The numerical modelling of DC electromagnetic pump and brake flow ...

The numerical modelling of DC electromagnetic pump and brake flow M. Hughes, K. A. Pericleous Centre for Numerical

Model&g

and M. Cross and Process Analysis,

University of Greenwich,

London,

UK

The interaction of an externally imposed magnetic and electric field on the laminar flow of a conducting fluid in a channel is studied using computational techniques. The Namer-Stokes equations and the equations describing the electromagnetic field are solved simultaneously in a single control volume-type computational fluid dynamic code, in a moderate Hartmann number and interaction parameter regime. The flow considered is two-dimensional, with an imposed magnetic field acting in the third dimension over the central region of the channel and decaying exponentially in the remainder. A pair of electrodes placed at right angles to the magnetic field exercises control ocer the resultant Lorentz force and hence the velocity profile shape. This configuration has application in direct-current electromagnetic pumps or, conversely, electromagnetic brakes. The initial parabolic flow profile acquires an M-shape / W-shape mode in the magnetic field fringe regions, corresponding to a pump/brake. A noeel coupled procedure is described to model magnetohydrodynamic phenomena and is used to explore the effects of the Reynolds number, interaction parameter, and applied Lloltage on the pump/brake configuration. Keywords:

DC magnetic pump, MHD flow, Lorentz force, CFD simulation

1. Introduction

Research into the development of a generic magnetohydrodynamic (MHD) solver, for use in a control volume (CV) computational fluid dynamic (CFD) framework, has been underway at Greenwich for some time.“16 The new solver enables the fully coupled solution of the equations characterizing fluid flow of electrically conducting fluids and those of induced magnetic fields. The first results of the study were presented recently’ for simple parabolic Couette and Hartmann flows under a constant magnetic field. The numerical results compared well against the analytical solutions that exist for these asymptotic flows. The present contribution shows the application of the method to a channel flow which is under the influence of a combined magnetic and electric field, each acting at right angles to the other, and at right angles to the flow. Channel flows of this type are encountered in practice in so-called direct current (DC) electromagnetic pumps used in liquid metal fast-breeder reactors and also in lithium blankets of future fusion reactors. The regime of interest in this situation involves moderate to high Hartmann numbers and interaction parameters. As a result of the Lorentz force, the

Address correspondence and reprint requests to Prof. K. A. Pericleous at the Computing and Mathematical Science Department, University of Greenwich, Wellington Street, London SE18 6PF, UK. Received 3 March 1995; revised 4 July 1995; accepted Appl. Math. Modelling 1995, Vol. 19, December 0 1995 by Elsevier Science Inc. 655 Avenue of the Americas, New York, NY 10010

27 July 1995

flowfield is distorted dramatically and the full solution of the Navier-Stokes equations is necessary since both inertial and viscous effects are important in the core region of the flow. Recent numerical studies of this problem have been presented by Ramos et a1.2,3 using both the Galerkin type finite element (FE) method and CV techniques. Winowich and Hughes4 seem to be among the first to investigate this type of flow numerically again using FE formulation but for nonconducting walls. These researchers predicted the M-shape velocity profiles developing on entry into and upstream of the electrode region in agreement with experimental data. The similar problem of circular ducts in transverse nonuniform magnetic fields has been studied recently by Hua’ who solved inertialess inviscid flow equations in a CV framework and by Tillack6 who looked also at the effects of duct inclination and duct radius variation. The modelling of the converse problem, where the system acts as a DC brake, appears to be a novel contribution for which previous publications do not exist. However, other related MHD flows were studied by Wu7 who investigated unsteady fully developed MHD channel flow by means of an FE method, Singh and Lal’ who modelled steady duct flows using FE methods, Gelfgat and Yu9 who used a stream-function vorticity method for MHD channel flows in nonuniform magnetic fields at low Reynolds and Hartmann numbers, and Yagawa and Masuda” who considered an incremental FE technique for MHD flows with low interaction parameter and Reynolds numbers.

0307-904x/95/$10.00 SSDI 0307-904X(95)00110-6

DC electromagnetic

pump and brake flow: M. Hughes et al.

