Free Convection in a Triangular Cavity Filled with Hybrid-Nanofluid along with Sinusoidal Heat

DOI: http://dx.doi.org/10.24018/ejers.2019.4.12.1613 48  Abstract—A numerical study on convective heat transfer of hybrid nanofluid packed in a right angled triangular cavity heated by a sinusoidal temperature maintained from lower side and subjected to a constant magnetic field have been studied in this work. The hypotenuse side of the triangular cavity has been kept in uniform cool temperature while the remaining side is insulated. The governing equations of the problem have been discretized numerically with help of finite element method. A fixed Prandtl number Pr=6.2 has been used for the numerical solution. Several values of Rayleigh number Ra=1010, and Hartmann number Ha=0-100 which are the nondimensional governing parameters have been examined. The volume fraction  =0.01, 0.05 and 0.1 and the heat generation


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Abstract-A numerical study on convective heat transfer of hybrid nanofluid packed in a right angled triangular cavity heated by a sinusoidal temperature maintained from lower side and subjected to a constant magnetic field have been studied in this work. The hypotenuse side of the triangular cavity has been kept in uniform cool temperature while the remaining side is insulated. The governing equations of the problem have been discretized numerically with help of finite element method. A fixed Prandtl number Pr=6. 2

I. INTRODUCTION
In recent years, convective flow in a cavity with numerous thermal boundary conditions have created an eagerness to several researchers due to its vast applications in industry and environment. Nonuniform temperature distribution keep on the active wall due to shading or other effects in the fields, such as solar energy collection and cooling of electronic components. Therefore, the study on convection heat transfer in cavity with sinusdoidal heated wall(s) is important in such situations. For efficient improvement of many thermal systems like micro electronic devices, heat exchangers, refrigerators, solar collectors, cooling of electronic components, nuclear reactors, transformers, and engine/vehicle etc by adopting heat transfer enhancement, one of the most resourceful strategy is using nonofluid/hybrid nanofluid. Nikbakhti and Rahimi [1] studied the double-diffusive natural convection with different heating sections and different parameters in a rectangular cavity with partially heated walls filled with air. Kwak and Hyun [2] studied the natural convection in a cavity with time-varying temperature on the sidewall. For improving heat transfer performance, geometrical arrangement may have the fruitful effect. Saidi et al. [3] presented numerical and experimental results of flow and heat transfer from a sinusoidal cavity. Sarris et al. [4] investigated the natural convection numerically in a rectangular enclosure with a sinusoidal temperature profile on the top wall. There was a clear effect related with Rayleigh number. When the increasing of Rayleigh number, the circulation patterns and their centers are move toward the upper wall corners. The term ''nanofluid'' was first used by Choi [5] refer to a fluid in which nanoparticles are suspended. Khanafer et al. [6] studied two-dimensional problem of buoyancy-driven heat transfer in a cavity.A nanofluid is a fluid comprehending nanoparticles (less than 100 nm-diameters) dispersed in a regular base liquid (water, ethylene glycol, or oil, etc.). Ho et al. [7] studied natural convection by using three sizes of vertically square cavity filled by water-Al2O3 nanofluids. Bilgen and Yedder [8] analyzed the natural convection with sinusoidal temperature profiles in a wall in a rectangular enclosure. They showed that when the heated section is in the lower half of the enclosure in this case the heat transfer is higher. Deng and Chang [9] studied numerically the natural convection with two spatially varying sinusoidal temperature distributions on the left and right sidewalls in a rectangular enclosure. They found that the nonuniform temperature distribution increases heat transfer as compared to the case of uniform wall temperature. Sheikholeslami et al. [10] studied natural convection heat transfer in a cavity with sinusoidal wall filled with CuO-water nanofluid in presence of magnetic field. B. Ghasemi, S.M. Aminossadati [11] studied natural convection of nanoparticles in a triangular enclosure using Brownian motion. Hybrid nanofluid can be found by appending more than one type of nanoparticles with base fluid. Hybrid nanoparticles mixture procedure has been reviewed widely by Sarkar et al. [12] in details. Botha et al. [13] did an experiment of hybrid nanofluid based on silversilica-oil. Chamkha et al. [14] analyzed natural convection in a semi-circular cavity for unsteady conjugate filled with a hybrid water-based suspension of Al2O3 and Cu nanoparticles. Rashad et al [15] studied natural convection in a triangular cavity filled with Cu-Al203/water hybrid nanofluid with MHD effect. They showed, when the natural convection is very small then results shows significant effect of increasing volume fraction of hybrid nanofluid. The objective of this paper is an effort to contribute in starting the groundwork of this research field. Therefore, this present study focused on triangular cavity heated sinusoidal from below with the presence of magnetic field and internal heat generation with a constant heat flux.

II. PROBLEM FORMULATION
The schematic diagram of the studied configuration is Free Convection in a Triangular Cavity Filled with Hybrid-Nanofluid along with Sinusoidal Heat Md. Rakibul Hasan, Md. Borhan Uddin, and Ahmed M. U. described in Fig 1. In this study, the steady two dimensional natural convection flow in a right angle triangular cavity of vertical length and horizontal length H has been considered in the presence of internal heat generation and uniform external magnetic field has been applied in the horizontal direction normal to the vertical wall. The enclosure has been filled with hybrid Al2O3-Cu/water nanofluid. The inclined wall of the enclosure has been cooled at a constant temperature Tc. A sinusoidal heat source has been located at the bottom wall. It has been also assumed that both the fluid and hybrid nanoparticles are in thermal equilibrium and there is no slip between them. The hybrid nanofluid used in the work has been considered as laminar and incompressible. It has also been assumed that the gravitational acceleration acts to the vertical downward surface. Fluid and hybrid nanofluid Properties has been given in Table 1, assumed constant except for the density variation, which is maintained on Boussinesq approximation.

III. MATHEMATICAL FORMULATION
By considering the problem described above the equations for the conservation of mass, momentum and energy in Cartesian coordinate system for hybrid nanofluid [15] for fluid and solid are given below:  are the Prandtl number, Rayleigh number and the magnetic Hartmann number respectively and Q is the dimensionless heat generation coefficient.
The equation (1) to (4) are made dimensionless by using the following relations The boundary condition for the problem is given below: , on the inclined wall: The effective properties of Hybrid nanofluid (Al2O3-Cu/water) is defined as follows: density:   3  3  2  2  1   3  3  2  2  2  1   3  3  2  2   3  3  2  2  1   3  3  2  2  3 1 1  is the overall volume concentration of two different types of nanoparticles dispersed in hybrid nanoparticles and is calculated as Nusselt at the heated wall of the enclosure is express as where H is length of the heated surface.

IV. COMPUTATIONAL PROCEDURE AND VALIDATION
To solve the dimensionless equation (1-4), numerical algorithm is applied using finite element method. In this method, the solution domain is discretized into meshes, which are fitted of non-uniform meshing elements (triangular, rectangular). Using the Galerkin weighted residual at the nodes of the elements, the governing partial differential equations are converted into a system of integral equations. This converted equations are used to find the basic unknown variables like velocity component U, V, temperature  and pressure P by utilizing the boundary conditions (6). One of the most essential part for numerical Results obtained in Rashad [16] Results obtained in this work Fig. 2 There was a good agreement shown in Fig 2, between present results and the numerical results of Rashad [16] for both the streamlines and isotherms inside the cavity. These validations boost the confidence that the procedure applied in this work is appropriate and help to stated objectives of the current investigation.