Jan Svoboda. Magnetic Techniques for the Treatment of Materials

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250 CHAPT ER 3. THEORY OF MAGNETIC SEPARATION

Allen found that the magnitude of the torque acting on a cubic conducting

particle by a rotating ﬁeld is

E

(3.194)

where O is the length of the cube edge, z is the magnetic ﬁeld rotation frequency

and (

is the resistivity of the material. For a sphere of diameter g

, the torque

is given by

E

60(

(3.195)

If such a torque is exerted on a particle by a rotating magnetic ﬁeld, and

the particle is resting on a surface, the particle will be rolled when the torque

becomes greater than the gravitational force holding the particle against the

surface. The gravitational torque preventing a cubic-shaped particle from rolling

can be written:



j (3.196)

Combining eqs. (3.195) and (3.196) an estimate of the minimum size of a cubic

particle that could be rolled at a given magnetic ﬁeld strength is obtained [A21]:

min

4

(

E

(3.197)

Equation (3.197) gives a reasonable estimate of the minimum size of a ”rollable”

cubic particle when the rotating magnetic ﬁeld has its rotation axis at the centre

of the particle. In a more practical situation of a rotating drum eddy-current

separator, the magnetic ﬁeld not only rotates the particle, but it is also moving

through the particle. This moving ﬁeld produces a ﬁeld which rotates in the

opposite direction, at half the ﬁeld rotation frequency [A21]. The minimum

”rollable” cubic particle size, ignoring inter-particle interactions, can, therefore,

be estimated as

min

8

(

E

(3.198)

The rotating frequencies of the magnetic ﬁeld, required to rotate and thus to

separate particles of various metals and various sizes in a magnetic ﬁeld of 0.45

T, are shown in Fig. 3.71 [A21]. It can be seen, for example, that particles of

gold, as small as 60 m, could be separated by rotating them at the frequency of

1000 s

1

= The concept has been demonstrated by separating aluminium particles

from their mixture with sand, with the recovery of 86% and the concentrate

grade of 95 % Al, at the feed rate of 5 t/h/m [A17].

Chapter 4

Elements of Design of

Magnetic Separation

Equipment

4.1 Introduction

The design of a magnetic circuit for a magnetic separator is based on the need

to optimize the magnetic force required to separate e!ciently, and selectively,

particles of dierent magnetic susceptibilities. As transpires from eq. (1.8),

the magnetic force is the product of the magnetic ﬁeld strength and of the

ﬁeld gradient. The desired magnetic ﬁeld strength, the ﬁeld gradient and their

combination are a function of the type of material to be separated, mode of

separation, required metallurgical performance and economic criteria. In this

chapter the basic design criteria and techniques of the generation of the magnetic

ﬁeld and its gradient, as used in magnetic separation, will be reviewed.

Magnetic circuit design has been traditionally di!cult because of the usual

non-analytic nature of the solutions for ﬁelds emanating from arbitrary sources.

Further, unlike currents in electric circuits, magnetic ﬂux is not conﬁned to the

wires but passes through various components of the circuit, including the air.

Over the years, several analytical techniques have been used to give approximate

solutions to magnetic design problems. In the simplest cases, the analytical so-

lution can be exact, but in general for any practical problem, analytical solutions

lead to large errors. Most designs are optimized by measurement and extrapo-

lation, and experience of working with a particular type of a magnetic circuit is

important.

The advent of powerful computational techniques, such as ﬁnite element

analysis, allows us to ﬁnd accurate solutions for two- or three-dimensional ﬁeld

distributions in complex geometries, which in turn may be used to predict de-

vice performance with similar precision. However, these modelling techniques

251

252 CHAPTER 4. DESIGN OF MAGNETIC SEPARATORS

require a detailed deﬁnition of the geometry and boundary conditions to be

solved, which assumes that an initial design already exists. While providing an

accurate ﬁeld solution for a given geometry, these techniques will not optimize

it. Suggestions for changes in dimensions, materials etc. must come from the

designer. Consequently, while computer-based ﬁeld analysis is an eective tool

for simulating a known device, it is too cumbersome for design optimization

[C16].

Field equations

Calculation of the ﬂux in a magnetic circuit is based on two Maxwell’s equations.

