J.
I
Phys. D
Appl.
Phys. 24 (1991) 658-1563, Printed in the UK
The induction melting of glass
Robert Ducharmet, Fred Scarfez, Phiroze Kapadiat and
John Dowdensll
t Department of
Physics, University of Essex, Colchester CO4 3SQ UK
$ Electroglass Ltd, 4 Brunel Road, Benfleet, Essex SS7 4PS, U K
5 Department of Mathematics. University of Essex, Colchester CO4 3SQ UK
Received 21 May 1990, in final form 26 October 1990
Abstract. The possibility of using a coupled system of an induction coil and a
metal susceptor for the purposes of melting glass is investigated. The magnetic
field strength and the current density in the different regions of the furnace are
calculated. Detailed consideration is then given to the influence of the glass on the
electrical Characteristics of the furnace and to the problem of magnetic forces on
the susceotor
1. Introduction
The ohmic heating of glass furnaces by the passage of
an electric current between carefully placed electrodes
in the melt is a well established and growing technology
in the glass industry (Scarfe 1980). Electroheat can also
he generated w i n g an induction mi! and this is a we!!
established method used by other industries. This
paper examines the possibility that if the large metal
susceptor is incorporated into the design of a glass
furnace, then an alternating magnetic field may be used
to raise its temperature and so melt t h e charge.
One important advantage of using a susceptor
coupled to an induction coil to melt glass, is that the
furnace can be started up from cold. This method is
therefore tailor made for applications where furnaces
are not run on a continuous basis. Other advantages
of electric over fuel-fired furnaces are that they are
easily sealed to prevent the leakage of volatile
emissions from the melt into the environment, are relatively maintenance free, and offer more control over
the convection currents in the melt.
A general mathematical formulation of the induction problem for a cylindrical tube in an axial magnetic
field has been given by McLachlan and Meyers (1935).
Their analysis has been developed further for particular
application to glass furnaces. An additional problem
considered here is the presence of a glass layer, which
serves to protect the outer surface of the susceptor
from oxidation. This layer is electrically conducting and
the eddy currents generated inside it must therefore be
taken into account. Maxwell’s equations are taken as
the starting point and it is assumed that the thickness
of the susceptor is at least several times larger than the
depth of penetration of the coil‘s magnetic field into
the susceptor. The magnetic field, the current density
1 To whom all correspondence should be addressed.
0022.3727191 1050658 + 06 503.50 0 1991 IOP
Publishing Ltd
and the power dissipation in the susceptor are calculated. Consideration is then given to the electrical
characteristics of the furnace and to the problem of
’
magnetic forces on the susceptor.
The problem of the eddy current heating of a composite cylindrical system has arisen before in connection with the manufacture of thermionic valves and
has received consideration from Wright (1Y37). There
are, however, two important differences between that
problem and the one treated here. Firstly, a complex
time dependence is introduced, and this results in considerable simplification of the analysis. Secondly, the
glass laycr in an induction furnace is relatively much
thicker than the outer envelope in a thermionic valve,
and the ‘thin cylinder’ approximation used by Wright
cannot, therefore, be invoked.
2. Mathematical formulation of !he preblen!
Suppose a molybdenum susceptor, in the shape of a
uniform coaxial cylindrical shell heated by an induction
coil, is to be used for the purposes of melting glass in
a cylindrical crucible. This is shown schematically in
figure 1. The electric and magnetic fields generated
inside the furnace by the coil are described by
Maxwell’s equations:
V 2 H = up,(aH/al)
where the electric field strength (in Vm-’) will be
denoted by E , H is the magnetic field strength (A m-I),
J the current density (Am-*), pn the permeability of
free space ( = 4n x lo-’ H m-’), U the electrical conductivity (Q-I m-l) and f the time ( s ) .
The induction melting of glass
J , and K O are complex and it is therefore useful to
represent them in the form
Ju(kri312)= F,(kr) = ber(kr) ibei(kr)
+
and
+
I
Ko(kri’lZ) = F2(kr) = ker(kr) i kei(kr).
