Jan Svoboda. Magnetic Techniques for the Treatment of Materials

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300 CHAPTER 4. DESIGN OF MAGNETIC SEPARATORS

Figure 4.50: A three-dimensional image of the magnetic ﬁeld around a sus-

pended magnet.

ﬁeld and, therefore, by the shape of the pole tips.

The shape of the pole tips is determined by several factors, namely by the

choice of the magnetic ﬂuid and the number of density fractions that are to be

produced. If paramagnetic liquids are to be used, the constant apparent density

of separation in the entire volume of the ﬂuid will be obtained if the force acting

on a particle is constant, I = const., which corresponds to EuE = const., as

transpires from eq. (3.144). This condition can be re-written as

= frqvw= (4.59)

where | is the vertical axis, as shown in Fig. 4.51. This requirement is the result

of the linear dependence of the ﬂuid polarization, namely 

E,ontheexternal

magnetic ﬁeld. The ﬁeld that satisﬁes condition (4.59) is the isodynamic ﬁeld.

As can be seen from eq. (3.145), the situation is somewhat dierent with

ferroﬂuids: if the ferroﬂuid is magnetically saturated, a constant density of

separation will be obtained by maintaining the constancy of the force acting

on the particles by satisfying the condition uE = const. In other words, the

condition

= frqvw= (4.60)

should be satisﬁed in a su!ciently large region of the magnetic ﬁeld.

Two basic shapes of pole tips have been used in ferrohydrostatic separa-

tors, namely a simple wedge geometry of the working space, and pole tips with

hyperbolic shape.

4.8. DESIGN OF A FERROHYDROSTATIC SEPARATOR 301

Iron yoke

Permanent

magnet

Figure 4.51: Schematic diagram of an FHS magnetic circuit with a wedge-

shaped working gap.

4.8.1 FHS separators with a wedge-shaped working gap

Wedge-shaped working space was used in separators with permanent magnets,

such as those built by Tohoku University [S50], Nittetsu Mining Co. (Japan)

[S51] and the Too Geos (Russia) [S52] machines. A schematic diagram of such

a magnetic circuit is shown in Fig. 4.51.

Since the gradient of the magnetic ﬁeld, in this design, is not constant along

the vertical (CE@C| 6= frqvw=), the apparent density of the ferroﬂuid along

the vertical will also vary. A typical distribution of apparent density of the

ferroﬂuid in a separator with wedge-shaped working space, used for recovery

of non-ferrous metals, is shown in Fig. 4.52. It can be seen that essentially a

continuous spectrum of densities is present in the pool of the ferroﬂuid. The

usefulness of this density distribution is very limited as inter-particle collisions

and interactions do not allow accurate and selective separation of individual

metals into individual density fractions.

Distributions of the magnetic induction and the apparent density of the fer-

roﬂuid along the vertical axis for the Too Geos separators with ferrite permanent

magnet-based wedge-shaped pole tips were reported by Solodenko [S52]. The

results are summarized in Fig. 4.53. Shimoiizaka et al. [S50] obtained a similar

pattern in a separator that used SmCo permanent magnets (see Fig. 4.54). It

can be seen again that a magnetic system with wedge-shaped poles creates a

plurality of values of the ﬁeld gradient, and, therefore, of the apparent den-

sity of the ferroﬂuid along the vertical axis of the magnet gap. It is clear that

such patterns of the apparent density can be used only for density separation

of mixtures of materials with large density dierential.

Speciﬁcations of the Tohoku University FHS separator, as reported by Shi-

302 CHAPTER 4. DESIGN OF MAGNETIC SEPARATORS

Surface of

ferrofluid

Bottom of

the magnetic

pole

/cm

Feed

oint

Pool of

ferrofluid

Figure 4.52: Distribution of the apparent density of ferroﬂuid in an FHS with

a wedge-shaped working space. Recovery of various non-ferrous

metals at dierent positions along the vertical axis is indicated.

Figure 4.53: The apparent density and magnetic induction in a permanent

magnet-based ferrohydrostatic separator with wedge-shaped pole

tips, as a function of distance from the base of the pole tips

(adapted from [S52]).

4.8. DESIGN OF A FERROHYDROSTATIC SEPARATOR 303

Figure 4.54: Magnetic induction and its gradient in the gap of a SmCo

magnet-based ferrohydrostatic separator with wedge-shaped pole

tips (adapted from [S50]).

Table 4.8: Speciﬁcations of the Tohoku University FHS separator for the recov-

ery of non-ferrous metals, for three dierent values of the width of

the magnet gap [F21].