The ﬁrst, in its dierential form is:



u ×



K =



m (4.1)

where m is the current density. After integrating over an arbitrary surface V we

obtain the above equation in the integral form:

(



u ×



K) · gd =



m ·gd (4.2)

Then, using Stokes’s theorem, the left-hand side of this equation can be trans-

formed into a line integral around a closed path F, which bounds the surface V,

and



K · g



o =



m · gd = L (4.3)

where L is the total current. This is Ampere’s circuital law: the line integral

of K around a closed path is equal to the total current crossing any surface

bounded by the line integral path. The left-hand side of eq. (4.3) is known as

magnetomotive force or ppi.

The second fundamental relationship required for a magnetic circuit design

expresses the fact that ﬂux must be conserved through any closed volume, or,

that the total ﬂux ﬂowing out of any volume must be zero. Therefore



E · gd =0 or



u ·



E =0 (4.4)

Equations (4.3) and (4.4) are two basic equations used for magnetic circuit

design.

4.2 Design of circuits with permanent magnets

4.2.1 Basic concepts and ﬁgures of merit

The general relationship between magnetic induction and intrinsic material mag-

netization or polarization is given by



E = 

(



K +



P)=



K +



M (4.5)

4.2. DESIGN OF CIRCUITS WITH PERMANENT MAGNETS 253

(BH)

J,B

(BH)

max

(BH)

Figure 4.1: The maximum energy products of permanent magnets.

The relationship between E and K is illustrated by the hysteresis curve shown

in Fig. 1.7. The second (demagnetization) quadrant of such a hysteresis curve is

important for description of the quality of a permanent magnet material. Rela-

tive contributions of M and K greatly inﬂuence the magnetizing, demagnetizing

and stability properties of permanent magnets. For instance, the coercive force

can never exceed the remanent magnetic induction E



 E

(4.6)

while the intrinsic coercive force K

is not under such a restraint.

There is clearly one point between E

and K

for which the product of E

and K has a maximum value. The optimum operation of a magnet corresponds

to conditions under which this EK product is maximum. This maximum value

of the energy product is given by

(EK)

lim

max

4



(4.7)

where M

(or P

) is the saturation polarization (or magnetization) of the magnet

material. It transpires from this equation that, in order to establish the maxi-

mum capability of a permanent magnet, one has to know only the value of the

saturation polarization. The maximum value of the energy product corresponds

to the largest rectangle which can be drawn under the demagnetization curve

corresponding to the magnetic induction, as shown in Fig. 4.1. Another impor-

tant ﬁgure of merit is the intrinsic energy product (EK)

max

.Thisquantityisof

importance when the magnet is subjected to large demagnetizing inﬂuences in

its operating environment. The maximum intrinsic energy product corresponds

to the largest rectangle that can be drawn under the demagnetization curve

254 CHAPTER 4. DESIGN OF MAGNETIC SEPARATORS

Table 4.1: Relative permeabilities of permanent magnet materials.

Material Permeability

Alnico 6 4.2

Ferrimag 5 1.06

NdFeB 1.08 - 1.1

Ferrite

or NdFeB

Alnico

≅

Figure 4.2: Flux leakage in two dierent types of permanent magnets.

corresponding to the material polarization. The situation is depicted again in

Fig. 4.1.

Dierent types of permanent magnets can dier considerably in their mag-

netic behaviour. For instance, the two energy products of an Alnico magnet

are very similar, namely (EK)

max



(EK)

max

, while for ferrite and rare-earth

magnets (EK)

max

AA (EK)

max

= The permeability of Alnico magnets diers

considerably from that of the air, namely 

pdwhuldo

A

dlu

, while for ferrite and

rare earth magnets these permeabilities are similar, so that 

pdwhuldo





dlu

Typical values of magnetic permeabilities are shown in Table 4.1.

As a result of the above dierences, the patterns of the magnetic ﬂux around

a magnet also dier. As is shown in Fig. 4.2, the ﬂux lines in Alnico magnets

leave the magnet limb perpendicularly, since the poles of the magnet are not at

the terminals, but appear to be spaced at about 0.7 of the length of the bar [P4].

