MHD free convection heat transfer having vertical fin in a square wavy cavity

MHD free convection heat transfer in a square wavy cavity with single vertical fin attached to its lower heated wall has been numerically simulated in this paper. The bottom wall is kept at a constant heat temperature Tn and the upper wall is kept at a constant cold temperature Tc, while the wavy vertical walls are kept at adiabatic. The magnetic field of strength is applied parallel to x-axis. Finite element method based Galerkin weighted residual technique is used to solve the governing equation. The Prandlt number for the flow inside the enclosure is 0.71. A parametric study has been carried out to investigate the effect of Rayleigh number and Hartmann number on the fluid flow and heat transfer characteristic inside the cavity. The obtained results indicated that the heat transfer rate is enhanced with the growth of Rayleigh number when Hartmann number is kept constant. A set of graphical results are presented in terms of streamlines, isotherms, velocity profiles, temperature profiles, local Nusselt number and average Nusselt number. The results are validated comparing with the previous published works.

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Introduction

Free convection is one of the most important phenomena in thermal systems because of its wide range of applications. The relevant research output has numerous applications such as solar energy collectors, cooling of nuclear reactors, ventilation of rooms, fire prevention, electronic cooling devices, biological and geothermal devices etc. A large number of research works have been done on different shapes of cavities. Bilgen [1] studied numerically natural convection heat transfer inside a differentially heated cavity with a horizontal fin attached to the hot wall. The result showed that the heat transfer rate was minimum with the fin attached to the middle or near the middle of the heated wall. Shi and Khodadadi [2] investigated the steady laminar natural convection heat transfer in a differentially heated square cavity due to a thin fin on the hot wall. They found that for higher value Rayleigh number, heat transfer rate was enhanced irrespective of the fin position or length. Ben-Nakhi et al. [3] studied numerically conjugate natural convection in a square enclosure with inclined thin fin of arbitrary length. Tasnim et al. [4] investigated numerically natural convection heat transfer in a square cavity with a baffle on hot wall. They observed that the effect of fin position on the heat transfer rate was depended strongly affected by Rayleigh number and the fin length. Jani et al. [5] carried out the effect of magneto hydrodynamic free convection fluid flow in a square cavity heated from below and cooled from other walls. They showed that free convection parameter and Hartmann number have notable effect on flow structure and temperature field. Xu et al. [6] reported the effect of fins and their height on natural convection flow transition cavity. They found that the flow near the finned wall changes from a steady to periodic unsteady flow at a critical Rayleigh number that is sensitive to the fin length. Sun et al. [7] reported the effect of triangular fins on mixed convection in a lid-driven cavity. They observed that the triangular fin is a good control parameter for flow structure, temperature field and rate of heat transfer. Pirmohammadi et al. [8] . Studied steady laminar free convection flow in presence of a magnetic field in an enclosure heated from left and cold from right wall.

The result showed that with increasing Hartmann number the rate of convective transfer and the average Nusselt number increase. Mahmud and Faster [9] investigated the magneto hydrodynamic free convection and entropy generation in a square porous cavity. They found that the fluid velocity is reduced with increasing the value of Hartmann number. Xu and Saha [10] studied the transition to an unsteady flow due to an adiabatic fin on the sidewall of a square cavity. Gdhaidh et al. [11] investigated analytically the enhancement of natural convection heat transfer within closed enclosure using parallel fins. They observed that as the fin number increases the maximum heat source temperature decreases. Elatar et al. [12] performed a numerical study on laminar natural convection inside square enclosure with adiabatic horizontal wall with a single horizontal fin at different length and positions attached to the hot wall. They investigated the effect of Rayleigh number, fin lengths and fin positions of the enclosure on fluid flow structure and heat transfer characteristics. Ali et al. [13] studied the magneto hydrodynamic mixed convection flow in a hexagonal enclosure. They found that Hartmann number and Richardson number have considerable effect on the flow field and temperature field. Rostami [14] numerically simulated the unsteady fluid flow and heat transfer behavior in a cavity with vertical wavy walls and horizontal straight walls. Abu-Nada and Chamkha [15] examined the mixed convection flow of nanofluid in liddriven cavity with a wavy wall. Sheremet et al. [16] studied the flow and convective heat transfer in a partially heated wavy porous cavity filled with a nanofluid. Alsabery et al. [17] reported the effect of rotating solid cylinder on entropy generation and convective heat transfer in a wavy porous cavity heated from below. Based on the above literature survey and to the best of author's knowledge, it appears that very little work reported on the wavy enclosure having vertical fin. There is no present study on MHD free convection heat transfer having vertical fin in a square wavy cavity will be numerically analyzed in this present study. The objective of this present study is to analyze the increase of the heat transfer rate in fluid through wavy cavity having vertical fin.

