Natural convection in a flexible sided triangular cavity with internal heat generation under the effect of inclined magnetic field
Introduction
Convection in triangular cavities is important in many engineering applications such as solar power, food processing, cooling of electronic devices, building roof applications and many others [1], [2], [3], [4], [5], [6], [7], [8]. In many specific applications, cooling and heating may be partial and non-uniformities in the thermal boundary conditions may also be of interest to the researchers [9], [10], [11], [12]. Convection in enclosures filled with heat generating fluids were extensively studied by many researchers [13], [14], [15], [16], [17], [18]. Internal heat generation may be seen in applications such as geothermal heat exchangers and nuclear reactors [19]. The effects of different obstacles installed in a nanofluid filled cavity on heat transfer and fluid flow with internal heat generation were studied by Selimefendigil and Oztop [20]. Acharya and Goldstein [16] numerically investigated the natural convection inside an inclined cavity with uniformly distributed internal energy source. Distinct flow pattern systems depending on the ratio of the external to internal Rayleigh number were observed.
The effects of magnetic field on heat transfer applications are important many technological applications such as coolers of nuclear reactors, micro-electronic devices, purification of molten metals and many others [21], [22], [23], [24], [25], [26], [27], [28], [29]. External magnetic field can be utilized to control the heat transfer characteristics in various applications [30], [31], [32], [33], [34], [35], [36]. Selimefendigil and Oztop [37] numerically studied the MHD mixed convection in a nanofluid field triangular enclosure having an inner adiabatic rotating cylinder with finite element method. It was observed that as the strength of magnetic field increases, average heat transfer and total entropy generation reduce. Mahmoudi et al. [38] numerically investigated the natural convection in a nanofluid filled triangular cavity partially heated from below with magnetic field effect. It was observed that heat transfer deteriorates with magnetic field and highest reduction in the averaged Nusselt number for high value of Hartmann number occurs at Rayleigh number of 106. Aminossadati [39] made a numerical study on natural cooling of a triangular heat source in a nanofluid filled triangular cavity under the effect of a horizontal magnetic field by using finite volume method.
Deformable walls or objects can be used to control the fluid flow and heat transfer characteristics inside the cavities [40], [41], [42], [43], [44], [45]. Selimefendigil and Oztop [41] numerically investigated the mixed convection in a nanofluid filled square cavity having flexible side walls with an inner rotating cylinder. The elastic modulus of the side was found to effect the heat transfer characteristics within the cavity. A numerical study of mixed convection in a lid driven cavity with a flexible bottom wall was performed by Khanafer [42]. It was observed that flexible wall significantly affects the heat transfer enhancement.
Based on the above literature survey and to the best of author's knowledge, natural convection in a partially heated triangular enclosure with internal heat generation and under the effect of an inclined magnetic field never been reported in the literature although is may found a lot of engineering applications in practice. This configuration may be encountered in practice as outlined above or the proposed novel passive method with flexible wall can be combined with magnetic field to control the fluid flow and heat transfer characteristics in triangular enclosures. The present numerical study aims at investigating the effects of external Rayleigh number, internal Rayleigh number, Hartmann number, inclination angle of the magnetic field, and Young's modulus of the flexible inclined wall of the triangular cavity the fluid flow and heat transfer.
Section snippets
Mathematical formulation
The physical domain and boundary conditions of the problem under consideration are demonstrated in Fig. 1. The left vertical wall of the triangular cavity is partially kept at constant temperature of Th while the flexible inclined wall is at constant temperature of Tc. The other walls of the cavity are assumed to be insulated and no-slip boundary condition is imposed for all walls of the cavity. The inclined wall of the triangular enclosure is flexible with Young's modulus of E, density of ρ
Solution methodology
The governing equations is solved by Galerkin weighted residual finite element method. Non-overlapping regions within the computational domain were constructed. Each of the flow variables within these regions are approximated by using the Lagrange finite elements of different orders. The approximations were substituted into the governing equations which results in residuals for each of the conservation equation. When the relative error for each of the variables Γ satisfy the following
Results and discussion
In this numerical study, natural convection in a partially heated triangular enclosure with internal heat generation is investigated under the effect of an inclined magnetic field. The influence of external Rayleigh number (between 104 and 106), internal Rayleigh number (between 104 and 107), elastic modulus of flexible inclined wall (between 500 and 105), Hartmann number (between 0 and 40) and inclination angle of the magnetic field (between 0° and 90°) on the fluid flow and heat transfer
Conclusions
Numerical simulation of MHD natural convection in a flexible sided triangular enclosure with internal heat generation was performed. The important conclusions from the numerical simulation results are:
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Averaged Nusselt number enhances with external Rayleigh number but local heat transfer is deteriorated for the higher values of external Rayleigh number in the vicinity of the upper location of the heater which is due to the inclined wall deformation with increasing external Rayleigh number.
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As the
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