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Magnetic traction force in an ordered matrix of a high gradient magnetic separator
Journal of Magnetism and Magnetic Materials 7 (1978) 293-295 0 North-Holland Publishing Company
MAGNETIC TRACTION FORCE IN AN ORDERED MATRIX OF A HIGH GRADIENT MAGNETIC SEPARATOR I. EISENSTEIN Department of Electronics,
The Weizmann Institute of Science, Rehovot, Israel
The magnetic traction force, experienced by a paramagnetic particle, inside an infinite ordered array of wires, that forms the matrix of a high gradient magnetic separator, is calculated. A comparison is made with the forces due to a single wire and to two wires.
A high gradient magnetic separator (HGMS) is a filtering apparatus that is capable of eliminating micron size weakly magnetic particles out of a slurry that is processed through it. The technique has first been used on a large scale, in the Kaoline industry and in the last decade it is a rapidly developing field with applications in the fields of benefaction of ores, sewage purification, and other pollution problems. The separating medium in a HGMS involves a ferromagnetic matrix that, when magnetized by an applied field, creates high field-gradient sites inside it that produce strong traction forces on the particles to be eliminated. Most of the availing matrices consist of a large number of filamentary wires. Yet, the magnetostatic aspect in theoretical treatments of such separators rely on models that involve one wire only. This is understandable in view of the difficulty in treating the prevalent “chaotic” matrices such as stainless steel wool. Recently, however, matrices which consist of an ordered array of wires were considered [ 1,2,3] which proved to be more promising than the disordered ones. Accordingly, it is the aim of this note to consider the magnetic traction force experienced by a paramagnetic particle inside a special type of ordered matrix, namely, an infinite rectangular array of wires. Such an infinite matrix might be a good approximation of an actual finite (ordered) one and should point at qualitative differences between the many-wires matrix and a single wire. To be more specific, the array we consider consists of an infinite number of parallel ferromagnetic wires that lie along the z-axis. Each wire is infinitely long, it 293
has a circular cross section with radius a, and it is saturated to a magnetization M along the x axis by an external field Ho along the same axis. The cross section of the array in the x-y plane forms a simple rectangular lattice with lattice constants al and a2 along the x and y axes respectively. Consider now a paramagnetic sphere with radius b inside the array. Its energy of interaction with the total field H = -VU inside the array is E = +(x,
-
x,)j-( vU)*du .
Here, xp and xm are the susceptibilities of the sphere and the inter-wire medium respectively; U is the total magnetostatic potential and the integral is over the volume of the sphere. Using V*U= 0 between the wires and Gauss’ divergence theorem, eq. (1) is transformed into [4] E =
-4(x, - x,) ;
$SU2dn
,
(2)
where the integral is performed at the surface of the sphere over the solid angle L? refered to the center of the sphere. The traction force 8’“’ is conveniently defined as
Fm = VE = ;n(xp -xm)(2M)*b2F,
(3)
where F is a dimensionless reduced force. The potential U is the sum U = U, + U1 of the external potential U, = -H$ and the lattice potential U1. The latter is the infinite lattice sum over the single wire potential
[51 u
1
=
27rA4a2x/(x2 + y’) .
(4)
294
I. Eisenstein /Magnetic
For an arbitrary field point this sum cannot be reduced to a single term expression and should be evaluated numerically. The problem is that the convergence is very slow and depends on the order of summation. In order to overcome the slow convergence, Poisson’s summation formula [6] is used to convert the sum along they direction into an exponentially convergent series over an index k(=(2rr/aa) times an integer). It is then found that the problem of the summation order, in the original sum, reduces to the question of how to evaluate the k = 0 term in the transformed sum. This question is answered by relating the k = 0 term with the demagnetizing factor D, (0 Q D, G 1) alongx, corresponding to the external shape of the lattice. Namely, a definite summation scheme corresponds to a definite external shape of the lattice which, in practice, is accounted for by the dependence on Dx of the k = 0 term. The newly expressed potential is then used, via (2) and (3), to obtain the energy and the traction [orce in terms of an exponentially convergent series in k. In the following we shall present some numerical results; a more detailed account of the calculations will be published elsewhere. Consider a square array of wires (ur = aa) and a paramagnetic sphere that is already captured by a wire and touches it on the x-axis. It is of interest to compare the traction F, (F,, vanishes), towards the wire, in the array and the force due just to a single wire [7]. The latter force is currently approximated by assuming b <
traction force in a HGMS
s - smg e
WlW
0.2
0.4
0.2
0.4
Reduced radius of ferromagnetic wire p=a/az
Fig. 1. Reduced traction force F, [eq. (3)] between a ferromagnetic wire and a touching (on the x-axis) paramagnetic sphere with different reduced radii Q = b/a,. All curves are drawn for H~/Zm+Z = 1. Curve i is calculated for al = a2 and D, = 0.5. (al and a2 are the distances between adjacent wires along x and y respectively. D, is the demagnetizing factor along x corresponding to the external shape of the array in the x-y plane).
