Jan Svoboda. Magnetic Techniques for the Treatment of Materials

Подождите немного. Документ загружается.

260 CHAPTER 4. DESIGN OF MAGNETIC SEPARATORS

Permanent

magnet

Mild steel

oke

Figure 4.6: A permanent magnet circuit for the calculation of the leakage coef-

ﬁcient.

note that a thin disc magnetized through its thickness produces no magnetic

ﬂux density although the magnetic ﬁeld strength K at point C is equal to M@

Calculation of the magnetic ﬁeld around a permanent magnet can be per-

formed either using the pole concept or the surface current models. In the

former approach a magnet is treated as having a uniform distribution of poles

on the ends of the magnet. Using this assumption, magnetic induction on the

} axis of a circular inﬁnitely long magnet, as a function of the distance from the

B C D B C D

A E A

Figure 4.7: Distribution of magnetic induction inside and outside magnets of

various shapes.

4.2. DESIGN OF CIRCUITS WITH PERMANENT MAGNETS 261

Table 4.2: Distribution of magnetic induction B inside and outside magnets of

various shapes [H18].

Point Bar Sphere Disc

A J/2 2J/3 0

B J/2 2J/3

C J 2J/3 0

D J/2 2J/3

E J/2 2J/3 0

(a)

(b)

Figure 4.8: Calculation of magnetic induction on the axis of a cylindrical mag-

net.

magnet (Fig. 4.8), is given by

}

1 

}@U

}

1@2

(4.26)

where U is the radius of the magnetic cylinder. By symmetry, E has no com-

ponents perpendicular to }= It can be seen that on the surface of the magnet

(} =0)the magnetic induction E = M@2, as shown in Table 4.2. Numerical

computations are required to determine the magnetic induction o the axis.

Expressions similar eq. (4.26) can also be obtained for the components of mag-

netic induction generated by rectangular surfaces [M19].

A uniform solenoid of rectangular or circular cross-section with surface

current density m

(A/m) is equivalent, for the purpose of calculating magnetic

induction, to a bar with uniform magnetization P = m

= M@

. If a solenoid

has a circular cross-section, magnetic induction E> at point S on its axis, is

262 CHAPTER 4. DESIGN OF MAGNETIC SEPARATORS

Figure 4.9: Deﬁnition of geometrical parameters of a cylindrical and a rectan-

gular permanent magnets for calculation of the magnetic ﬁeld.

given by

E = 

(cos 

 cos 

) (4.27)

It can be seen that on the surface of an inﬁnitely long solenoid (

=0,

) E = 

P@2, as could have been expected.

Selvaggi and Selvaggi [S48] developed mathematical equations that allow

the external and internal magnetic ﬁelds of cylindrical and rectangular perma-

nent magnets of ﬁnite dimensions to be determined. These magnets are shown

schematically in Fig. 4.9. The authors assumed that the surface charge density

at the poles is equal to M

and that only two parallel faces contribute to the

ﬁeld.

The equation for the external ﬁeld at distance } from the face of a circular

magnet of radius U and length O is

h{w

(

} + O

+(} + O)

]

1@2



}

+ }

]

1@2

)

(4.28)

The equation of the internal ﬁeld, at the centre of the magnet is

lqw

= {

)

} (4.29)

Clearly, for O $4, E

lqw

$ M

It can be seen that both terms in parentheses in eqs. (4.28) and (4.29) are

dimensionless and thus any system of units can be used.

4.2. DESIGN OF CIRCUITS WITH PERMANENT MAGNETS 263

Similar expressions were derived for a rectangular permanent magnet of ﬁnite

dimension, as deﬁned in Fig. 4.9. The equation for the external ﬁeld is

h{w



;

arcsin

(

+ }

)(

+ }

)

1@2



arcsin

(

+(} + O)

)(

+(} + O)

)

1@2

(4.30)

while the expression for the internal ﬁeld is given by

lqw



;

  2 arccos

4[(

)(

)]

1@2

(4.31)

Considerably more complicated expressions for the o-centre magnetic ﬁeld can

also be found in [S48]. Good agreement between the measured values and those

determined from eqs. (4.28) to (4.31) was found.

4.2.6 Magnetic forces

Permanent magnet circuits can be designed either for holding magnetic bodies

or for attracting them. The distinction is that the holding magnet has to exert

a magnetic force on a material with which it makes a good contact, while the

attracting magnet has to exert a force on a magnetic body at a considerable

distance. The distinction is important, because the optimum design for the two

purposes diers considerably.

While the attraction force is essential in many forms of magnetic separators,

such as suspended magnets or wet low-intensity drum magnetic separators, in

other types of separators it is the holding force that plays a decisive role. The

correct compromise between holding and attracting particles from a distance

must be achieved.

