Investigation of entropy generation in a square inclined cavity using control volume finite element method with aided quadratic Lagrange interpolation functions
Introduction
The study of natural convection in inclined cavities is an issue that has broad applications in many industrial areas such as cooling of microelectronic apparatus, solar energy collection, air conditioning of buildings and so on. An additionally valuable stage in the examination of these problems is the calculation of generated entropy. Former, numerous studies are motivated on the cavity analysis. Yilbas et al. [1] analyzed the natural convection in a square cavity with top and bottom wall temperature by considering the entropy generation. They concluded that the total entropy generation increases with increasing the wall temperature. Baytas studied the entropy generation in an inclined porous cavity [2]. In 2002, a numerical prediction of heat transfers and fluid flow characteristics inside an enclosure bounded by two isothermal wavy walls and two adiabatic straight walls presented by Mahmud et al. [3]. Magherbi et al. [4] investigated the effect of the non-reversible distribution ratio on the maximum entropy generation in an enclosure. Misirliogluin et al. [5] examined the natural convection phenomena inside a cavity with two horizontal smooth walls and two vertical wavy walls. Their study indicates that for large Rayleigh numbers and adequate values of the aspect ratio the generated heat in the porous cavity cannot be transferred from the hot wall to the cold wall. Kandaswamy et al. [6] numerically studied the heat transfer in a square cavity for heated plate placed horizontally/vertically. They concluded that heat transfer is higher in the vertical situation. In 2008, the influence of aspect ratio on entropy generation in a quadrilateral cavity investigated by Ilis et al. [7]. Their study confirmed that the total entropy generation initially increases with increasing of aspect ratio, gets to a maximum value, and then decreases. In 2008, Pirmohammadi et al. [8] studied the steady, laminar, natural-convection flow in the presence of a magnetic field in a cavity that heated from left and cooled from the right. The results showed that the conduction heat transfer mechanism becomes dominant as the Hartmann number amplified. Moreover, the mechanisms of heat transfer and the flow characteristics depend strongly upon both the strength of the Rayleigh number and the magnetic field. Pirmohammadi and Ghassemi [9] investigated the effect of magnetic field on convective heat transfer inside a tilted square cavity. They proved that at a given inclination angle, by increasing the Hartmann number the convection heat transfer reduces. El Jery et al. [10] studied the influence of an external oriented magnetic field on the rate of generated entropy in a cavity. The outcomes show that increasing Hartmann number induces the decrease of entropy generation magnitude. Bouabid et al. [11] numerically examined natural convection and entropy generation in an inclined square cavity where the vertical walls were at different constant temperatures while the horizontal walls were insulated. In 2012, free convection and entropy generation of nanofluid inside an enclosure with different patterns of vertical wavy walls were investigated by Esmaeilpour and Abdollahzadeh [12]. In 2012, Chang Cho et al. [13] performed a numerical investigation into the natural convection heat transfer characteristics within an enclosed cavity that filled with nanofluid. The results indicated that the average Nusselt number increases with an increasing volume fraction of nanoparticles for an extensive range of Rayleigh number. In 2013, entropy generation in steady MHD flow due to a rotating porous disk in a nanofluid was investigated as a result of Rashidi et al. [14]. Sanatan Das and Rabindra Nath Jana [15] investigated the effects of magnetic field on the entropy generation through an incompressible viscous flow between two infinite horizontal parallel plates. They concluded that the entropy generation is proportional with the magnetic parameter. In 2014, entropy generation for the laminar natural convection in a square inclined cavity was numerically investigated by Shavik et al. [16]. Their results illustrated that with the increasing of inclination angle the total entropy generation due to fluid friction increases but total entropy generation due to heat transfer and average Bejan number decreases. Arici et al. [17] evaluated the effect of aspect ratio on heat transfer and natural convection in a wavy walls cavity. Srinivasacharya and Hima Binduthe [18] considered entropy generation due to heat transfer, fluid friction and magnetic field for the incompressible micropolar steady fluid flow in a four-sided channel with constant wall temperatures. Jangili et al. [19] analyzed the entropy generation in a magnetized-micropolar fluid that flows among two vertical concentric rotating cylinders of infinite length. They concluded that increasing the couple stress effect increases the temperature and micro-rotation and decreases the velocity. In 2016, the numerical computation of unsteady laminar three-dimensional natural convection and entropy generation in an inclined cubical trapezoidal air-filled cavity was performed by Kadhim Hussein et al. [20]. Their results show that when the Rayleigh number increases, the flow patterns change especially in three-dimensional results and the flow circulation increases. Also, the inclination angle effect on the total entropy generation becomes insignificant when the Rayleigh number is low. Moreover, by increasing the Rayleigh number the average Nusselt number increases. In 2017, Shi et al. [21] have numerically investigated the natural convection and entropy generation of air in a two-dimensional square enclosure under a magnetic quadrupole field. Their result indicated that the magnetic buoyancy force has potential applications for enhancing the heat transfer, but this leads to increasing of total entropy generation. In 2017, natural convection and entropy generation due to the heat transfer and fluid friction irreversibilities in a three-dimensional cubical cavity with partially heated and cooled vertical walls were numerically investigated by Rashed et al. [22]. They concluded that the total entropy generation rate increases when the Rayleigh number increases while the Bejan number decreases as the Rayleigh number increases.
