Optimal electrical parameters

Author : Luis Ricardo Jaccard
Between 1923 and 1975, Andreae, Morkramer, Kelly, Persson and
many others who have studied the operation of submerged arc
furnaces concluded that the values of voltage and current best
suited to the operation of these furnaces depended strongly on
the diameter of the electrodes.
But Westly, the author of factor C3, in 1975, presented a technical
paper on which he showed that had not found relationship
between ideal current and voltage with the diameter of the
The main objectives of this presentation are: to show that the
factor C3 of Westly does not reflect reality, to rescue the
concept of Andreae´s factor k and contribute with a more
simple formula to be applied to find the values of optimal
voltage and current.
Submerged arc furnace
I (A)
V transformer
Ore and coal
cold charge
V = Electrode to hearth voltage
Power and resistance
• The active power for each electrode is: P = V x I
• The resistivity r of the charge depends
especially on the percentage of coal
necessary for each process. The higher the
percentage of coal, the lower the load resistivity
and for a specific position of the electrode,
the lower the electrical resistance of the load.
•The load resistance is proportional to
resistivity and electrode-distance : R  r. H
• The distance H at which the electrode will be positioned will be:
HR/r, but, R = V/I, so to have a predetermined position H of the
electrode it will be needed a certain V/I value, which will depend
on the resistivity r of the charge: V / I H.r. The greater the
resistivity of the material, the higher the V/I to keep certain H.
Electrode optimal position
It can be proved that for each material there is an
electrode position H in which the chemical
reactions are carried out more efficiently. When the
electrode is in that position, the specific power
(kW/ton) transferred to the load, in the reaction
zone, is the ideal. If H is less than ideal, the kW/ton
are higher than necessary, and when H are higher
than ideal, the kW/ton are less than the ideal value.
In both cases, the energy consumption increases
and there is deposition of undesirable material on
the hearth.
The question is what are the values of V and I (or R = V / I) needed to
achieve the electrode optimal position for each material and for each
value of power P = V. I.
Optimal V and I - Andreae
In 1923, Andreae found that the values of V/I
appropriate to achieve the optimal position
depended on the diameter of the electrodes. For
equal power, a larger diameter electrode required
to operate with a V/I also higher.
Andreae called power density “pd” the relation
“power/electrode cross section” and found that by increasing
the power density was necessary to reduce V/I.
Andreae defined a factor "k"= (V/I) . D . p representing the
values of V and I that allowed to operate the furnace with the
electrode in the ideal position for each raw material and for each
power density at the electrode tip.
Optimal V and I - Kelly
• Between 1940 and 1952, Kelly charts the optimum operating
points for different values of power density on the electrode
(pd = P/SE ) and for different materials, where SE = the
electrode section = p. D² /4
• Kelly graph for FeSi75
K = V/I . D . Pi
pd (kW/pol2)
Optimal V and I - Kelly
• The graph shows that as the power density is increased, to
maintain ideal H is necessary to reduce R = V / I.
• For a furnace that has a certain diameter electrodes, operating
with higher values of P requires lower values of R (increasing
current and reducing voltage).
• For each power, increasing the diameter of the electrodes allows
operation with higher voltages and lower currents.
Optimal V and I – Our formula
• After studying the work of Andreae, Kelly, Morkramer, Westly and
Persson, we decided to conduct tests on cassiterite (tin production)
furnaces and, between 2005 and 2006, we found that the optimum
position of the electrode was obtained with values of V  D/P1/4 and
also that this formula is almost perfectly suited to graphics by Kelly,
especially for FeSi75 and CaC2.
• The former means that in a given furnace, to maintain the optimal
position of the electrode, the increase in power should be performed
with reduced electrode-to-hearth voltage (V  D/P1/4 ) and with
increasing current (I  P5/4).
• The formula shows that the values of V and I, needed for optimum
positioning of the electrode, depend on D, confirming what was
predicted by Andreae, Kelly, Persson, Morkramer and others who
studied this subject between 1923 and 1975.
VD/P1/4 formula deduction
The goal is to position the electrode at a certain height H
for different values of R = V / I.
Dc is the diameter of reaction zone
Dc2 is proportional to the power P.
Therefore, Dc  P1/2 (1)
If Dc>> D, the load resistance R is inversely proportional to
Dc: Rr.H/Dc (2).
Substituting (1) in (2): Rr. H/P1/2 (3)
But it was proven that the resistivity r is inversely
proportional to power density: r1/(P/D²) (4)
From (3) and (4): RD² . H / P3/2 . And, for certain H:
RD²/P3/2 (5). But, R = V²/P (6).
Then, from (5) and (6):
V  D / P1/4
VD/P1/4 formula explanation
Conceptually, the formula can be explained as follows:
By increasing the power of a furnace that operates
with a certain diameter electrodes, the electrical
resistance of the load decreases for two reasons: a)
because the resistivity of the load decreases due to
increased power density in the area of contact with
the electrode tip (pdP/D²) and b) because the
diameter of the reaction zone increases with P1/2 .
By increasing the diameter of the electrode of a furnace that
operates with a certain power, the resistivity of the load increases
because the power density in the area of contact with the electrode
decreases (pdP/D²). If the load resistance increases, to maintain
equal H is necessary to increase V/I.