Parameter l

=60mm l

=80mm l

= 100 mm

Magnetic ﬁeld [T] 0.362 0.314 0.265

Field gradient [T/m] 3.5 3.0 2.5

Apparent density [kg/m

] 9600 8200 6900

moiizaka et al. [S50] and Fujita [F21], are summarized in Table 4.8. SmCo

permanent magnets were used to generate the magnetic ﬁeld in the working

space. The height of the ferroﬂuid pool was 120 mm and the angle  between

thepoletipsamountedto30

. Water-based ferroﬂuid of density approximately

1400 kg/m

and magnetic polarization of 0.028 T was used.

Another disadvantage of the permanent magnet-based FHS is that the value

of the magnetic induction and of the ﬁeld gradient cannot be easily varied. The

magnetic induction can be varied only by changing the distance between the

pole tips using a mechanical device that can slide one of the poles against the

other. The ﬁeld gradient can be varied by adjusting the angle =

As a result of the absence of constancy of the apparent density of the fer-

roﬂuid and of any meaningful control of the operating magnetic ﬁeld and its

gradient, the permanent magnet-based wedge-shaped ferrohydrostatic separa-

tors are suitable only for separation of materials that dier in density by at

304 CHAPTER 4. DESIGN OF MAGNETIC SEPARATORS

Figure 4.55: Hyperbolic pole proﬁle with a mirror plate.

least 2000 kg/m

. Recovery of platinum-group metals [S53, S54, G4] and pos-

sibly separation of non-ferrous metals [S50] are applications where the limited

accuracy of such separators can be tolerated. For more accurate separation of

materials with small density dierences, hyperbolic pole tips must be used.

4.8.2 Separators with hyperbolic pole tips

A common objective of sink-and-ﬂoat separation is usually a production of two

density fractions selectively and accurately. In order to achieve such a goal

in ferrohydrostatic separation, it is necessary for the apparent density of the

ferroﬂuid to be constant throughout the volume of the ferroﬂuid. As follows from

eq. (3.145), this can be achieved by ensuring that the ferroﬂuid is magnetically

saturated and by designing the shape of the pole tips in such a way that the

ﬁeld gradient along the vertical axis is constant, namely CE@C| = const. An

additional constraint on the desired ﬁeld pattern is that the horizontal gradients,

i.e. CE@C{ and CE@C}, are as small as possible. Therefore, for ideal density

separation in ferrohydrostatic separators, the following conditions should be

met:

= frqvw 6=0>

=0 (4.61)

Such a pattern of the magnetic ﬁeld can be achieved by a hyperbolic shape of

the pole tips [K18], as shown in Fig. 4.55. In deriving the above conditions (eq.

(4.61)), it was assumed that the branches of hyperbola forming the pole proﬁle

extend to inﬁnity and that their asymptotes ({-axis) are at equal potential

[G14]. The latter condition is satisﬁed in magnetic systems, in which the upper

4.8. DESIGN OF A FERROHYDROSTATIC SEPARATOR 305

Figure 4.56: GMUO ferrohydrostatic separator with hyperbolic pole tips.

half of the hyperbolic surface is replaced by a ferromagnetic plate, coincident

with the { axis, and perpendicular to the | axis, as shown in Fig. 4.55 [K18].

It was shown [K18, G15] that conditions given by eqs. (4.61) cannot be

satisﬁed at all points in the working zone of the magnetic system. These condi-

tions can be met only in the plane of symmetry passing through the centre of

the interpolar gap deﬁned by the | axis in Fig. 4.55. The situation is further

complicated by the fact that, in practice, the poles have ﬁnite dimensions. The

hyperbolic surfaces are, therefore, terminated at some ﬁnite values of {> | and

}. Such a truncation of the ideal hyperbolic shape necessitates corrections to

the shape of the pole tips, in order to meet the conditions expressed by eqs.

(4.61).

The surfaces of the poles are shaped in such a way that they coincide with

one of the elementary solutions of the Laplace equation. A powerful method

for the analytical solution of Laplacian ﬁelds, namely conformal transformation,

was used to calculate the magnetic ﬁeld generated by poles of ﬁnite size and

of hyperbolic proﬁle [G14, G15, G16, S55]. This approach is the basis of the

design of the GMUO ferrohydrostatic separators, as described in Section 2.8 and

306 CHAPTER 4. DESIGN OF MAGNETIC SEPARATORS

Figure 4.57: The gradient of the magnetic ﬁeld along the vertical axis of the

GMUO FHS-40M separator, as a function of distance from the

base of the magnet pole tips.

shown schematically in Fig. 4.56. Similar design of the pole tips was used by the

US Bureau of Mines [R9, K17] in their equipment for separation of non-ferrous

metals.

The distribution of the magnetic ﬁeld gradient along the vertical | axis in

the GMUO FHS-40M separator is shown in Fig. 4.57. It can be seen that in

a 40 mm long region along the vertical the gradient is constant. The apparent

density of the ferroﬂuid is, therefore, also constant and selective and accurate

density separation is possible in that region.