The first derivatives of these functions can similarly be
expressed as
i3/*J;(kri3/*) = F ; (kr) = ber‘(kr)+i bei’(kr)
and
i’/2Kt)(kri”2)= F ; ( k r ) = ker’(kr)+ikei’(kr)
. .
.
CloY sectlo“
figure 1. Cross section of an induction-heated glass
furnace.
Equation (1) can also be written as
VZH= iwup,H
when a separable solution of the form
H = ffs(r, 8, z ) exp(iwf)
,
,
.
---
is assumed; (r, 0, z ) are cylindrical polar coordinates
and w denotes the angular frequency of the coil.
If the ratio of the length of the crucible wall to the
inner radius c is a large number (of the order of IO),
then a long cylinder approximation can be made and
it may be assumed that the magnetic field lines run
parallel to the axis. Under these conditions equation
(3) can be expressed in the form
d2H
dr2
1dH .
+ -=I U ~ ~ W H
r dr
-
(4)
This is a special form of Bessel’s equation and its general solution is
H = AJ,(kri’i2)
+ BKo(kri’/2)
The theory of Bessel functions has been worked out in
considerable detail (Watson 1966) and their values are
therefore readily accessible from standard tables.
In order to solve equation (4) for a composite cylindrical system consisting of N layers, a solution of the
form of expression ( 5 ) must in general be written for
each layer. The 2N arbitrary coefficients can then be
evaluated from the conditions that the electric and
magnetic fields are continuous across the N boundaries
(McLachlan 1935, Wright 1937). If, however, the thickness b - a of the susceptor is large compared with the
skin depth of molybdenum (as must be the case to
ensure the best possible coupling between the coil and
the susceptor) then the magnetic field strength in the
region r < a will be negligible. The conduction currents
in the glass inside the susceptor can therefore he
ignored.
The approximations
dZH H,
I d H Hb
and
d r 2 ?i2
r dr
6b
(5)
where J, and KOdenote Bessel functions of the first and
second kind (McLachlan 1961, ch.7), the parameter k
is defined by k = g2/2/6and
where Hb= H ( b ) , show that for large values of the
parameter b / 6 , the contribution of the middle term in
expression (4) can be neglected. It is shown below that
this condition is generally satisfied in the susceptor, for
all coil frequencies of practical interest. Use of the
coordinate transformation x = b - r thus results in the
simplified equation
describing the magnetic field in the susceptor. This has
the general solution
H = Cexp[-(I-i)x/6]+Dexp[(l-i)x/6]
(7)
where C and D are arbitrary complex coefficients.
However, the term exp[(l - i)+/6] does not attenuate
with depth. Therefore, since it has already been argued
that H is negligible where r a , it is clear that the
coefficient D must be set equal to zero.
The foregoing considerations suggest the following
solution to expression (4) for the induction furnace
H=
is called the penetration or skin depth. The values of
----
{
HO
r,>r>c
A F , ( k r )+ B F 2 ( k r )
car> b
(8)
r< b
C exp[ - (1 - i)x/S,]
where 6, denotes the skin depth in the susceptor.
659
R Ducharme et a/
In order to complete the solution to Maxwell's
equations for this problem, the current density and the
electric field in both the glass layer and the susceptor
must now be calculated. The only non-zero component
of the current density J runs parallel to the 0 direction.
Equation (2) can therefore be expressed in the much
simplified form
dH
dr
J = --
(9)
On substituting expression (6) into this equation and
recalling x = 6 - r, it can be seen that
J = - rk&W+Bk&i(kr)
czr>b
C[(1- i)/~3~]exp[
-(1 -i)x/6$]
IC'
rsb
where the subscript G denotes the glass layer. The
corresponding electric field can be obtained by a simple
application of Ohm's law and this gives
.=-[
[AF;(kr) +BFi(kr)]kG/oG
czr>b
[C(1-i)/oS6,]exp[ - (1 -i)x/aS]
r66.
(10)
The values of the unknown coefficients A, B and C
may now be calculated from the boundary conditions
on the electric and magnetic fields at the surfaces r =
b and r = c. Clearly three conditions are required and
ihcse may 'De cxprcssed as Ioliows.
(i) The magnetic field H i s continuous at the boundary r = c
H,
= AF,(kc)
+ BF,(kc).