In 2017, nanofluid flow and heat transfer between non-parallel plates in the presence of a magnetic field were analyzed by Dogonchi and Ganji [23]. Their results show that an increase in Schmidt number leads to a rise in the temperature profile and Nusselt number. In 2017, Dogonchi and Ganji [24] investigated the MHD nanofluid flow and heat transfer between parallel walls considering Cattaneo-Christov heat flux model. In 2017, Go-water nanofluid flow and heat transfer in a porous channel in the presence of thermal radiation and magnetic field impacts were studied by Dogonchi et al. [25]. They proved that the temperature profile and the Nusselt number are in direct relation with solid concentration and are inverse with the radiation parameter. They showed that the temperature and Nosselt number are directly related to solid concentrations and are inversely related to the radiation parameter. In 2018, Alsabery et al. [26] studied the steady conjugate mixed convection in a double lid-driven square cavity filled with water- nanofluid comprising a solid inner body. In 2018, Marangoni natural convection in a cubical cavity filled with a nanofluid was examined by Sheremet and Pop [27]. In 2018, entropy generation analysis for an inclined square porous enclosure considering magnetic field was performed by Rashad et al. [28]. They considered a nanofluid as working fluid and explored the impacts of size and location for a heat sink and a heat source. In 2018, Izadi et al. [29] investigated the natural convection of a nanofluid between two eccentric cylinders saturated by porous material. 1n 2018, radiative nanofluid flow and heat transfer between parallel disks with penetrable and stretchable walls was studied by Dogonchi et al. [30]. In 2019, the effects of shape of the nanoparticles on entropy generation in a semi-annulus nanoliquid-filled cavity was analyzed by Seyyedi et al. [31]. In 2019, Shafqat et al. [32] examined the influence of magnetic field on entropy generation owing to mixed convective nanoliquid flow. In 2019, Dogonchi et al. [33] investigated CVFEM analysis for Fe3O4-H2O nanofluid in an annulus subject to thermal radiation. In 2019, Magneto-hydrodynamic flow and heat transfer of a hybrid nanofluid in a rotating system among two surfaces in the presence of thermal radiation and Joule heating was studied by Chamkha et al. [34]. More relevant publications can be found in [[35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54]].
The aim of the present study is the investigation of the natural convection characteristics and the entropy generation in a square inclined cavity in the presence of a magnetic field at different Rayleigh. The advantages of this study can be summarized as follows:
- (1)
Proposing new correlations for entropy generation number using QLIFs.
- (2)
Investigation simultaneously a longitudinal magnetic field and inclination angle for the cavity.
Section snippets
Problem statement
Consider a two-dimensional square inclined cavity. The left and right walls are maintained at the uniform temperatures Th and Tc (Tc < Th), respectively. The top and bottom walls are adiabatic. Fig. 1 shows the physical model and coordinate system considered in this study. Steady laminar natural-convection flow in the presence of a longitude magnetic field in a square cavity of length L was considered. In the figure, β denotes the inclination angle. The cavity is filled with an electrically
Basic governing equations
The density of electric current and the electromagnetic force are defined in following, respectively:
It is assumed that the electric field is zero [10]. Then, Eqs. (1), (2) can be shortened as:
The governing equations, mass, x-y-momentum, and energy equations are as follows, respectively:
Numerical method
In the present study, an in-house FORTRAN program is written based on the CVFEM (Control Volume Finite Element Method) [[59], [60], [61], [62], [63], [64], [65], [66]]. In the program, firstly the vorticity, dimensionless temperature, and stream function are obtained by numerically solving of Eqs. (12), (13), (14). Finally, Eq. (19) is solved to obtain the entropy generation number.
Results and discussion
The effects of Hartmann number, inclination angle and Rayleigh number have been investigated in the cavity. Table 4, Table 5, Table 6 show the values of Nuave, Ngen, Beave, and ECOP at Ra = 104, Ra = 5 × 104, and Ra = 105, respectively. Discussion about these tables will be performed in the next subsection. As a novelty in the numerical calculation, the median Ra = 5 × 104 is considered in quadratic Lagrange functions for a full range interpolation. The detail of the mathematical procedure is
Conclusion
In this study, a two-dimensional square inclined cavity was considered in the non-dimensional form. The entropy generation number and the average Nusselt number were calculated by CVFEM solution by considering the magnetic field as Hartmann number. The impacts of Hartmann number, inclination angle and Rayleigh number on the entropy generation number, the average Nusselt number, the ECOP, and the average Bejan number were examined. Some correlations for the entropy generation number were
Declaration of Competing Interest
It is declared that we have no conflict of interest.
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