A force between two magnetic charges T

and T

separated by a distance u

is given by Coulomb’s magnetostatic law:



I =

4

u

(4.32)

where u

is the unit vector in the direction of u= For a thin cylindrical magnet

T = MD where M is the magnetic polarization and D is the cross-sectional area

of the magnet. We can then write, for semi-ﬁnite magnetic bars of the same

cross-sectional area D



I =

4

u

(4.33)

264 CHAPTER 4. DESIGN OF MAGNETIC SEPARATORS

where M

and M

are magnetic polarizations of bars d and e, respectively. When

these bars are in contact, eq. (4.33) reduces to:

I =

2

D (4.34)

As an example, magnetic polarization of 1 T produces a pull of 40 N/cm

,or

approximately 4 kg/cm

, while 0.2 T creates a pull of a mere 0.16 kg/cm

Equation (4.34) is often simpliﬁed even further, under the assumption M

= E> into the form

I =

2

D (4.35)

which is incorrect in many instances [M19]. For instance, if M

=0, then cor-

rectly I =0> according to eq. (4.34). The simpliﬁed formula would incorrectly

yield I = M

D@8

. Generally, eq. (4.35) is valid only when bars d and e are

of the same material and when there is no source of external magnetic ﬁeld.

It is obvious from eqs. (4.34) and (4.35) that the holding force is proportional

to the square of ﬂux density and only to the ﬁrst power of the magnet area. By

using tapered soft-iron pole-tips it is possible to concentrate the ﬂux to a higher

density than the magnets themselves can provide. Although the cross-sectional

area of the magnet would decrease somewhat, the holding force will increase

nevertheless.

Although by increasing the magnetic ﬂux it is possible to increase the holding

force for zero air gap, the reach of the magnetic force would decrease once the

air gap is introduced. The force of attraction would thus decrease. Therefore,

in those magnetic separators in which a pull at a distance (traction force) is

important, wide pole-pieces working at a low induction and allowing high force

penetration into the air gap, are required. On the other hand, closely spaced

poles that produce high holding force are incorporated into those magnetic sep-

arators that rely on the holding force rather than traction force. Dry drum and

roll magnetic separators are an example of such machines.

4.2.7 Soft iron components of the magnetic circuits

Pole-pieces of a solid or laminated soft iron or mild steel , or occasionally cobalt-

iron, are an essential part of magnet design. These ﬂux-conducting members

are used to complete a return path for the ﬂux and to concentrate or change

the ﬂux density in a circuit. Cost is usually an important factor in the choice

and majority of magnetic circuits use ordinary cold-rolled steel for these com-

ponents. Cobalt-iron alloy is used in some high-performance circuits. It is a

good practice to operate these materials near their maximum permeability, as

shown in Fig. 1.9. This will ensure the maximum ﬂux-carrying capacity of the

circuit. Magnetization curves for some high-permeability materials is shown in

Fig. 4.10.

Relative position of permanent magnets and mild steel components is also

of signiﬁcant importance. Figure 4.11 illustrates three possible arrangements

4.2. DESIGN OF CIRCUITS WITH PERMANENT MAGNETS 265

Figure 4.10: Magnetization curves for several high-permeability materials.

(a)

(b)

(

)

Three positions of permanent magnets in a circuit

Figure 4.11: Three arrangements of a circuit with permanent magnets.

266 CHAPTER 4. DESIGN OF MAGNETIC SEPARATORS

Table 4.3: Reversible temperature coe!cients of permanent magnetic materials.

Material (B

) [%/

C] (H

) [%/

Ferrite -0.20 +0.40

SmCo

-0.04 -0.30

NdFeB -0.12 -0.60

of a circuit with permanent magnets. Arrangement (a) is the least e!cient

as the leakage between the circuit elements is very high. On the other hand,

arrangement (c) is the most e!cient as a result of low leakage. Analysis of these

circuits with an Alnico 5 magnet showed that the ratio of the gap ﬂux to the

total ﬂux in the arrangement (c) is three times higher than in arrangement (a)

[P12]. It is clear that permanent magnets should be placed as close to the air

gap as possible. A valuable advantage can be obtained by placing a magnet,

magnetized through its thickness, on a ferromagnetic plate. Then the magnet

behaves like one of twice its thickness.

4.2.8 Stability of permanent magnets

Permanent magnets are expected to maintain a constant ﬂux output over a

long period of time, although any magnet, after being fully magnetized, is likely

to lose strength slowly with time. In nearly all applications the magnet is

subject to inﬂuences that tend to alter the magnet ﬂux. However, in contrast

to early permanent magnets, the modern materials exhibit remarkable stability,

sometimes as high as 10

6

Reversible changes

The magnetization changes fall into two classes, namely reversible changes and

irreversible changes. The reversible changes in the magnetic properties, as

a function of temperature, originate in the change in spontaneous magneti-

zation. Reversible changes are functions of temperature and circuit loading

and are time-independent. They disappear completely without need for re-

magnetization when the permanent magnet is returned to its initial tempera-

ture. Figure 4.12 illustrates the dependence of the remanent magnetization on

temperature for three types of permanent magnet materials.