On the other hand, in ferrite and rare-earth magnets the ﬂux lines tend to reach

the terminal and the pole spacing is, therefore, the full length of the bar. In a

long bar of Alnico 5, only 45% of the lines in the neutral section of the magnet

reach the end of the bar. Therefore, a high percentage of the total energy is

4.2. DESIGN OF CIRCUITS WITH PERMANENT MAGNETS 255

Iron pole-pieces

Permanent magnet

Figure 4.3: Magnetic circuit with a permanent magnet.

stored in leakage ﬁelds. On the other hand, in a ferrite bar approximately 95%

of the ﬂux at the neutral section reaches the end of the magnet. These magnets,

therefore, store most of their energy within their volume and their useful energy

approaches (EK)

max

[P4].

4.2.2 The static gap design

In a circuit with permanent magnets there are no currents, L = 0, and eq. (4.3)

reduces to



K · g



o =0 (4.8)

In a simple magnetic circuit of Fig. 4.3, with a permanent magnet and an air

gap, under static conditions, in which the dimensions of the air gap are constant,

eq. (4.8) can be re—written

jds



K · g



o +

pdjqhw



K · g



o =0 (4.9)

Therefore, the sum of the magnetomotive force (mmf) in the magnet and the

magnetomotive force in the air gap equals zero:

+ K

=0or K

= K

(4.10)

assuming that the soft-iron parts have inﬁnite magnetic permeability. Also,

because the ﬂux is continuous

= E

(4.11)

256 CHAPTER 4. DESIGN OF MAGNETIC SEPARATORS

In the above equations O

, D

and E

are the length, cross-sectional area and

the magnetic induction of the air gap, respectively, while O

, D

and E

are

the length, cross-sectional area and magnetic induction of the magnet. It can

be seen from eq. (4.9) that the magnetomotive force inside the magnet is in

the opposite direction to the magnetomotive force in the air gap. Permanent

magnets thus operate in the top left-hand quadrant (demagnetization quadrant)

of the hysteresis curve.

By multiplying eq. (4.10) by eq. (4.11) we get

= K

(4.12)

where Y

and Y

are the volumes of the gap and magnet, respectively. This

equation can be re-written as



= K

(4.13)

These equations illustrate that the larger the product K

and/or the vol-

ume of the magnet, the greater the magnetic induction in the air gap. It is

also possible to determine, from the above equation, the volume of the magnet

required to generate a desired ﬂux density in a given air gap:



(4.14)

By combining eqs. (4.12) and (4.13) we can also write

= 

= S

(4.15)

where S

= 

is the so-called permeance of the air gap, while the perme-

ance of the magnet can be deﬁned as S

= 

. Permeance is a measure

of magnetic conductance which, if known, enables an accurate prediction to be

made concerning how much of the total ﬂux of a magnet will reach the air gap.

The reciprocal of permeance is called reluctance,  =1@S.

It can be seen from eq. (4.15) that by increasing the air-gap area and/or

reducing the air gap length the ease with which ﬂux travels across this gap will

increase.

Equation (4.15) can be plotted as a straight line with a negative slope on

a demagnetization curve. The intersection of this so-called load line with the

demagnetization curve represents the operating point of the magnet, as shown

in Fig. 4.4. The slope of the load line is equal to 

= For a

very small air gap, in the limit O

$ 0> the magnet is short-circuited and

@

$4> as shown in Fig. 4.4. In the opposite case, if the soft iron

pole pieces are removed and the magnet stands alone, the air gap becomes large

and the magnet operates in the open circuit regime. The optimum operation of

the magnet is obtained when the working point on the demagnetization curve

corresponds to the maximum value of the product (EK), as illustrated in Fig.

4.1.

4.2. DESIGN OF CIRCUITS WITH PERMANENT MAGNETS 257

Load line

Open circuit

Operating point

Figure 4.4: The load lines of permanent magnets operating in dierent regimes.

4.2.3 Leakage factors

In all practical situations the pole pieces will have ﬁnite magnetic permeability

and even more importantly, there will be a considerable leakage of the magnetic

ﬂux. Allowance for these imperfections can be made by introducing correction

factors N

(leakage coe!cient) and N

(loss factor) [P4]:

Total ﬂux in circuit

Useful ﬂux

Useful ﬂux + Leakage ﬂux

Useful ﬂux

(4.16)

magnet mmf

useful mmf

(4.17)

The equation for the load line then becomes

= 

(4.18)

It is of interest to note that the E

ratio is entirely determined by the

physical relationships between the permanent magnet and the entire magnetic

circuit within which it functions. This ratio is independent of the material from

which the magnet is made.