It is found, using eq. (3.145), that a ﬁeld gradient of 2.5 T/m generates an

apparent density of approximately 4200 kg/m

, using, for example, a kerosene-

based ferroﬂuid with magnetic polarization of 0.02 T. The apparent density can

be varied by varying the electrical current in the coils and thus varying the

magnetic ﬁeld in the working gap.

The availability of computer electromagnetic modelling software has changed

the speed and accuracy of the design of suitable proﬁles of the pole tips con-

siderably. Although the initial proposal and design of the proﬁles still has to

be worked out from ﬁrst principles of the magnet design, computer-based ﬁeld

analysis can be used to verify and ﬁne-tune the design. Computer modelling,

therefore, allows the pole proﬁles to be optimized so that the required density

range and its accuracy are obtained. An example of the optimized hyperbolic

proﬁle of pole tips of FHS is shown in Fig. 4.58 and a general view of a ferro-

hydrostatic separator with hyperbolic pole tips is shown in Fig. 4.59.

4.8. DESIGN OF A FERROHYDROSTATIC SEPARATOR 307

Figure 4.58: Hyperbolic pole tips in FHS, as obtained by computer modelling.

Figure 4.59: Ferrohydrostatic separator with hyperbolic pole tips.

308 CHAPTER 4. DESIGN OF MAGNETIC SEPARATORS

4.8.3 Calculation of proﬁles of pole tips in FHS

Although computer modelling facilitates optimization and ﬁne-tuning of the pole

proﬁles, the ﬁrst step in the design of a magnetic system of a ferrohydrostatic

separator is, as a rule, the calculation of the ﬁeld and of the pole proﬁle from

ﬁrst principles. The magnetic ﬁeld between the poles of the magnet can be

represented by a scalar magnetic potential X :

K = uX (4.62)

Since the electrical currents are absent in this system, the magnetic potential

satisﬁes the Laplace equation

X = X =0 (4.63)

which can be written in polar coordinates

+ u

C

=0 (4.64)

A possible solution of the Laplace equation is the potential function

X(u> )=U(u)V() (4.65)

where U is a function of u only and V is a function of  only. It can be shown

that these functions have the following form [B25]:

U = fu

+ gu

p

(4.66)

and

V = j cos p + k sin p (4.67)

where f> g> j and k are constants.

A particular solution to the Laplace equation can, therefore, be written in

the form of a circular harmonic of order p as

X(u> )=(fu

+ gu

p

)(j cos p + k sin p) (4.68)

Of practical interest are, for example, solutions of the form [G17, Z7]:

= D

 (4.69)

and

= D

sin p (4.70)

If the surfaces of the poles of the magnet are made to coincide with a given

pair of equipotential surfaces (the proﬁle of which is independent of the third

dimension), the ﬁeld between them and the proﬁle of the pole tips are described

by eqs. (4.69) and (4.70).

4.8. DESIGN OF A FERROHYDROSTATIC SEPARATOR 309

Table 4.9: Properties of pole proﬁles.

Value of m U

/A Comment

2 u

sin 2 = frqvw= constant gradient

3/2 u

3@2

sin

 = frqvw= isodynamic ﬁeld, I = frqvw=

 = frqvw= wedge conﬁguration

Taking into account that K = uX , the modulus of the magnetic ﬁeld

intensity is

|K| = pDu

p1

(4.71)

The gradient of the magnetic ﬁeld along the vertical (| axis) then has the

form

=(p  1)pDu

p2

sin p (4.72)

while the product of the magnetic ﬁeld and its gradient (force index) are ex-

pressed as

=(p  1)p

2p3

sin p (4.73)

These equations can be used to evaluate and design various shapes of the

pole proﬁles. For p =1,KCK@C|, or the magnetic force, is zero. The sim-

plest practical case of a wedge pattern is expressed by eq. (4.69). The wedge

conﬁguration is represented by a straight line described by the equation  =

const=

The isodynamic proﬁle, in which KCK@C| = const=> can be obtained from

eq. (4.73) when p = 3/2. The isodynamic proﬁle of the pole tips is then given

= Du

3@2

sin

 = frqvw= (4.74)

and

= D

sin

 (4.75)

Pole proﬁles that generate a magnetic ﬁeld whose gradient in constant along

the vertical axis can be obtained from eq. (4.72) when p = 2andtheequations

of the proﬁle are as follows:

= Du

sin 2 = frqvw= (4.76)

and

=2D sin 2 (4.77)

Equations (4.74) to (4.77), in which p determines the geometrical features of

the ﬁeld and D describes the magnetic characteristics of the system, provide the

full description of the proﬁle of the pole tips. Properties of these conﬁgurations

are summarized in Table. 4.9.