wire force is less satisfactory but is reasonable. Fig. 2 shows the traction force for a fixed value of q and for three values of D,, as a function of the wire radius. It is seen that as Dx increases, the force decreases which is a result of an increasing demagnetizing field. Table 1 shows the maximum lattice force (F,) and its location (p, q) for different values of pertinent parameters. A comparison is made with the maximal singlewire force. It is seen that for H,,/2nM = 1, the variance
1.5
I
Reduced
I
I
Dx = 0.0
wire radius
I
p=a/a2
Fig. 2. Reduced traction force F, between a ferromagnetic wire and a touching (along x) paramaanetic sphere with reduced 4 = b/a2 = 0.07, inside an infinite array of wires. Forar,a2andD,seefig. 1.
I. Eisenstein /Magnetic traction force in a HGMS
295
Table 1. The location (p,q) and value (F,) of maximum reduced traction force [eq. (3)] between a ferromagnetic wire (with radius a) and a touching (alongx) paramagnetic sphere (with radius b) inside an infinite array of wires. For al, a2 and D, see fig. 1. 6 = al/a2
DX
HO/2nM
1
0.0 0.5 1.0 0.0 0.5 1.0 0.5 0.5
1 1 1 0.1 0.1 0.1 1 1
1 1 1 1 1 0.5 2
single wire single wire
-____
P =a102 0.177 -0 -0 0.257 -0 4l 0.074 -0
0.072 -0 +O 0.081 -+O +O 0.031 -+O
1 0.1
of the maximal lattice force with DX is small and insig nificant. Its location is more sensitive to the value of D,; but it so happens that Fx is rather flat, at the vicinity of the maximum, along the p/q = 2.34line (where the maximum of the single wire force occurres) so that the exact location is again not very significant. It is further observed that the single wire presents a good approximation both to the maximal force (F,= 1.266) and the corresponding p/q (= 2.34) value of the lattice. It is expected that as Ho/27rM exceeds 1, these conclusions come even closer to the truth since the role played by the demagnetizing field diminishes with respect to the role played by Ho. They are expected to be less reliable when Ho/2nM becomes less than 1 as is exemplified by the data for Ho/2nM = 0.1, in the table. Presumably, the dependence on the ratio a l/a2 (see table 1) also becomes stronger as Ho/2rM decreases.
2.46 2.34 2.34 3.17 2.88 2.88 2.39 2.34
1.307 1.266 1.266 0.671 0.568 0.568 1.282 1.266
2.34 2.88
1.266 0.568
References
[II R.R. Birss, R. Gerber and M.R. Parker, IEEE Trans. Magn. Vol. MAGl2 (1976) 892. [21 S. Uchiyama, S. Kondo, M. Takayasu and I. Eguchi, IEEE Trans. Mag. Vol. MAC-12 (1976) 895. [31 R.R. Birss, R. Gerber and M.R. Parker, IEEE Conf. Pub. No. 142 (1976) 74. [41 I. Eisenstein, IEEE Trans. Magn., Vol. MAG-13 (1977) 1646. 151 C.A. Coulson, Electricity, (Oliver and Boyed, London, 1961). 161 R. Bellman, A Brief Introduction to Theta Functions (Halt, Reinhart and Winston, New York, 1961). I71 A. Aharoni, IEEE Trans. Magn. Vol. MAG12 (1976) 234.