The reversible changes resulting from temperature changes can be repre-

sented by temperature coe!cients, expressed as a percentage change in M or E

per

C. A summary of the temperature coe!cients  of E

and K

for sev-

eral permanent magnet materials is given in Table 4.3. In addition, Table 1.18

summarizes the Curie temperatures and maximum operating temperatures of

various permanent magnet materials.

In many instances the temperature coe!cients of magnetization and coer-

cive force do not give enough information about how a magnet will respond to

temperature changes. Changes in the shape of a demagnetization curve and the

4.2. DESIGN OF CIRCUITS WITH PERMANENT MAGNETS 267

Figure 4.12: Temperature dependence of remanent magnetization of various per-

manent magnetic materials (adapted from [P4]).

position of the load line and the working point can only be seen from a complete

set of demagnetization curves, measured at dierent temperatures. Figures 4.13

and 4.14 illustrate families of demagnetization curves at various temperatures

for ferrite and NdFeB permanent magnets. Such curves are useful for graphical

design of magnets operating under various temperature conditions.

Complete sets of such demagnetization curves are available either in the

literature (e.g. [P4]) or the most recent data can be obtained directly from

magnet manufacturers.

Irreversible changes

In the case of irreversible changes, after the removal of the disturbing inﬂuence,

the magnetization does not return to its original value. In certain types of

changes however, the magnetization can be fully restored by re-magnetization.

Such changes are, for instance, ambient temperature changes and exposure to

an external magnetic ﬁeld. Upon exposure to an external ﬁeld, the operating

point moves down the demagnetization curve and the intrinsic magnetization

of the magnet shows a corresponding change. When the external ﬁeld is re-

moved, the magnetization recovers only partially. This loss is recoverable by

re-magnetization. The loss will be greater when a magnet is operating in the

steepest part of the demagnetization curve.

Although mechanical shock and vibration contribute energy to a permanent

magnet, changing its magnetization, modern materials with high coercive forces

seem to be resistant to changes by mechanical eects. At the same time, how-

ever, ﬁeld exposure is not able to completely eliminate changes that might have

occurred.

268 CHAPTER 4. DESIGN OF MAGNETIC SEPARATORS

Figure 4.13: Demagnetization curves of a ferrite magnet at dierent tempera-

tures.

Figure 4.14: Demagnetization curves of an NdFeB magnet at various tempera-

tures.

4.2. DESIGN OF CIRCUITS WITH PERMANENT MAGNETS 269

Table 4.4: Magnetic induction required to magnetize permanent magnet mate-

rials.

Material B

vdw

[T]

Alnico 5 0.3

Alnico 8 0.8

Ferrite 1.0

RECo

2.0 to 3.0

NdFeB 2.5 to 3.0

While Alnico and ferrite magnets are stable and not subject to corrosion,

rare-earth magnets have oxidation problems that limit their maximum operating

temperature. If no coating is provided for protection, oxygen diuses into such a

magnet, causing a metallurgical change in a surface layer. The surface layer will

have a lower K

, which will allows this region of the magnet to be demagnetized

more easily. Surface oxidation thus leads irreversibly to a reduction in the

magnetic properties.

Irreversible changes, resulting from permanent changes in the structural or

metallurgical state, are usually time-dependent and re-magnetization does not

restore the original state of magnetization. The temperature at which changes

in properties ﬁrst become noticeable corresponds to the maximum operating

temperature. Temperature-induced irreversible losses in permanent magnets

are shown in Table 1.17.

4.2.9 Magnetization of permanent magnets

It is generally accepted that to magnetize a permanent magnet to saturation,

an external magnetic ﬁeld three to ﬁve times its intrinsic coercivity K

must be

applied. Typical values of the magnetic induction required to magnetize various

permanent magnet materials are summarized in Table 4.4.

Simple bar or ring-shaped magnets may be magnetized by a larger perma-

nent magnet, a solenoid or an electromagnet. To magnetize rare-earth magnets,

very high ﬁelds are required and magnetization is often performed with small

coils or solenoids energized by large current pulses of short duration. Large mag-

nets cannot be magnetized by a pulse of short duration, because eddy currents

prevent the ﬂux reaching the centre of the magnet. In general, the frequency i

of the pulse must be chosen so that the magnetizing pulse lasts longer than the

eddy current. The depth of penetration, or the skin depth ,isgivenby

 =

(



i

1@2

(4.36)

where ( is the resistivity of the material.

Resistivity plays a very important role since most permanent magnets have

very low permeabilities . Ceramic ferrite magnets have high resistivity (A

4

m and full penetration of a pulse is easy. On the other hand, Alnico and