Physical Configuration

The physical model and boundary conditions of the present problem under consideration are shown in Fig. 1. Which represents of two dimensional square wavy enclosure with side length W. The bottom wall is kept at a constant heat temperature 𝑇 ℎ and the upper wall is kept at a constant cold temperature 𝑇 𝑐 (𝑇 ℎ > 𝑇 𝑐 ), while the wavy vertical walls are kept at adiabatic. A heated fin of length (𝑙) and thickness (𝑏) is attached to the bottom wall at a position (ℎ) from the left surface. The gravitational force acts in the vertically downward direction and a uniform external magnetic field of strength (B0) is applied along the horizontal direction.

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Schematic description of the physical model and boundary conditions

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Mathematical Formulations

The physical domain is shown in Fig. 1. The flow is considered steady, laminar, incompressible and two-dimensional, which is filled with electrically conducting fluids 𝑃𝑟 = 0.71 interact.The governing differential equations for free convection flow using conservation of mass, momentum and energy can be written as:

where 𝑢 and 𝑣 are the velocity components along x and y directions respectively, 𝑝 is the pressure, 𝜌 is the density, 𝜇 is the dynamic viscosity, 𝛽 is the coefficient thermal expansion, 𝜎 is the electrical conductivity, 𝐵 0 is the magnetic field, 𝑇 is the temperature, 𝑔 is the gravitational force, 𝑣 = 𝜇 𝜌 is the kinematic viscosity and 𝛼 is the thermal diffusivity.

Boundary conditions

The boundary conditions for the present problem are specified as follows:

On the top wall: 𝑢 = 0, 𝑣 = 0, 𝑇 = 𝑇 𝑐 On the left vertical wall: 𝑢 = 0, 𝑣 = 0, Using the following dimensionless parameters, the governing equations can be converted to the dimensionless forms:

where, X and Y are the coordinates varying along horizontal and vertical directions, U and V are the velocity components in the X and Y directions respectively, 𝜃 is the dimensionless temperature and P is the dimensionless pressure. After substitution of the dimensionless variables into equations ( 1) -( 4), we get the following dimensionless equations:

In the above equations, 𝑃𝑟 is the Prandtl number, 𝐻𝑎 is the Hartmann number or magnetic parameter and 𝑅𝑎 is the Rayleigh number defined by:

Nusselt Number

First, the heat transfer by conduction was equated to the heat transfer by convection

where 𝑛 is the normal distance to coordinate surface. By introducing the dimensionless variables, defined in equation ( 5) into equation (11), local Nusselt number is defined as:

The average Nusselt number on the cold wall is obtained as follows:

Numerical Technique

The nonlinear governing partial differential equations, i.e., mass, momentum and energy conservation equations are transformed into a system of integral equations by using the Galerkin weighted residual method of finite-element formulation. The nonlinear algebraic equations so obtained are modified by imposition of boundary conditions. These modified nonlinear equations are transferred into linear algebraic equations with the help of Newton's method. Lastly, these linear equations are solved by applying Triangular factorization. For numerical computation and post processing, the software COMSOL Multiphysics is used.

Program Validation and Comparison with Previous Work

In order to check the accuracy of the numerical results obtained in this problem, a comparison is made between results of the average Nusselt number obtained from this study with other published results for enclosure without a fin as shown in Table 1. The maximum derivation found was 1.55% for the case of 𝑅𝑎 = 10 6 .

