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The present work is devoted to investigating heat transfer and fluid flow in a two dimensional square open cavity containing a heated circular cylinder at the centre. A constant heat flux is set at the left sidewall; high and low temperatures are fixed at the bottom and top walls of the cavity respectively. The right side of the cavity is open. Galerkin Weighted Residual Finite Element Analysis is used to visualize the heat transfer and fluid flow solving two-dimensional governing mass, momentum and energy equations for steady state, heat transfer and fluid flow in presence of magnetic field in side an open square cavity. A uniformly heated circular cylinder is placed at the centre of the cavity as a heat source. To find the effects of Rayleigh number (Ra) on the thermal fields and fluid flow in presence of magnetic field and a heated circular cylinder as heat source by visualization and line graphs is the objective of this study. Numerical results are presented in graphical and tabular form. The study is conducted for different values of Rayleigh number, some fixed Hartmann numbers (Ha) and heat flux (q). In the conclusion it has been observed that the temperature field and fluid flow pattern are functions of the parameters Rayleigh number and Hartmann number.

The convective heat transfer and the natural convection flow with the presence of magnetic field of the fluid are of great importance in scientific and engineering research, because these types of complicated geometries have applications in engineering and industrial fields. This type of problems of heat transfer attracted very attention of researchers since its numerous uses in the areas of energy conservations, cooling of electrical and electronic equipments, design of solar collectors, heat transfer devices etc. Recently many research work have been done in this field which influence the convection flow nature. It is difficult to solve the natural convection problem in complicated geometry like it. MHD acceleration can use to develop supersonic air craft technology.

The numerical study on MHD natural convection heat transfer and fluid flow in a two-dimensional square open cavity containing a heated circular cylinder has been done. The left vertical wall is kept at A constant heat flux is set at the left vertical wall, different high and low temperature are set at bottom and top walls of the cavity and remaining side wall is open. Two dimensional Naviar Stoke’s equations with energy equation are used to govern the problem and Galerkin’s Weighted Residual Finite Element Method is used to solve the problem. In the result it is observed that all isotherm lines are concentrated at right lower corner of the cavity and the heat transfer rate is suppressed in suppressed of Rayleigh number in the cavity when Hartmann number is fixed. Recirculations are created in the flow field. Recirculation region is increasing with increasing Rayleigh number (Ra). It is a good agreement with the existing Heat Transfer Theory.

The terms magneto hydrodynamic, hydrodynamics, magneto gas dynamics and magneto aerodynamics all are the branches of fluid dynamics that deals with the motion of electrically conducting fluids in presence of electric and magnetic fields. The influence of the magnetic field on the boundary layer is exerted only through induced forces within the boundary layer itself, with no additional effects arising from free stream pressure gradient.

Literature ReviewChan and Tien [

Ostrach [

The study related to heat absorption or rejection in the confined rectangular enclosures has been well discussed in the literature C. Taylor and P. Hood [

The two-dimensional diagram of the system considered in the present study is shown in

consists of an open square cavity with sides of length L. A heated circular cylinder of diameter D is located at the center of the cavity. The two-dimensional Cartesian co-ordinate system with origin at the lower left corner of the computational domain is considered here. A constant heat flux q is considered at the left vertical wall of the cavity. The bottom wall is kept high temperature T_{h} and top wall is kept at low temperature T_{c}. The remaining right side wall is open. The temperature at the cylinder T_{h}_{1} is less than that of the bottom wall. A magnetic force of strength B_{0} is applied horizontally normal to the side walls.

The two-dimensional Naviar-Stoke’s equations together with the energy equation are used to govern the MHD natural convection heat transfer and fluid flow. In this case the flow is considered as steady, laminar, incompressible, two-dimensional and with buoyancy force. To relate the density changes to temperature changes in the fluid properties the Boussinesq approximation is used to here. The following system of differential equations can govern the steady MHD natural convection.

At Bottom wall:

At Top wall:

At the Left wall: u = v = 0, and heat flux q = 150 w/m^{2}, 200 w/m^{2}, p = 0.

At the right and open side: Convective Boundary Condition (CBC), pressure p = 0, u = v.

At the circular cylinder:

In this work x and y denote the distances measured along the bottom and vertical wall respectively from the origin; u and v denote the velocity components in the x and y direction respectively; T represents the temperature in Kelvin scale; g and a denote the kinematic viscosity and the thermal diffusivity respectively; p denotes the pressure and r is the density of the fluid in the cavity.

The governing equations are nondimensionalized using the following scales.

Continuity equation:

Momentum equations:

Energy equation:

where

At Bottom wall:

At the left wall: U = V = 0; Heat Flux q = 125 w/m^{2}, 150 w/m^{2}, 200 w/m^{2}, p = 0

At the right and open side: Convective Boundary Condition (CBC), p = 0.

The Nusselt number for natural convection is a function of the Grashof number only. We can obtain the local

Nusselt number Nu from the temperature field by applying the function

The overall or average Nusselt number was calculated by integrating the temperature gradient over the heated wall as follows:

Here Pr is a nondimensional heat transfer characteristic in the flow field of natural convection. Here Pr = 0.73.