(ii) The magnetic field H is continuous at the
boundary r = 6
C = AFl(k6)
+ BF,(kb).
(11)
(iii) The electric field E is continuous a t the boundary r = 6
C5-I = 4 F ; ( k b )
+ BF; ( k b )
where
5 = (1 + i)(o,/20c)"Z.
(12)
The solution to this set of equations for the coefficients
A and B is given by
and
'g
05
k
0
660
O
lcml
b
6
lrml
Figure 2. The magnetic and electric fields (in
dimensionless units) as a function of distance from the
boundafy of the glass layer for different values of the
phase: (a) w t = 0; (b) ut = in; (c) wf = in.
with time. These graphs are based on the following
values for each of the relevant furnace parameters.
6=0.35m
~G =
c=0.4m
w=2nx106s-'
,
i(jc2-:m-:
a,=i.5xi06R ' m
,
L .
A coil frequency of 1 MHz has been selected in order
to ensure that the variations in E and H are as marked
as possible. The graph illustrates that even with this
high coil frequency value, the variation in the coil's
magnetic field across the glass layer is quite small compared with its maximum value H,,, ( A H / & < 0.15).
This result follows as a consequence of the fact that
the thickness of the glass layer is several times smaller
than the penetration depth of the coil's magnetic field
in the glass. It is clear from the second set of curves,
however, that a large variation in the electric field does
occur. Indeed. the value of E can be up to eighty times
larger in the glass than its maximum value in the metal.
This indicates the existence of significant eddy currents
in the melt and shows why the conductivity of the glass
cannot be ignored.
3. The magnetic flux inside the furnace
The magnetic flux QF can be calculated from the magnetic field strength H , using the integral expression
Q = p,
The value of the coefficient C may thus be obtained
by substituting these results into expression (11).
Figure 2 shows the development of the real parts
of the electric and magnetic fields in the glass layer
''Oh
0.
.
H dS
(13)
where S denotes the surface area penetrated by the
flux. The total flux will be the sum of its components
inside the crucible, the glass layer and in the susceptor.
Given that H = H o across the air gap separating the
coil from the crucible and inside the crucible itself,
The induction melting of glass
expression (13) can be evaluated throughout this region
and gives
(PA = n(rz
~
- c2)pnHn
where z ( r : - 6’) is the area of the air gap and the
crucible perpendicular to the H field. The flux QG in
the glass can be found by substituting expression (8)
into (13) and integrating between the limits r = b and
r = c . Thus
QG= 2 n p n
lbc
[AF,(kr)+BF,(kr)]rdr.
This expression is then evaluated to show that
Im operators take the real and imaginary parts of whatever follows. Thus, on substituting expression (14) into
equations (15) and (16) the following set of results is
obtained.
(i) The effective resistance of the glass layer
where
G = ( A / H o ) [ c F (; k c ) - bF; ( k b ) ]
+ ( B / H o ) [ c F i ( k c )- b F i ( k b ) ] .
(ii) The effective resistance of the susceptor
A similar procedure shows that the flux in the susceptor
is
n p n w N 2 ( d- c’)
I
(iv) The inductive reactance of the glass layer
XA
where the integration in this case was between the
limits r = b and r = a , and it has been assumed that
H ( a ) 0.
[(L)
2npowN2b
Re
(1 + i)]
kJd2
Hn
(iii) The combined inductive reactance of the air
gap and the crucible
R, =
+B[cF;(kc)-bF;(kb)]).
(17)
=
-
4. The electrical characteristics of the furnace
The magnetic field H o induced inside the walls of a
glass furnace by a coil of length I, having N turns and
carrying a current I, is given by the expression
Ho = N I $ .
