MHD Free Convective Heat Transfer in a Triangular Enclosure Filled with Copper-Water Nanofluid

Two-dimensional time independent free convective flow and temperature flow into a right-angled triangle shape cavity charged by 2 Cu H O  nanofluid has been performed. The horizontal side of the enclosure is warmed uniformly h T T  whilst the standing wall is cooled at low temperature c T T  and hypotenuse of the triangular is insulated. The dimensionless non-linear governing PDEs have been solved numerically employing the robust PDE solver the Galerkin weighted residual finite element technique. An excellent agreement is founded between the previous, and present studies. The outcomes are displayed through streamline contours, isotherm contours, and local and average Nusselt number for buoyancy-driven parameter Rayleigh number, Hartmann number, and nanoparticles volume fraction. The outcomes show that the temperature flow value significantly changes for the increases of Rayleigh number, Hartmann number , and nanoparticles volume fraction. Average Nusselt number is increased for the composition of nanoparticles whereas diminishes with the increase of Hartmann number.


INTRODUCTION:
Temperature transfer and fluid flow of nanofluids into the triangle shape enclosure have a wide applications in numerous industrial and engineering systems like heat exchangers, fire prevention, solar collectors, home ventilation systems, and refrigeration units etc. Cu -water, 23 Al O -water, and 2 TiO -water are very common nanofluids. These nanofluids are used widely for the augmentation of temperature transfer. Enhancement of warmed-up conductivity into nanofluids was studied by Choi and Eastman (1995). Manca et al. (2010) performed heat transfer in nanofluids. Influence of external magnetic field on natural convective flow into rectangle shape enclosure using warmed up and cooled neighbor walls was investigated by Ece and Buyuk (2006). Convective temperature disposal within a cavity heated partially was performed by Ozotop and Abu-Nada (2008). Free convective temperature flow value into nanofluids within an inclined was performed by Ghasemi and Aminossadati (2009). Impact of inclined angle using copper-water nanofluids within an enclosure was investigated Abu-Nada et al. (2009). Varol et al. (2007) investigated free convectional flow within a triangle shape cavity using isothermal heater. Numerical computations of FEM on convectional temperature flow into nanofluids within a rectangle shape enclosure was performed by Rahman et al. (2009). FEM numerical computations on free convection into nanofluids within a triangle shape cavity in account of both uniform and non-uniform warmed boundary condition was investigated by Basak et al. (2008). A comprehensive review on convectional flow was researched Kamiyo et al. (2010). Aydin and Yesiloz (2011) investigated usual convectional flow into a quadrantal enclosure using warmed and cooled neighbor walls.
Magnetic field which creates an external force on the thermal and energy systems that highly attack the liquid flow and temperature flowing. Sheikholeslami et al. (2014) researched about CuO-water nanofluid effect on convectional flow taking into account Lorentz force in flow domain. Computational study using Al 2 O 3 -water nanofluids on natural convectional flow was investigated by Rashmi et al. (2011). Bhardwaj and Dalal (2013) investigated convectional temperature flow as well as entropy generation into a right-angled shape triangle cavity. Natural convectional flow into a nanofluid using horizontal warmed plate was studied by Arani et al. (2011). Combined convectional flow and temperature flow characteristics into lid-driven square enclosure using heated blocks was investigated by Boulahia et al. (2016). Magneto hydro dynamics free convectional temperature transfer into nanofluid within isosceles triangle shape cavity was performed by Rahman et al. (2016). Convectional temperature flow into nanofluid within a triangular shape cavity was performed by Uddin et al. (2018).
From the literature review, the principle intention is to investigate the temperature flow and fluid flow within a right-angled triangle shape cavity charged by copper-water nanofluid. The impact of Rayleigh number, volume fraction of nanoparticles and Hartmann number are performed numerically and discussed them from physical point of view.

Physical Model:
For present study, schematic spectacle for right-angled triangular enclosure has been presented in insulated. The enclosure charged as water 2 () HO for base fluid and copper () Cu for nanoparticles. The gravitational acceleration g acts in downward approach towards y  axis and a uniformly external magnetic field including power 0 () B is also imposed towards horizontal approach. The nanofluid is calculated as incompressible and laminar. Two dimensional time independent flow is processed for current study . The characteristics of nanoparticles and base fluid is given in Table 1. All solid boundaries are announced as no slip walls.

Boundary conditions:
on horizontal wall: on perpendicular wall:

Dimensional Analysis:
The dimensionless variables are listed below: Using the dimensionless variables in equations (1)-(4), the non-dimensional equations are: Dimensionless boundary conditions are: on horizontal wall: on perpendicular wall: Local Nusselt number is defined towards horizontal hot wall of the cavity as Average Nusselt number on horizontal hot side can be calculated:

COMPUTATIONAL PROCEDURE:
Governing dimensionless equations (7)- (10) including boundary conditions (11a)-(11c) are concluded numerically by using finite element technique of Galerkin weighted residual type that is very powerful tool to solve these kinds of nonlinear equations. This method is expressly explained by Zienkiewicz and Taylor (1991). The computational procedure for the present problem is accomplished using excellent PDEs solver Comsol Multi physics. For grid independent solution, a comprehensive mesh testing technique is calculated to guarantee the numerical solution when 4 10 Ra  . The subsequent five variant distinct grids designated as normal, fine, finer, extra fine, and extremely fine including of 1187, 1757, 5067, 13333 and 16773 elements in resolution field respectively. Average Nusselt number is considered for checking the grid delicacy for the aforementioned elements. Table 2 displays that average Nusselt number for 13333 elements exhibits less difference form compared to other elements. Consequently, grid size of 13333 elements is considered of the grid independent solution. in Fig 2. The results show a strong agreement and gives us the confidence for using the current numerical scheme.  that have a character of opposing of its propagation which weakens the streams within the enclosure.

CONCLUSION:
We have investigated numerically free convectional flow as well as temperature flow of a right-angled triangle charged by cupper-water nanofluid considering with the help of Galerkin weighted residual finite element analysis. The influence of various parameter are presented using streamline contours, isotherm contours, and Nusselt number and interpreted. The following main findings are listed: i.
Rayleigh number play significant roll on flow field and temperature transfer value. ii.
The fluid flow enhances for rising Rayleigh number whereas diminishes for rising Hartmann number.