MAGNETIC TRACTION FORCE IN AN ORDERED MATRIX OF A HIGH GRADIENT MAGNETIC SEPARATOR I. EISENSTEIN Department of Electronics,
The Weizmann Institute of Science, Rehovot, Israel
The magnetic traction force, experienced by a paramagnetic particle, inside an infinite ordered array of wires, that forms the matrix of a high gradient magnetic separator, is calculated. A comparison is made with the forces due to a single wire and to two wires.
A high gradient magnetic separator (HGMS) is a filtering apparatus that is capable of eliminating micron size weakly magnetic particles out of a slurry that is processed through it. The technique has first been used on a large scale, in the Kaoline industry and in the last decade it is a rapidly developing field with applications in the fields of benefaction of ores, sewage purification, and other pollution problems. The separating medium in a HGMS involves a ferromagnetic matrix that, when magnetized by an applied field, creates high field-gradient sites inside it that produce strong traction forces on the particles to be eliminated. Most of the availing matrices consist of a large number of filamentary wires. Yet, the magnetostatic aspect in theoretical treatments of such separators rely on models that involve one wire only. This is understandable in view of the difficulty in treating the prevalent “chaotic” matrices such as stainless steel wool. Recently, however, matrices which consist of an ordered array of wires were considered [ 1,2,3] which proved to be more promising than the disordered ones. Accordingly, it is the aim of this note to consider the magnetic traction force experienced by a paramagnetic particle inside a special type of ordered matrix, namely, an infinite rectangular array of wires. Such an infinite matrix might be a good approximation of an actual finite (ordered) one and should point at qualitative differences between the many-wires matrix and a single wire. To be more specific, the array we consider consists of an infinite number of parallel ferromagnetic wires that lie along the z-axis. Each wire is infinitely long, it 293
has a circular cross section with radius a, and it is saturated to a magnetization M along the x axis by an external field Ho along the same axis. The cross section of the array in the x-y plane forms a simple rectangular lattice with lattice constants al and a2 along the x and y axes respectively. Consider now a paramagnetic sphere with radius b inside the array. Its energy of interaction with the total field H = -VU inside the array is E = +(x,
-
x,)j-( vU)*du .
Here, xp and xm are the susceptibilities of the sphere and the inter-wire medium respectively; U is the total magnetostatic potential and the integral is over the volume of the sphere. Using V*U= 0 between the wires and Gauss’ divergence theorem, eq. (1) is transformed into [4] E =
-4(x, - x,) ;
$SU2dn
,
(2)
where the integral is performed at the surface of the sphere over the solid angle L? refered to the center of the sphere. The traction force 8’“’ is conveniently defined as
Fm = VE = ;n(xp -xm)(2M)*b2F,
(3)
where F is a dimensionless reduced force. The potential U is the sum U = U, + U1 of the external potential U, = -H$ and the lattice potential U1. The latter is the infinite lattice sum over the single wire potential
[51 u
1
=
27rA4a2x/(x2 + y’) .
(4)
294
I. Eisenstein /Magnetic
For an arbitrary field point this sum cannot be reduced to a single term expression and should be evaluated numerically. The problem is that the convergence is very slow and depends on the order of summation. In order to overcome the slow convergence, Poisson’s summation formula [6] is used to convert the sum along they direction into an exponentially convergent series over an index k(=(2rr/aa) times an integer). It is then found that the problem of the summation order, in the original sum, reduces to the question of how to evaluate the k = 0 term in the transformed sum. This question is answered by relating the k = 0 term with the demagnetizing factor D, (0 Q D, G 1) alongx, corresponding to the external shape of the lattice. Namely, a definite summation scheme corresponds to a definite external shape of the lattice which, in practice, is accounted for by the dependence on Dx of the k = 0 term. The newly expressed potential is then used, via (2) and (3), to obtain the energy and the traction [orce in terms of an exponentially convergent series in k. In the following we shall present some numerical results; a more detailed account of the calculations will be published elsewhere. Consider a square array of wires (ur = aa) and a paramagnetic sphere that is already captured by a wire and touches it on the x-axis. It is of interest to compare the traction F, (F,, vanishes), towards the wire, in the array and the force due just to a single wire [7]. The latter force is currently approximated by assuming b <
traction force in a HGMS
s - smg e
WlW
0.2
0.4
0.2
0.4
Reduced radius of ferromagnetic wire p=a/az
Fig. 1. Reduced traction force F, [eq. (3)] between a ferromagnetic wire and a touching (on the x-axis) paramagnetic sphere with different reduced radii Q = b/a,. All curves are drawn for H~/Zm+Z = 1. Curve i is calculated for al = a2 and D, = 0.5. (al and a2 are the distances between adjacent wires along x and y respectively. D, is the demagnetizing factor along x corresponding to the external shape of the array in the x-y plane).