The updated version of the software on Finite Element Method as numerical technique based on the Galerkin weighted residual method of finite element method is used in this study. This technique is well described by Tailor and Hood [

Computer simulation of Finite Element Method is used to perform the analysis of laminar natural convection heat transfer and fluid flow in an open square cavity with a heated circular cylinder. Effects of various Rayleigh number (Ra) with some fixed Hartmann number (Ha) and Heat flux q are studied on heat transfer and flow inside the cavity by visualizations and line Graphs. The visualizations describe the temperature distributions and flow fields by isotherms and streamlines for different parameters. The values of Ra and Ha in this study vary from 5 × 10^{4} to 5 × 10^{6} and 25 to 150 respectively while Pr = 0.73 & heat flux q = 125, 150, 200. The results are also presented in the tabular form.

The flow with all Ra in this work have been affected by the buoyancy force. Figures from ^{6} in

^{4}, 10^{5}, 5 × 10^{5}, 10^{6}, on isotherms as well as on streamlines for the present configuration at Ha = 75, q = 200. ^{5}, 10^{6}.

^{5}, 5 × 10^{5}, 10^{6}, 5 × 10^{6} on isotherms as well as on streamlines for the present configuration at Ha = 25, q = 125. In this case all most all isothermal lines are concentrated at the right lower corner of the cavity. ^{6}. The recirculations region are increased for Ra = 5 × 10^{5}, 10^{6}, 5 × 10^{6}.

^{4}, 5 × 10^{4}, 10^{5}, 5 × 10^{5} on isotherms as well as on streamlines for the present configuration at Ha = 25, q = 150. In ^{5}, 5 × 10^{5}. In

^{5}, 5 × 10^{5}, 10^{6}, 5 × 10^{6} on isotherms as well as on streamlines for the present configuration at Ha = 75, q = 125. In ^{5}, 10^{6}, 5 × 10^{6}. The isothermal lines are concentrated at the lower right corner of the cavity. One small vortex is formed above the cylinder for Ra = 5 × 10^{6}. ^{5}, 5 × 10^{5}, 10^{6}, 5 × 10^{6} on isotherms as well as on streamlines for the present configuration at Ha = 100, q = 125. In ^{6}. The recirculations region are increased for Ra = 5 × 10^{5}, 10^{6}, 5 × 10^{6}. All most all isothermal lines are concentrated at the lower right corner of the cavity for every

Ra.

^{5}, 5 × 10^{5}, 10^{6}, 5 × 10^{6} at Ha = 125 and heat flux q = 125. The isotherm lines are concentrated at right lower corner of the cavity for every Ra and the isotherm lines are located in the right half of the cavity. Here recirculations are formed around the cylinder for every Ha. One small vortex is formed above the cylinder for Ra = 5 × 10^{6}. The recirculations region are increased for Ra = 5 × 10^{5}, 10^{6}, 5 × 10^{6}. The isothermal lines are concentrated at the lower right corner of the cavity for every Ra. Here the highest temperature remains at upper half of the open side for Ra = 5 × 10^{6} and Ha = 125.

^{4}, 10^{5}, 5 × 10^{5}, 10^{6} on isotherms as well as on streamlines for the present configuration at Ha = 100, q = 250. Here isotherm lines concentrate at lower right corner of the cavity for every Ra. Here recirculations are formed around the cylinder for every Ra. The recirculation regions are increased for Ra = 5 × 10^{5}, 10^{6}.

The various Raleigh numbers and magnetic force affect the heat flux along the sides of the cavity. These are observed in figures from ^{4} to 5 × 10^{5} and decreasing for Ra = 10^{6} to 5 × 10^{6}.

Ra | Heat Flux at Top Wall | Heat Flux at the Cylinder | Heat Flux at Open Side |
---|---|---|---|

5 × 10^{4} | 6000 | 28,000 | 200,000 |

10^{5} | 9500 | 30,700 | 200,000 |

5 × 10^{5} | 35,500 | 42,000 | 220,000 |

10^{6} | 60,000 | 40,000 | 320,000 |

5 × 10^{6} | 140,000 | 26,500 | 1,100,000 |

The numerical study on MHD natural convection heat transfer and fluid flow in a two-dimensional square open cavity containing a heated circular cylinder has been done.

The flow with all Ra in this work has been affected by the buoyancy force. Temperature fields are illustrated in the flow region.

The high temperature region remains at the lower half of the open side for all Ra except for Ra = 5 × 10^{6} in

The isothermal lines are nonlinear for all Ra used in this work and they occupied more than right half region of the cavity.

The significant findings of this work are that for all cases of Ha and Ra the isothermal lines concentrated to the right lower corner of the cavity and there are recirculations around the cylinder and one small vortex has been created above the cylinder in the cavity for highest Ra.

The heat transfer is increasing with increasing Ra at top wall and open side of the cavity.

The recirculation region is increasing with increasing Ra.

The result of this work is a good agreement with the existing heat transfer theory.

Sheikh AnwarHossain,Mohammad AbdulAlim,Satrajit KumarSaha, (2016) Effects of Rayleigh Number on MHD Heat Transfer and Fluid Flow in Two-Dimensional Open Square Cavity with Heat Generation. American Journal of Computational Mathematics,06,1-16. doi: 10.4236/ajcm.2016.61001