Some of the surprising flow patterns predicted by theory were confirmed by experiments. Early experiments on round ducts in a transverse magnetic field, which reduces by half in the axial direction, were performed by Holroyd.” More recently, research in fusion power reactors has produced a wealth of relevant experiments in rectangular and round ducts in the USA, the former USSR, and other countries. Examples include the work of Picologlou and Reed’* from the Argonne “ALEX” facility and Barleon et a1.13 in the European Torus MEKKA program. The main objectives of this paper are to present a detailed picture of the electric and hydrodynamic fields encountered in DC pumps and “brakes” using a primitive-variable formulation and to study the influence of the various parameters characterizing the flow, including the Reynolds number, the interaction parameter, and the electrode voltage. By critical evaluation of the findings we hope to demonstrate the suitability of the technique used for such complex flows. The next section deals with the general equations of MHD, and this is followed by the description of the problem to be solved, a summary of the numerical method adopted, and a description of the results obtained.

(a)

I I

Cd) Figure 2. (a) B= 0.25 units; (b) B= 0.50 units; (c) B= 0.75 units; (d) B= 1.00 units: 0.24 A/m2 units reference.

2. MHD model The formulation of an appropriate model representation of the systems under consideration have been developed by others. 2,3 In summary, the MHD equations for an incompressible Newtonian fluid with isotropic properties applies here. These can be summarized as follows:

+ I

ix

WtNfTI(Nli

magnetic

line closure V. B = 0

magnetic

induction

= V X (U X B) + L

g

UP

continuity

8. U = 0

Navier-Stokes

pg

x

=kV*T+

+J X B

-

u

+ a,,

J=a(E+uXB)

Ohm’slaw CIUNNEL

= - Vp + #‘U J2

DT

energy

V *B

WALL

(la-e) On a laboratory scale, the VX (U X B) term is negligible in comparison with other terms in the induction equation (lb); then, the induction equation is decoupled from the velocity field, and the system of equations breaks down into three parts14: (i) an electromagnetic system based on V.B=O dB

1

at

UP

_=-

(2a-c)

V*B

J=a(E+UXB)

(ii) a fluid mechanics

problem based on

v.u=o Figure 1. (a) Schematic of a DC electromagnetic pump. (b) Schematic of a two-dimensional MHD channel flow configuration and applied magnetic field.

714

Appl.

Math.

Modelling,

1995, Vol. 19, December

pg=

-VptpV’UfJxB

(3)

DC electromagnetic and (iii) a thermal problem


Substituting equations results as follows:

based on

(4)

(l)-(5),

a system

v.LJ*=o N_‘U * . Vu*

The present application is isothermal. However, the algorithm developed contains the thermal equations also. For this reason and for completeness they have been included here.

(6) into equations

= -VP*

V2+* = V.(U* J*=

-V4*

+Mp2V2U*

+J*

xB*

xB*)

+U*

xB* (7a-d)

2.1 Electric field potential Finally, replacing E in Ohm’s law by its scalar potential 4, an elliptic equation for the electric potential is obtained as below v2+=

V.(UXB)

(5)

Equations (l)-(5) along with the boundary conditions listed below govern the problem under consideration.

2.2 Magnetic parameters The MHD equations may be put into dimensionless form, using certain parameters, which will be referred to later (the subscript ‘0’ denotes a characteristic value of the field variable and L represents a characteristic dimension of the geometry under consideration). These are: R, the magnetic Reynolds number, given by c+J,,L. This parameter measures the extent to which magnetic field convection dominates over magnetic field diffusion. When R, is small, diffusion dominates and the magnetic transport equation uncouples from the velocity equation. M, the Hartmann number, given by (aBiL2/pf)‘.‘, where pu, denotes the dynamic viscosity of the fluid. This parameter measures the ratio of the Lorentz force to viscous forces. N, the interaction parameter. For finite conductivity N is taken as aBiL/pU, and represents the ratio of the Lorentz force to inertial forces.

3. The problem considered The problem considered is shown schematically in Figure I together with the applied magnetic field distribution. The fluid in effect enters and leaves a magnetic region which can cause severe distortions to the velocity profile. The configuration shown in Figure 1 represents a DC electromagnetic pump, or alternatively a brake depending on the sign of the electrode potential. It has been modelled two-dimensionally (2D) by considering a channel of height 2d and length 2L, + L,. L, is the length of the wall on either side of an electrode which is of length L,. For small magnetic Reynolds numbers, the induced magnetic field can be neglected and, hence, the magnetic field is uncoupled from the velocity field. The applied magnetic field is shown in Figure I and is constant in the electrode region, with exponential decay on either side representing the fringing effect at the magnetic pole edges. An electrically conducting fluid flows through the duct under steady2 laminar flow conditions. The fluid has a parabolic profile far upstream of the fringe region. The laminar assumption is consistent with the flow Reynolds number used. Table 1.

