Finite difference 2D model of a magnetohydrodynamic inertial actuator M. Mesurolle1, Y. Lefèvre1, C. Casteras2 1
LAPLACE, Université de Toulouse, CNRS – INPT-ENSEEIHT, 2 Rue C. Camichel, BP 7122, 31071 Toulouse Cedex 7, France 2
CNES, Space Center of Toulouse, 18, Av. Edouard Belin, 31401 Toulouse Cedex 9, France E-mail:
[email protected] –
[email protected]
Abstract A magnetohydrodynamic inertial actuator uses a liquid metal rather than a solid mass. The liquid, inside an annular channel, is accelerated thanks to transverse electric and magnetic fields. This paper proposes a 2D steady state numerical approach to compute the angular momentum’s device under the assumption of a laminar flow.
I.
with σ the electrical conductivity. In steady state and after introducing the electric scalar potential , following equations are obtained from equations (1)-(3): ρ ∙ ∇ ∇ ∇p µ ˄ ∆φ ∇ ∙ ˄
INTRODUCTION
Attitude control systems are necessary to satellites in order to complete earth or astronomical observation missions. Most of the time, reaction and momentum wheels are used to perform the attitude control in space missions. These wheels consist of DC brushless motors. Despite significant improvements on electronic parts, they have many drawbacks, directly linked to their ball bearings [1]. To overcome this situation, that means using no moving solid component, a solution consist in using a conductive liquid accelerated by the Lorentz force thanks to transverse electric and magnetic fields [2]. A magnetohydrodynamic (MHD) reaction wheel has been studied using a dynamic lumped parameter model based on a 1D steady state analytical model [3]. In this paper a steady state approach is proposed in order to derive a 2D MHD model of an inertial MHD actuator [4]. The first section presents the set of MHD equations. The second section describes the studied device. The third section concerns the steady state MHD 2D model. At last the finite difference scheme is proposed before showing some of the obtained results.
(5) (6)
(4)-(6) constitute the MHD set of equations at steady state. II.
STUDIED DEVICE
The studied device is composed by an annular channel, with rectangular section. A liquid metal can flow in it. A DC voltage drop U is imposed between two electrodes creating an axial electric field, as illustrated on Fig. 1. A DC radial magnetic field is imposed by means of an electromagnet for instance.
II. MHD SET OF EQUATIONS The Navier-Stokes (NS) equation describes the motion of fluids submitted to a driving force density . Considering an incompressible fluid, NS equation yields:
Fig. 1 A cross section of the annular channel
III. 2D AXISYMMETRIC MODELLING ρ
ρ ∙ ∇ ∇ ∇p µ
(1)
with ρ the mass density, μ the dynamic viscosity, p, the non-hydrostatic pressure and , the fluid speed. We use the hypothesis of quasi-static approximation, the Maxwell’s equations are given by the set of equations (2): ∇. 0
∇˄
∇˄ μ
(2)
with , the magnetic field density, , the electric field, the current density and , the magnetic permeability. Equations (1) and (2) are coupled through the Lorentz force equation and the generalized Ohm’s law: ˄
˄
(3)
The geometry, the distribution of the imposed electric scalar potential and the magnetic field are invariant by rotation around the axis Oz. 2D axisymmetric assumptions can be applied. In this case, there is no pressure gradient along eθ and all variables do not depend on θ. Therefore the speed is assumed to be ortho-radial and is of the form: vr, z!"
(7)
Due to Ampere’s law, the current density induces a magnetic field density #. As (O, r, z) is a plane of symmetry for the current density, # is ortho-radial. Assuming that the imposed magnetic flux density is radial and divergence free the magnetic field density in the channel is given by: r, z B &!' br, z. !"
Acknowledgement: This work has been performed thanks to the French space center (CNES) and “la direction générale de l’armement” (DGA)
(8)
with :
[ 1
& ) & )* +*
)* is the imposed magnetic flux density at the middle of channel & +* . These assumptions give the steady state model that can be deduced from [5]. The NS equation (5) yields: /
Δv -1 )* +* 1 0
2
34
/ 56 76 89 0
3
8:
0
(9)
) + , * *
V.
56 76 2 3
;
0
) + * *
The same type of equation can be obtained by discretizing equation (10). All equations around each node are assembled to constitute a system of linear equations. The unknowns are the values of the electric potential scalar on each node and its velocity. The right hand side of this system is obtained by taking into account the boundary conditions (11)-(14).
and from equation (6) one can obtain: Δφ
\
RESULTS
(10)
To these equations, the boundary conditions must be added. We assume that the fluid has no velocity at the vertical wall and at the horizontal electrodes: vRm > g, z 0
(11)
vr, >w 0
(12)
The electric potential is imposed at the electrodes: φr, >w ∓
B
(13)
At the vertical non-conductive walls the current density is tangential and the electric field as well, then: 8
8C
IV.
Rm > g, z 0
(14)
FINITE DIFFERENCE DISCRETIZATION
The 2D axisymmetric model is constituted by the coupled equations (9) and (10) and the boundary conditions (11)-(14). As the study domain is very simple (Fig. 1), finite difference method is chosen to discretize the study domain and these partial derivatives equations. The study domain of Fig.1 is then discretized by a grid of D* vertical lines and of E* horizontal lines. If we consider a node D, E. It is numbered F . The nodes around it are numbered FG , for D 1, E, F , for D, E 1, FH , forD 1, E and FI , for D, E 1. Let JK the distance between node F and node FK . The difference equation of equation (9) around node F is of the form: -L
MN
CO MT CN
P4 CN4
Q 1 vR -LG
1 UH LI UI
V Q CN
MS
1 vG L v -LH
CN V
Q CO I I
0
(15)
where & is the radius of node F , vK is the velocity at node FK and LW , XW , QW are coefficients that depend on JY , for instance: 2 2 L
,L JG JH G JG JG JH The coefficients [ and \ are physical coefficients defined by equation (9):
Fig. 2: Distribution of velocity in the study domain
The type of results is shown on Fig. 2. From the distribution of velocity and the electric scalar potential in the study domain the electromechanical quantities can be computed. Some of them are compared to the ones obtained from the 1D model presented in [3]. TABLEAU I. COMPARISON 1D AND 2D MODELS
Torque Mean velocity Angular Momentum VI.
1D model 0.0043 N.m 0.229 m/s 0.1960 kg.m2/s
2D model 0.0047 N.m 0.225 m/s 0.1947 kg.m2/s
CONCLUSION
The results of the table I validate the 1D model compared to the 2D model. In the final version of the paper we will show that the 2D model allows getting more results such as the equivalent charge density defined in (10). REFERENCES [1] K. D. Kumar, “Satellite Attitude Stabilization Using Fluid Rings,” Acta Mechanica Online, Vol. 208, No. 12, 2009. [2] D. R. Laughlin, H. R. Sebesta, and D. E. Ckelkamp-Baker, “A Dual Function Magnetohydrodynamic (MHD) Device for Angular Motion Measurement and Control,” Advances in the Astronautical Sciences, Vol. 111, 2002. [3] A. Salvati and F. Curti, "MHD Reaction Wheel for Spacecraft attitude control : configuration and lumped",The International Academy of Astronautics and Astrodynamics, 24-26 March 2014, Rome , Italy. [4] C. Casteras, Y. Lefevre, and D. Harribey, “Magnetohydrodynamic inertial actuator,” 2013. WO Patent App. PCT/EP2013/053,101 [5] R. Moreau, Magnetohydrodynamics. Norwell, MA: Kluwer, 1990.