𝑹𝒂	𝟏𝟎𝟒	𝟏𝟎𝟓	𝟏𝟎𝟔
Shi and Khodadadi [2]	2.247	4.532	8.893
Tasnim and Collins [4]	2.244	4.524	8.855
Elatar et al. [12]	2.234	4.517	8.948
Present result	2.245	4.522	8.835

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Comparison of average Nusselt number obtained from the present study and those reported in the literature.

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Table 1:

Comparison of average Nusselt number obtained from the present study and those reported in the literature.

𝑹𝒂 𝟏𝟎 𝟒

𝟏𝟎 𝟓 𝟏𝟎 𝟔 Shi and Khodadadi [2] 2.247 4.532 8.893 Tasnim and Collins [4] 2.244 4.524 8.855 Elatar et al. [12] 2.234 4.517 8.948 Present result 2.245 4.522 8.835

In addition, a comparison of the streamlines and isotherms is made for the present results with those of Tasnim and Collins [4] and Elatar et al. [12] at (𝑅𝑎 = 10 5 , 𝐵 = 0.01, 𝐿 = 0.5 𝑎𝑛𝑑 𝐻 = 0.75) as shown in Fig. 2. The shapes of streamline contours are almost identical. For the isotherms, one can see the strong agreement of the present results with those by Tasnim and Collins [4] and Elatar et al. [12] as seen in the figure.

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Comparison of Streamlines and isotherms with Ra = 105, B = 0.1, L = 0.50 and H = 0.75

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Tasnim and collins [4] Elatar et al. [12 Present results

Effect of Rayleigh number

The results are presented in terms of streamlines, isotherms in Fig. 3 and in terms of velocity profiles, temperature profiles and local Nusselt number along the bottom heated wall are shown in Fig. 4. In Fig. 3(a) shows that when 𝑅𝑎 between 10 3 and 10 5 the strength of buoyancy inside the cavity is significant and two elliptic-shaped eddies inside the cavity, Again when 𝑅𝑎 = 10 6 the strength of buoyancy inside the cavity is more significant and four elliptic-shaped eddies appear inside the cavity. Conduction dominant heat transfer is observed from the isotherms in Fig. 3

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Variation of Rayleigh number(103 ≤𝑅𝑎≤106) on (a) Velocity profiles (b) Dimensionless temperature and (c) Local Nusselt number

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Streamlines and isotherms for different values of Rayleigh number (103 ≤𝑅𝑎≤106) at 𝜆= 2, 𝑃𝑟= 0.71 and 𝐻𝑎= 10.

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Effect of Hartmann number

The results are presented in terms of streamlines, isotherms in Fig. 5 and in terms of velocity profiles, temperature profiles and local Nusselt number along the bottom hot wall are shown in Fig. 6. Fig. 5(a) shows that when 𝐻𝑎 = 0 the strength of buoyancy inside the cavity is significant and two elliptic-shaped eddies appear inside the cavity. Again when Hartmann number increases, the strength of the buoyancy inside the cavity is more significant and two elliptic-shaped eddies move downward and close to the bottom wall appear. Conduction dominant heat transfer is observed from the isotherms in Fig. 5

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Variation of Hartmann number(0 ≤𝐻𝑎≤50) on (a) Velocity profiles (b) Dimensionless temperature and (c) Local Nusselt number with

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Streamlines and isotherms for different values of Hartmann number(0 ≤𝐻𝑎≤50) at 𝜆= 2, 𝑅𝑎= 104 and 𝑃𝑟= 0.71.

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Conclusion

The effect of magnetic field on the flow structure and heat transfer behaviors for free convection flow and heat transfer within a square wavy cavity with single vertical fin has been studied numerically by using finite element method. The numerical procedure

Figures and tables from the paper

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Variation of the average Nusselt number of the fin surface versus Hartmann number (0 ≤𝐻𝑎≤50) for different values of Rayleigh

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