Boundarv conditions

Met (a) u(--,

(b) v(--,

y)=o

(c) P( -a,

y) = /mown (8)

Outlet (d) ;b,

2.3 Nondimensional

statement

of the problem

To facilitate parametric studies and comparison with other researchers, a nondimensional system of equations is introduced:

y) = 1.5 U,Cl - I)2 d

y) = 0

(e) vb, y) = 0 (f) Pb, y) = 0 Symmetry plane

(a) $(x,0)=0 (b)~(x,O)=O (9)

(6)

Top wall (c) u(x, d)=v(x, d)=O (d) 4(-m, d)=+(m, d)=O Electrode L (a)+(x,d)=+,Ixl and transverse plane (yz) of the duct. The expected current behavior across the transverse plane is illustrated in Figure 2. This shows the computed current density field in the transverse plane of a 3D insulated duct for varying magnetic field strength. For the channel considered in this paper, I J, I -x I J, I, IJ, I, while the magnetic field rises steeply from a zero magnitude to a constant value within the magnet pole region, the problem can, therefore, be treated as 2D, giving great savings in computing effort.

Figure 3.

716

Math.

Modelling,

Inside the region of interest, given that there are no electrodes and since the walls are insulators, the electromo tive and electrostatic field will coexist, acting in opposite directions. The electromotive field dominates around the region where the magnetic field is strong, while outside this area, where the magnetic flux density falls quickly to zero, the electrostatic field will dominate. This causes the current to flow in closed loops and the system will naturally act as a brake. In the upstream approach to the magnetic field region, the axial current component is in the positive x-direction for the lower part of the duct and in the negative x-direction in the upper half; hence, it has a value close to zero in the central region. The opposite happens in the downstream fringe region. The current flux lines are, therefore, spread out near the channel center and compressed nearer to the walls. The velocity profile that

mQ

-

fl”VCATlO”

(a)

m40

PUMP

t*mJl_C\TloN

@I

(a) Velocity vectors and electric field potential.

Appl.

et al.

1995,

Vol.

(b) Flow streamlines.

19, December

(c) Lorenz force.

DC electromagnetic develops is then M-shaped, since resistance to the axial component of velocity is weakest near the walls, but the fluid nevertheless obeys the no-slip condition of the wall itself. Provided that the applied magnetic field is strong and the electrical conductivity of the fluid high enough, the process, of entry and exit from such a magnetic region will cause the velocity profiles to distort dramatically, the severity of distortion will be influenced by the conductivity of the walls. The introduction of electrodes in this region can serve to influence the magnitude and direction of the electrostatic field. In the case of a pump the strength of the existing electrostatic field is boosted by the presence of the electrode to such a degree that it dominates over the electromotive field. The resulting current distribution now acts in such a direction as to make the system act as a pump. Reversing the polarity of the electrodes causes the electrostatic field to act in the same direction as the electromotive field causing the system to act as a brake.

Figure 4.

and brake flow: M. Hughes

Although the governing equations of an MHD system have been summarized above, the characteristics of a particular problem need to be established through the boundary conditions. Table I summarizes the relevant equations for the problems considered here. The inlet relations define a parabolic profile into the channel. The outlet pressure is assumed constant, and a no-slip velocity condition exists at the walls. The electric potential at the far ends of the duct is fixed to zero, a consequence of the electric field decaying rapidly in the far regions of the duct as the electric current flows along the paths of low resistance centered around the electrode region. A plane of symmetry exists along the center of the channel. Equation (lOa) specifies the electrode potential, while (lob) gives the boundary condition for a nonconducting wall.

PUHP

sl”uLeTIOW

cn=lo;“=70.7;R.=%Jo>

(a)

mm

PW

tl?lul_~TIoN

(b) Flow stream lines. (c) Lorenz force.


