The Study of the Electroconductive Liquids Flow in a Conduction Magnetohydrodynamic Pump

Abstract

This paper deals the study of a linear MHD pump solution used to eliminate and to avoid the dangers of the mercury appearing through pollution and contamination. The formulation of the magnetohydrodynamic phenomena is derived from Maxwell and Navier-Stokes equations are solved using the finite volume method. Simulation results highlight the performance of the pump such as the electromagnetic force, the velocity, and the pressure, the application of Ansys-Fluent software validation these results.

1. INTRODUCTION

Magnetohydrodynamics (MHD) is the theory of the interaction of electrically conducting fluids and electromagnetic field [1]. Application arises in astronomy and geophysics as well as in connection with numerous engineering problems, such as liquid metal cooling of nuclear reactors, electromagnetic casting of metals, MHD power generation and propulsion [1].

The pumping of liquid metal may use an electromagnetic device, which induces eddy currents in the metal. These induced currents and their associated magnetic fields generate the Lorentz force, and allow the pumping of liquid metal [2,3]. Magnetohydrodynamics is widely applied in various domains, such as metallurgical industry, to transport or the liquid metals in fusion and the marine propulsion [4,5].The advantage of these pumps, which ensure the energy transformation, is the absence of moving parts.

The interaction of moving conducting fluids with electric and magnetic fields allows for a rich variety of phenomena associated with electro-fluid-mechanical energy conversion [6,7].

In our case, a linear MHD pump is studied to eliminate and to avoid the dangers of the mercury appearing through manipulation and pollution. Metal pollutants and contamination at the Azzaba Algeria plant have led to its closure [8]. A 2D study is sufficient to highlight the performances of the pump. A bibliographic study [9-11], of mercury linear MHD pumps shows that there are many phenomena that require further study in the dynamic state. In previous work, a 2D electromagnetic phenomena in a MHD pump using the finite volume method has been obtained [9]. This paper presents the numerical modeling of the coupling electromagnetic hydrodynamic phenomena using the finite volume method in a DC MHD conduction pump. The resolution of the equations is obtained by introducing the magnetic vector potential A, the vorticity ζ and the stream function Ψ.

 

2. MATHEMATICAL ANALYSIS OF ELECTROMAGNETIC AND HYDRODYNAMIC PROBLEMS

The schematic structure of the pump is shown in Fig. 1. In the pump, the electromagnetic forces are obtained from the Lorentz forces are induced by interaction between the applied electrical currents and the magnetic fields [3,5].

Fig. 1.Scheme of a DC MHD pump [3].

The basic design concept is to apply electrical currents across a channel filled with electrically conducting liquid (mercury) and magnetic fields orthogonal to the currents via electromagnetic circuit [9-11].

The properties of the mercury are given in Table 1.

Table 1.Fluid properties.

The electromagnetic model of the MHD pump is as follows:

The magnetic induction and the electromagnetic force are given as:

Following the two-dimensional (2D) development in Cartesian coordinates, where the current density and the magnetic vector potential are perpendicular to the longitudinal section of the MHD pump, the equation becomes:

The equations describing the pumping process in the channel are the momentum conservation equation for laminar incompressible flows are:

Where V the velocity vector, P the pressure, ρ the density of the liquid, ν the kinematic viscosity and the Laplace force which is given by equation (3).

The difficulty is that in the previous equations there are two unknowns: the pressure and the velocity. The elimination of the pressure from the equations leads to a vorticity-stream function which is one of the most popular methods for solving the 2-D incompressible Navier-Stokes equations [9,12]:

Where Vx and Vy are the components of the velocity V.

Using these new dependent variables, the two momentum equations can be combined (by eliminating pressure) to give:

After substituting equation (7) into equation (10) we obtain an equation involving the new dependant variables ξ and ψ such as:

To determine the pressure, the resolution of an additional equation is necessary. This equation is obtained by differentiating equation (5) and using the continuity equation (6). This equation is referred as the Poisson equation for pressure:

 

3. NUMERICAL METHOD AND RESULTS

The method consists of discretising differential equations by integration on finite volumes surrounding the nodes of the grid.

In this method, each principal node P is surrounded by four nodes N, S, E and W located respectively at North, South, Est and West (Fig. 2) [13,14].

Fig. 2.Discretisation in finite volume method.

We integrate the electromagnetic and Navier Stokes equation in the finite volume method delimited by the surfaces E, W, N and S [15]. Finally we obtain the algebraic equation which is written as:

After integration, the final algebraic equation will be:

The matrix of this system of equations is written in the form:

where:

[M+jL] : Matrix Coefficients,

[A] : Vector Potential Matrix,

[F] : Vector source Matrix.

With the Dirichlet boundary and the Neumann conditions are . The resolution is done according to an iterative process.

The resolution of the electromagnetic and the hydrodynamic equations allows to determine the magnetic potential vector, the magnetic induction , the electromagnetic force , the velocity and pressure in the channel of a conduction MHD pump.

Fig. 3.Computation algorithm.

Figure 4 represents the distribution of the magnetic vector potential in the MHD pump.

Fig. 4.Magnetic vector potential in a Dc MHD pump.

Figure 5 represents the magnetic flux density in the channel, and shows that, the magnetic induction reaches its maximum value in the inductor.

Fig. 5.Magnetic induction in the MHD pump.

Figure 6 represents the electromagnetic force in the channel, demonstration that the maximum value is in the middle of the channel of the MHD pump.

Fig. 6.Electromagnetic force in the MHD pump.

Figure 7(a) represents the velocity in the channel of the MHD pump. The velocity of the fluid flow passes by a transient mode, then is stabilized as in all electric machines and the steady state is obtained after approximately five seconds. The obtained results are almost identical qualitatively to those obtained in previous reports [6,16].

Fig. 7.Velocity in the channel of the MHD pump. (a) Using finite volume method and (b) using ansys.

Figure 8(a) shows the pressure in the channel of the MHD pump, demonstration that the pressure increases as the time increases.

Fig. 8.Pressure in the channel of the MHD pump. (a) Using finite volume method and (b) using ansys.

Software ANSYS is a tool for electromagnetic fields simulation and calculation of the physical systems, allowing electromagnetic, hydrodynamic and thermal analysis.

The results from the ANSYS Fluent are exploited to represent the velocity and the pressure in the channel of the DC MHD conduction pump, Figs. 7(b), 8(b).

 

4. CONCLUSIONS

This paper presents the numerical modeling of the coupling electromagnetic hydrodynamic phenomena using the finite volume method in a DC conduction MHD pump. Various characteristics such as the distribution of the magnetic vector potential, the magnetic flux density, the electromagnetic force, the velocity and the pressure are described. The obtained results using finite volume method are the same to those obtained by ANSYS Fluent.