient of the active wall reduces as the corrugation amplitude increases. The thermal performance follows the flow inhibition near the corrugated region where the flat wall is the best option. Isotherm changed (see Figure 4) because the Brownian motion parameter went up, which led to an unnecessary drop in the average Nusselt number.
Figure 10 summarizes the variations of average Nusselt number versus the Rayleigh number from 103 to 106 for various undulation number at and . The Nusselt number is reduced by applying the undulated cylinder at any heating condition. Increasing undulated number decreases the heat transfer rate constantly from low to moderate Rayleigh number. The heat transfer rate decreases by increasing the waviness, and it is being significant at a high heating intensity. Increasing the waviness frequency to the thermal performance is equivalent with increasing the amplitude as mentioned in the previous case. At higher buoyancy force, increasing the undulation number decreases the thermal gradient, but the free space becomes larger. It is also supported by the fact that the nanoparticle isoconcentrations are distributed heterogeneously at a relative large buoyancy force.
Differences in the average Nusselt number and Rayleigh number are shown in Table 3 for different values of the controlling parameters, , , and . The average Nusselt number increases significantly as the solid volume fraction increases. As can be seen from the table, the amplitude goes up, and the average rate of heat transfer goes down. The maximum rate of heat transfer takes place at , , and . The heat transfer rate varies almost linearly with the solid volume percentage. Increasing the solid concentration from 0 to 0.01 results in a 13% improvement in heat transfer rate at and . Increasing the solid concentration from 0 to 0.01 results in a 9% improvement in heat transfer rate at and .
5. Conclusions
Free convection of nanofluids induced by different temperature levels between cold walls and a hot sinusoidal cylinder was studied numerically. The continuity, Brinkman-Forchheimer flow, and energy equation inside the annulus are formulated in the dimensionless form. Two-phase Buongiorno nanofluid model was applied for the nanofluid layer. These equations were solved by the Galerkin FEM. Computed results of the flow fields and temperature distributions, as well as the heat transfer rate, are depicted graphically. The undulation frequencies, alumina volume fraction, and Rayleigh number affected the shape of streamlines, isotherms, isoconcentrations, and Nusselt number plots. Some important points from the study are given below: (1)The intensity of the flow movement reduces as the undulated number increases but is stagnant with varying the undulated amplitude(2)Shape, position, and number of the vortices depend on the waviness frequency and Rayleigh number. The nanoparticle isoconcentrations are not symmetric along the geometric centerline while the streamlines and isotherms are symmetric along the centerline(3)Homogeneous nanofluid model could be valid for low heating intensity with higher amplitude and number of corrugation conditions(4)The higher the alumina concentration, the higher the heat transfer rate. The heat transfer rate can be boosted by up to 13% by suspending 1% alumina particles. The heat transfer enhancement decreases with increasing the amplitude and/or increasing the waviness number
Abbreviations
Nomenclature| : | Undulation amplitude (m) |
| : | Heat capacity (J kg-1 K-1) |
| : | Forchheimer constant |
| : | Darcy number |
| : | Brownian diffusion coefficient (kg m−1 s−1) |
| : | Diameter of the nanoparticle (nm) |
| : | Thermophoretic diffusivity coefficient (kg m−1 s−1)) |
| : | Gravitational force (m s−2) |
| : | Thermal conductivity (W m−1 K-1) |
| : | Permeability (m2) |
| : | Length of enclosure (m) |
| : | Undulation number |
| : | Pressure (Pa) |
| : | Perimeter (m) |
| : | Prandtl number |
| : | Cylinder radius (m) |
| : | Rayleigh number or heating intensity |
| : | Temperature (K) |
| , : | Velocity components in the - and -directions (m/s) |
| , , , : | Space coordinates (m) and dimensionless space coordinates. |
| : | Thermal diffusivity (/s) |
| : | Thermal expansion coefficient (1/K) |
| : | Porosity |
| : | Dimensionless temperature |
| : | Kinematic viscosity (m2/s) |
| : | Density (kg/m3) |
| : | Solid volume fraction |
| : | Stream function (m2/s) |
| : | Dynamic viscosity (kg m−1 s−1). |
| : | Cold |
| : | Base fluid |
| : | Hot |
| : | Nanofluid |
| : | Alumina nanoparticles |
| : | Sinusoidal cylinder. |
Data Availability
The data used to support the findings of this study are included in this paper and available without any restrictions.
Conflicts of Interest
The authors declare that there is no conflict of interests regarding the publication of this research article.