(b)

DC electromagnetic


distribution at the center of the duct for the channel with no magnetic restrictions, against those from a pump and brake type configuration.

malized pressure in the electrode region is not significantly influenced by the Reynolds number and hence plots of pressure distribution for Re are not included.

5.2 M-Reynolds

5.3 N-Interaction

/ Hartmann

number effects

Figures 7, 8, and 9 show profiles of axial velocity at various axial locations along the duct, for Reynolds numbers of 500, 1,000, and 2,000 and for interaction parameters of 5 and 10, respectively. The figures indicate that axial velocities are distorted into M-shapes in a similar manner for each case. However, it is clear from the diagram that the distortion takes longer to recover its parabolic shape as the Reynolds number increases, although the change in the M-shape profile around the electrode region is not particularly distinct. Similarly, nor-

MHDBRAE

Ouct

Figure 12.

“Ill

a-8on -uct

SIWTION

parameter

effects

Comparisons of the flow profiles in Figures 8 and 9 show that the distortion of the axial component of velocity increases as does the recovery distance when the interaction parameter increases, with the peak of the velocity profile increasing quite considerably in each case. The figures indicate that a small change in interaction parameter appears to give a much larger distortion of the axial velocities around the electrodes than a relative change in the Hartmann number and, hence, the Reynolds number. Again, the pressure when normalized does not appear

~“=10:“(r70.7;R.e1oo,

w

Ellctrodm h-m>

+1.su

(a) Total current. (b) Electric field current. (c) Current due to magnetic field (U

Appl.

x

Math. Modelling,

B term).


721

DC electromagnetic


sensitive to the values of N chosen. Figure 7 shows the results in a form that can be compared against Ramos et al.* The agreement is very good indeed. 5.4 Electromagnetic

brake

It is interesting to consider the behavior of the flow once the externally imposed electric field is removed, or even reversed, so that it enhances instead of counteracting the induced field. The effect on pressure can be seen in Figures 6 and 10. Figure lla shows velocity vectors and electric field contours. The parabolic velocity profile is pinched toward the axis ahead of the electric/magnetic field region, with reverse flow developing close to the wall. The parabolic shape is recovered quickly downstream of the magnetic field. The superimposed contours of electrostatic field potential 4 are positive close to the electrode, decaying toward zero at the axis, and become negative immediately outside the electrode region. Figure lib shows the corresponding flow streamlines. The fluid is pulled away from the duct wall by the electromagnetic Lorentz force, and two recirculation vortices appear. The fluid accelerates along the axis to preserve continuity. Figure Ilc shows vectors of the Lorentz force. The components of current density J and their resultant are shown in Figure 12. In this case, the components act in the same sense, augmenting each other. 5.5 M-Reynolds

/ Hartmann

number effects

Figures 13 and 14 show profiles of axial velocity at various axial locations along the duct, for Reynolds numbers of 500, 1,000, and 2,000 and for interaction parameters of 5 and 10, respectively. The figures indicate that axial velocities are distorted into W-shapes which increase

Figure 14.

Axial velocity profiles,

N = 10.

slightly as the Reynolds number is increased. The figures indicate that the distortion takes slightly longer to recover its parabolic shape as the Reynolds number increases, although the recovery distance is significantly shorter than that of the distortion from the pump, due to the recirculatory flow profile. Normalized pressure in the electrode region is not significantly influenced by the Reynolds number and, hence, plots of pressure distribution for Re are not included. 5.6 N-interaction

parameter

effects

Comparisons of the flow profiles shown in Figures 13 and 14 indicate that the distortion of the axial component of velocity in the electrode region increases along with a corresponding increase in N number. However, unlike the pump configuration, the recovery distance does not increase significantly with increasing N number. The results suggest that the fluid remembers more the Lorenz force when the duct acts as a pump than when it acts as a brake. This, coupled with steeper velocity gradients away from the walls in the brake case, leads to a faster rate of transfer of axial momentum. As a consequence, the parabolic velocity profile is recovered faster in brakes than in pumps. Again, the pressure when normalized does not appear sensitive to the values of N chosen.

0.

6. Conclusions

Figure 13.

722

Axial velocity profiles,

Appl.

Math. Modelling,

N = 5.