Although the correction factor N

is rarely smaller than 2 and has no upper

limit, for reasonably e!cient permanent magnets it is in the range from 2 to 3.

The loss factor N

is generally much smaller than N

, usually within the range

1.05 to 1.45 [H18].

258 CHAPTER 4. DESIGN OF MAGNETIC SEPARATORS

Dynamic operation

If the magnetic circuit geometry does not remain unaltered, then the load line

slope in eq. (4.18) will change through any of the parameters O

, D

, N

and N

The operating point of the magnet may undergo an excursion within the short

and open-circuit limits shown in Fig. 4.4. A portion of the ﬂux changes from

the leakage paths to the useful paths and back again, as the useful permeance

is changed. The operation of a lifting magnet is a typical situation in which the

magnetic system undergoes dynamic changes.

4.2.4 Determination of permeance and leakage factors

The primary requirement for magnetic circuit design, and therefore the design of

the basic magnet used in the circuit, is to be able to calculate the relationships

between ﬂux paths. As pointed out in the previous section, the easiest approach

to such calculations is by means of permeances. In general, the permeance of the

ﬂux paths varies from a minimum near the neutral section of the magnet to a

maximum for zones at the end of the magnet or close to the air gap. Relatively

simple formulae for various circuit geometries are available in the literature

[T6, P4, M19, H18, M20]. Adding the gap and leakage permeances using these

relationships is one of the methods of estimating the total permeance.

For instance, permeance of the air gap between rectangular surfaces of width

d and thickness e (Fig. 4.5) is given by

S =



(4.19)

This equation is a special case of a more general expression

S =



(4.20)

which is valid for any shape of the air gap. The leakage permeance of the

geometry shown in Fig. 4.5 is given by an approximate formula [H18]

S =





ln(

+1) (4.21)

In order to demonstrate the di!culties facing a designer, Parker and Studders

[P12] quote, for the same geometry, another formula that leads to values of S

about 20% to 30% higher.

For a bar magnet that operates in the open-circuit mode, the permeance is

the permeance from one limb of the magnet to the other, since there are no pole

pieces or a well-deﬁned air gap. Permeance of such magnets, of various shapes

such as bars, rods and rings, is

S =1=77

D (4.22)

4.2. DESIGN OF CIRCUITS WITH PERMANENT MAGNETS 259

Figure 4.5: Geometry of a magnetic circuit for calculation of the leakage perme-

ance.

where D is the cross-sectional area of the pole. This relationship is applicable

to ferrite and rare-earth magnets, while for Alnico magnets the relation is

S =1=77 × 0=7

D (4.23)

This equation takes into account the fact that for Alnicos the eective spacing

of the poles is only 0.7 of the total length, as shown in Section 4.2.1.

It is clear that the calculation of individual permeances is very laborious and

various empirical formulae have been developed for the determination of the

total leakage. A very simple example of such a formula [H18] is an expression

for the leakage factor N

for a gap of length O

between circular pole-pieces of

diameter g :

=1+

(4.24)

Comprehensive sets of formulae, usually attributed to Tenzer [e.g. T6], are

available in the literature [e.g. H18]. For instance, in a magnetic circuit shown

in Fig. 4.6, in which the permanent magnet material is adjacent to the gap,

leakage from the sides and base, which are at the same potential, can be ignored.

The correction factor N

is then given by [H18, M19]:

=1+

(1=1X

0=67d

0=67d + O

)(1 +

) (4.25)

where X

is the perimeter of a part shown as length d= Similar equations are

available for other geometries [H18, M19].

4.2.5 Magnetic ﬁeld around a permanent magnet

The distribution of the magnetic induction at various points of magnets operat-

ing in the open-circuit regime is shown in Fig. 4.7 and Table 4.2. The long thin

bar is magnetized along its long axis while the thin disc is magnetized along

its short axis. The sphere is magnetized along its diameter. It is of interest to