This equation is not exact, but does follow as an automatic consequence of the long cylinder approximation
discussed earlier. The coil current I, is alternating and
therefore the magnetic flux through the furnace
QF = @ A + Q G + Q s
(14)
(v) The inductive reactance of the susceptor
The back EMF generated by the furnace does not
equal the total voltage drop across the coil, because
there is t h e coil’s own resistance and inductive reactance to be taken into account. Coils in this application
usually consist of long lengths of copper tubing wound
into the form of cylinders. It is found in practice that
the resistance and inductive reactance of a coil for a
given frequency of alternating current can be calculated
approximately from the formulae
is also time dependent. Subsequently a back EMF - V,
is generated across the coil and this is related to the
rate of change of flux linkage by Faraday’s law
aQF
V, = N - .
at
On multiplying the right-hand side of this expression
by the normalizing factor (NIc/lHo) and making the
standard replacement aiai-. iw, the transformed
equation can then be expressed as
(see Paschkis and Persson 1960, p 487). The formulae
have the same mathematical form as the equivalent
expressions for the resistance and inductive reactance
of the susceptor (see equations (17) and (U)),
although one difference is that a correction factor
n(=1.1) has been included in equation (19) to take
account of the spaces between the windings.
VB = (RE + ~ X E ) I ,
where
5. The power dissipation
is the effective resistance of the glass layer and the
susceptor referred to the coil, and
The instantaneous power dissipation in the susceptor
can be evaluated by integrating the Poynting vector
over the susceptor’s outer surface S.This integral may
be expressed as
P=
is the inductive reactance of the furnace. The Re and
I,
[(Re E)
X
.
(Re H ) ] d S
(20)
where it is necessary to take real parts because two
661
R Ducharme et a1
Table 1. The real and imaginary parts of C/Hofor several differentfrequencies.The assumed values for
the parameters b, c, U,, uo are those quoted for figure2.
~
Frequency (MHz)
0.1
0.5
0.25
1 .o
0.75
~~
Re(C/Ho)
Im(C/Ho)
0.999
0.999
-1.92 x 10-2
-4.66 x
0.998
-4.26
X
0.995
-7.57 x 10-2
10.'
0.988
-0.117
the power dissipation in the glass substantially exceeds
that in the susceptor. It is notable that this effect would
be even more pronounced if the glass layer were
thicker or characterized by a higher conductivity. The
power dissipation in the glass is important because
it influences both the electrical characteristics of the
furnace and the convection currents in the melt.
6. The force on the susceptor
Cal frequency IMHzl
A current element J d r flowing in a magnetic field H
will experience a force dF given by the Lorentz force
law as
Figure 3. The power ratio R,/R. as a function of
frequency.
.
L'LI!CIK!S~
,I
ce-pirx
....
quanr!i!rs
I
o r c e ~ rmixed (!errain
1970, p 665). Given that E and H are constant over
the surface r = b, expression (20) is easily integrated
to give
P = Re H(b) Re E(b)2nbl.
(21)
The values of R e H(b) and R e E(b) may always be
calculated directly from expressions (8) and (10). If,
however, the penetration depth of the glass is at least
four or five times larger than (c - b), it is reasonable
to make the approximation C/H,- 1 (see table 1).
Thus, recalling that Ho varies as exp iwr, it can be seen
that
Re H ( b j = H , cos mr.
A similar calculation based on expression (10) shows
that
Re E(b) = ( H m k , / o , ~ 2 ) ( c o wr-sin
s
or)
..
where ni :s
d F = [(ReJ) x (ReH)]p,, d r
_.
differenri-i vciuz-e &:e=:.
:.?e h c e
on the susceptor for the problem depicted in figure 1
is directed radially inwards towards the cylinder axis.
The magnitude of this force can be found by integrating
over the volume of the susceptor. Consequently
B
b
F=npol/
d
(-(ReH)')rdr
dr
(23)
where equation (9) has been used to eliminate J . The
penetration depth in molybdenum for all coil frequencies greater than 5 kHz can be shown to be less
than 4 mm. This implies 6 Q b and, on the strength of
this result, it would seem a good approximation to set
r = b in expression (23). Evaluation of this integral,
subject to the approximation C/H, = 1, then gives
F = n b l p u H i cos2 mt
(24)
where in accordance with expression (22)
Integration of expression (21) over one complete time
cycle shows that the time-averaged power dissipation
(P)has the value
The ratio of the power dissipation in the glass layer
IZRG to the power dissipation in the susceptor IZR,
may be expressed as RG/R,. A graph of this power
ratio as a function of coil frequency is shown in figure 3.