New algorithms to represent coupled MHD flows have been employed to investigate the behavior of a conducting fluid passing through a duct with an electrode placed in the channel. The flow behavior predicted for the pump configuration showed a close quantitative agreement with the


.__-______^.- ..-

DC electromagnetic results of Ramos and Winowich even though the meshes used are considerably coarser than previously employed. The behavior of the system subject to variations in the Hartmann number, interaction parameter and Reynolds number have been investigated. The pump configuration causes the flow to deform into an M-shape profile which is influenced by the various system parameters. If the voltage of the controlling electrode is reversed, the configuration changes to that of a brake. The simulation then demonstrates the expected pressure drop across the magnetic region, which through the Lorentz force, effectively tries to arrest the fluid. The resulting velocity profile then becomes W-shaped, with a pronounced flow reversal close to the wall. Finally, it is worth noting that the algorithm, which is 3D can be employed in other flow situations where a fully coupled MHD simulation is required.”

et al.

solutions of three-dimensional MHD flows in transverse magnetic fields. Liquid Metal Mugnetohydrodynamics, eds. J. Lielpeteris and R. Moreau, Kluwer Academic Publishers, 1989, Vol. 10, pp. 13-20 6. Tillack, M. S. Application of the core flow approach to MHD fluid flow in geometrical elements of a fusion reactor blanket. Liquid Metal Magnetohydrodynamics, eds. J. Lielpeteris and R. Moreau, Kluwer Academic Publishers, 1989, Vol. 10, pp. 47-54 7. Wu, S. T. Unsteady MHD duct flow by the finite element method. strong non-uniform

Int. .I. Num. Meth. Eng. 1973, 6, 3 8. Singh, B. and Lai, J. Finite element problem in magnetohydrodynamic channel flow problems. Int. J. Num. Merh. Eng. 1982, 15,

1104 9. Gelfgat,

10.

11.

References

84,313

and brake flow: M. Hughes

5. Hua, T. Q. Numerical

12.

Hughes, M., Pericleous, K. A. and Cross, M. The CFD analysis of simple parabolic and elliptic MHD flows. Appl. Math. Modelling 1994, 18, 150-155 Ramos, J. 1. and Winowich, N. S. Finite difference and finite element methods for MHD channel flows. In?. J. Num. Meth. Fluids 1990, 11, 907-934 Ramos, J. I. and Winowich, N. S. Magnetohydrodynamic channel flow study. Phys. Fluids 1986, 29(4), 992-997 Winowich, N. S. and Hughes, W. F. A finite element analysis of two-dimensional MHD flow. AIAA Prog. Astronaut. Aeronaut. 1983,

pump

13.

14. 15. 16.

Y. M., Peterson, D. E. and Shcherbinin, E. V. Velocity structure of flows in nonuniform magnetic fields, Part 1: Numerical calculations. Magnetohydrodynamics 1978, 14(l), 55-61 Yagawa, G. and Masuda, M. Finite element analysis of magnetohydrodynamics and its application to lithium blanket design of a fusion reactor. Nucl. Eng. Design 1982, 71, 121 Holroyd, J. R. An experimental study of the effects of wall conductivity, non-uniform magnetic fields and variable area ducts on liquid metal flows at high Hartmann number. Part I. Ducts with non conducting walls. J. Fluid Mech. 93, 609-630 Picologlou, B. F. and Reed, C. B. Experimental investigation of 3-D MHD flows at high Hartmann number and interaction parameter. Liquid Metal Magnetohydrodynamics, eds. J. Lielpeteris and R. Moreau, Khtwer Academic Publishers, Netherlands, 1989, Vol. 10, pp. 71-77 Barleol, Experimental and theoretical work on mhd at the Kemforschungszentrum Karlsruhe. The Mekka-Program Liquid Metal Magnetohydrodynamics, eds. J. Lielpeteris and R. Moreau, Kluwer Academic Publishers, 1989, Vol. 10, pp. 55-61 Patankar, S. V. Numerical Heat Transfer and Fluid Flow. Hemisphere Pub. Corp., Washington D.C., 1980 Moreau, R. Magnetohydrodynamics, Kluwer Academic Publishers, Netherlands, 1990, Vol. 3. Hughes, M. Computational magnetohydrodynamics. Ph.D. Thesis, University of Greenwich, 1994

Appl. Math. Modelling,


723