C3 Fator - Westly
• In 1975, Westly presented a paper in which concluded that there is
no relationship between the optimal electrical parameters of the
operation and the diameter of the electrode. He said verbatim:
"When a furnace is operated on say 20 MW, the operating
resistance will be the same whether the electrode diameter is
1250 mm (49 in.) or 1550 mm (61 in.) provided the raw material
are the same. Apparently in conflict with the Andreae concept
this conclusion certainly gave rise to concern. But we have to
accept it as experience confirmed that it was really so. Then what
about the Andreae concept?".
• After the presentation, at the discussions, Westly was harshly
questioned by J. A. Persson and in the end, Westly seemed to agree
with Persson.
C3 Fator - Westly
• Westly concluded that the optimal voltage and current
depended only on power, arriving to the following
relationships: IP2/3 and V1/P1/3 . He called the coefficient
“I/P2/3” factor C3.
• If the Westly (factor C3) formula were correct, a FeSi75
furnace that operated with 23 MW and 83 kA could use
700 mm graphite electrodes, since they bear a current of
83 kA. However, according to all the theory before the year
1975 if the current of 83 kA were used with the electrode of
700 mm for the 23 MW (92 V) power, the electrode tip would
be too far from the furnace hearth (high H) causing deposition
of material and high specific energy consumption.
C3 Fator - Westly
• But why is believed that the factor C3 formula is correct and,
sometimes, when used, are not noticed large discrepancies with
1. One reason is the fact that most furnaces operate with the maximum
admissible current through the electrodes. Westly in his work
mentions that factor C3 suitable for FeSi75 furnaces is 10.8. This
value is correct when operating with maximum power densities at
the electrode and the voltages and currents coincide with the k
factor value found by Kelly, for the power density of 3.1/3.2 kW per
square inch, but are completely different when the electrode
diameter is increased or the power is diminished (lower power
C3 Fator - Westly
2. Similarly, because of the furnaces operate with maximum current
density we have the false impression that the relationship I P2/3 is
correct. Here´s an example:
A FeSi75 furnace with 1150 mm electrodes operates correctly with
70 kA and active power of 17.7 MW. Want to increase power to
23 MW, and applying the formula of C3 factor, it is concluded that
the current must be increased to 83 kA. By our formula, if the
diameter of the electrode remained the same, the current should be
increased to 91 kA. However, since the currents of 83 kA or 91 kA are
too high for the electrode of 1150 mm, will probably be decided to
increase the diameter, for example, to 1250 mm. Thus, by our
formula with this diameter, the current to maintain the optimal
position of the electrode should be of 83.7 kA, similar to that
calculated with the factor C3.
Comparison with Kelly
• For FeSi75 production, our formula VD / P1/4 comes to results
almost identical to the representation of the factor k performed by
Kelly. The formula of factor C3, for low power densities, shows
completely different results to those found by Kelly.
Fator k de Andreae, conforme gráfico original de Kelly,
comparado com k baseado em J e em C3
Fator k de Andreae
k J
k Kelly
k C3
Densidade de potência ( kW / pol² )
Optimal V and I – Comparison C3 and J
• We compare the values of V and I that would be calculated with the
Westly formula (C3), where V = P1/3 / C3 , and those calculated with
our formula (J), where V  D / P1/4 .
We start from a known optimal operation in which the electrode
diameter is 1150 mm (45 inches), the power is 15 MW
(3 phases), current of 65 kA and power factor 0.69. Shown are the
values of V and I that would be calculated with C3 and J for two
cases: a) equal power (15 MW), larger diameter electrode (1350
mm) and, b) lower power (7,5 MW) without change the electrode
diameter (1150 mm). Are calculated the power factors that would
be achieved in each case for a reactance of 1.23 mOhm.
See table at the next slide
Optimal V and I – Comparison C3 and J
P (MW)
D (mm)
V c/C3
kA c/C3
V c/J
kA c/J
FP c/C3
FP c/J
• It notes that according with Westly after increasing of the electrode
diameter, the furnace, for equal power, should continue operating with
the same electrical parameters. By our formula and by the k factor to
maintain the optimal position of the electrode after the increase in
diameter, the electrode-to-hearth voltage should be increased and the
current decreased.
• By reducing the power, keeping the diameter of the electrode,
according to our formula or with the factor k, the current should be
decreased to a greater extent than predicted by the formula of Westly
and voltage should be increased rather than decreased.
1. The C3 factor formula (I = C3.P2/3) of Westly, in our assessment does
not correspond to reality.
2. The Andreae k factor = (V/I).D.p) and graphics on this factor by Kelly,
performed for different materials and different power densities, more
accurately represent the points of optimal operation of the furnaces.
3. The formula we derived and called factor J (VD / P1/4 ) obtain results
similar to those found by Kelly for the factor k with the advantage of
being simpler to understand and apply.
4. It can be concluded that the operation with electrodes of larger
diameter allows optimal positioning with higher voltage and lower
current values, and therefore greater power factors, with the
following advantages:
4.1. Greater electrical efficiency.
4.2. Lower consumption of electrodes.
4.3. Minor deviations from the ideal position of the electrode.
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