This curve is based on the same characteristic furnace
parameters as were used to obtain figure 2 . The striking features of the graph is that at high frequencies,
662
It is known from experience in working with rod
shaped electrodes of the type used in conventional
electric glass furnaces, that molybdenum is an
extremely brittle metal and great care must be taken
in its handling. It has happened on occasion, for
example, that a molybdenum rod 1m long and 4 cm in
diameter has fractured inside an electrode holder as a
result of stresses produced only by. its own weight
(120N). Since forces must be scaled in proportion to
cross sectional area for shear stresses to be held
The induction melting Of glass
./loo
‘YO0 kW
I
Fwquerry IkHzl
Figure 4. Dependence of the magnetic force (F,)
experienced by the susceptor on frequency.
constant, it is therefore to he expected that a molybdenum susceptor, with a cross sectional area fifty times
greater than that of a rod shaped electrode, is at some
risk from forces of the order of, or exceeding, 6000 N.It
should he noted that damaging stresses are most likely to
arise in the vicinity of the bolts and brackets which fix the
susceptor in place inside the crucible.
The peak force on the susceptor is given by
and this result follows from expression (24). A plot of F,
against coil frequency f( = w / 2 n ) , for three different
values of (P)is shown in figure 4. It can he seen from the
graph that the magnitude of F,,, can exceed 6000 N at
low coil frequencies. Under these conditions there is
therefore the danger that the time varying forces on the
susceptor will cause it to fatigue and ultimately to crack.
7. Summary
~
An analytical solution to the problem of eddy current
heating of a composite cylindrical system has been provided in the case when the outer cylinder is glass and
the inner cylinder is molybdenum. The metal susceptor
is in fact hollow, but this is not important to the solution since the glass inside is well shielded from the
coil’s magnetic field by the metal. The coil induces
eddy currents in the unshielded glass and in the susceptor. These currents in turn produce fields outside
themselves and therefore react hack on both the outer
cylinder and on the coil. This back induction has been
calculated and used to determine the electrical characteristics of the complete system.
The flow of conduction currents in the glass layer
becomes most significant when the coil frequency is
high. Under these circumstances it has been shown that
the greater proportion of the power dissipation in the
furnace takes place in the glass and not in the susceptor. The problem here is that this only becomes true
when the glass is heated and ceases to he an insulator.
The melting of the glass is therefore accompanied by
a sudden change in the hack potential generated by
the furnace across the coil. Since power supplies are
necessarily limited in the range of operating conditions
to which they can adapt, it is clear that the wrong
choice of equipment may result in a furnace becoming
inefficient or even inoperable at high temperatures.
A susceptor in an induction coil will experience a
force owing to the action of the coil’s magnetic field
on the current flowing inside it. An expression for this
force has been obtained and its magnitude calculated
under a variety of operating conditions. This is an
important consideration because molybdenum, like
most refractory metals, becomes very brittle at the high
temperatures encountered in glass furnaces. There is,
therefore, the danger that the susceptor will sustain
damage if the forces on it are too large. It has been
demonstrated that this is most likely to he a problem
at very low coil frequencies (of the order of magnitude
of 1OOOHz or less), when large, time varying forces
u p to several thousand Newtons could cause fatigue
damage in the vicinity of the bolts and brackets that
fix the susceptor in place inside the crucible.
Acknowledgment
One of us (R Ducharme) is indebted to the Science
and Engineering Research Council for the award of a
Case Studentship.
References
Lorrain P 1970 Electromagnetic Fields and Waues (San
Francisco: Freeman)
McLachlan N W 1961 Bessel Functions for Engineers
(Oxford: Oxford University Press)
McLachlan N W and Meyers A A 1935 Eddy current loss
in a tube in axial alternating magnetic field Phil. Mag.
29 846
Paschkis V and Persson J 1960 Industrial Elecrric Furnaces
and Appliances (New York: Wiley Interscience)
Scarfe F 1980 Electrical melting, a review Glass Technol.
21 (1) 37-50
Watson G N 1966 A Treatire on [he Theory of Bessel
Functions (Cambridge: Cambridge University Press)
Wright D A 1937 Eddy current heating of composite
cylindrical systems Phil. Mag. 24 1
663
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