wire force is less satisfactory but is reasonable. Fig. 2 shows the traction force for a fixed value of q and for three values of D,, as a function of the wire radius. It is seen that as Dx increases, the force decreases which is a result of an increasing demagnetizing field. Table 1 shows the maximum lattice force (F,) and its location (p, q) for different values of pertinent parameters. A comparison is made with the maximal singlewire force. It is seen that for H,,/2nM = 1, the variance
1.5
I
Reduced
I
I
Dx = 0.0
wire radius
I
p=a/a2
Fig. 2. Reduced traction force F, between a ferromagnetic wire and a touching (along x) paramaanetic sphere with reduced 4 = b/a2 = 0.07, inside an infinite array of wires. Forar,a2andD,seefig. 1.
I. Eisenstein /Magnetic traction force in a HGMS
295
Table 1. The location (p,q) and value (F,) of maximum reduced traction force [eq. (3)] between a ferromagnetic wire (with radius a) and a touching (alongx) paramagnetic sphere (with radius b) inside an infinite array of wires. For al, a2 and D, see fig. 1. 6 = al/a2
DX
HO/2nM
1
0.0 0.5 1.0 0.0 0.5 1.0 0.5 0.5
1 1 1 0.1 0.1 0.1 1 1
1 1 1 1 1 0.5 2
single wire single wire
-____
P =a102 0.177 -0 -0 0.257 -0 4l 0.074 -0
0.072 -0 +O 0.081 -+O +O 0.031 -+O
1 0.1
of the maximal lattice force with DX is small and insig nificant. Its location is more sensitive to the value of D,; but it so happens that Fx is rather flat, at the vicinity of the maximum, along the p/q = 2.34line (where the maximum of the single wire force occurres) so that the exact location is again not very significant. It is further observed that the single wire presents a good approximation both to the maximal force (F,= 1.266) and the corresponding p/q (= 2.34) value of the lattice. It is expected that as Ho/27rM exceeds 1, these conclusions come even closer to the truth since the role played by the demagnetizing field diminishes with respect to the role played by Ho. They are expected to be less reliable when Ho/2nM becomes less than 1 as is exemplified by the data for Ho/2nM = 0.1, in the table. Presumably, the dependence on the ratio a l/a2 (see table 1) also becomes stronger as Ho/2rM decreases.
2.46 2.34 2.34 3.17 2.88 2.88 2.39 2.34
1.307 1.266 1.266 0.671 0.568 0.568 1.282 1.266
2.34 2.88
1.266 0.568
References
[II R.R. Birss, R. Gerber and M.R. Parker, IEEE Trans. Magn. Vol. MAGl2 (1976) 892. [21 S. Uchiyama, S. Kondo, M. Takayasu and I. Eguchi, IEEE Trans. Mag. Vol. MAC-12 (1976) 895. [31 R.R. Birss, R. Gerber and M.R. Parker, IEEE Conf. Pub. No. 142 (1976) 74. [41 I. Eisenstein, IEEE Trans. Magn., Vol. MAG-13 (1977) 1646. 151 C.A. Coulson, Electricity, (Oliver and Boyed, London, 1961). 161 R. Bellman, A Brief Introduction to Theta Functions (Halt, Reinhart and Winston, New York, 1961). I71 A. Aharoni, IEEE Trans. Magn. Vol. MAG12 (1976) 234.