Budd, C.

Chapman, S.J.

King, J.

Lacey, A.A.

Larson, D.

Peletier, M.

Please, C.P.

Riley, D.

Wheeler, A.

Wood, G.

English

Mathematics in Industry Information Service Eprints Archive

Temperature variations and control of a Calciner
Elkem
1 Background
The electrodes used in arc-furnaces are made by heating anthracite to about 2500°C in
a 'calciner', thereby removing volatiles and producing some graphite. The calciner is a
stationary cylinder 8m high and 2m in diameter, see figure 1. Raw anthracite is fed con-
tinuously from bins around the top electrode and removed by two rotating scrapers at the
bottom, after residing in the calciner for about 15 hours. The anthracite is heated ohmi-
cally by passing an electric current of 14kA between the electrodes, which are separated
by about 3m. The anthracite conducts electricity poorly below 1500°C.
Although the particulate flow of the anthracite is stable and near to a uniform plug
flow, the temperature distribution seems not to be axisymmetric within the calciner.
Moreover if the particulate flow is disturbed (temporarily reduced or enhanced), the
cold side becomes significantly colder for about 10 hours. The Study Group was set the
problem of explaining the temperature variations cause by changes in the flow rate, and
to consider the effect of different standard control strategies.
In the following, the governing partial differential equations are introduced in sec-
tion 2. Then section 3 develops a slender body theory for the heated plume downstream
of the top electrode, while section 4 looks at the thermal boundary layer on the top elec-
trode which pre-heats the anthracite. Simplified lumped-parameter models governed by
ordinary differential equations are considered in sections 5 & 6, and finally in section 7
the control of one of these through flow rate and current is studied.
2 Governing equations
The quasi-steady conduction of electric current is governed by
where a is the electrical conductivity and fjJ is the electric potential. We will approximate
the electrical conductivity by zero below a critical temperature Tc = 1500 OK and the
constant a = 2500 mho m-I above Tc.
The heat equation including advection, diffusion and ohmic heating is
1
~------- Gas stock
Figure 1 Elkem Ca1ciner
Feedirq bins
Electrode holder
Secoodory bus bars
FUrnace cover
Top electrode
Furnace shell with lining
Bottom electrode
To the bag filter
Discharge mechanism
Dischorqirq chutes with the
weighing container
Conveyor
Figure 2 Discharge mechanism
2
Typical values for the anthracite in the calciner are :- density p = 900 kg m -3, specific
heat C = 1700 J kg-1 oK-I, temperature T = 1500 "K, time t = 6x 104 s, advection velocity
U = 1.4 X 10-4 m s-1, (axial) length L = 5m, heat conductivity K, = 10-5 m2 s-1, electrical
conductivity a = 2500 mho m-I, and electrical potential </J= 70 V. From these values we
can form two important non-dimensional groups, the Peclet number Pe = UL/ K, = 70
and the ohmic heating number a = a</J2 / pCK,T = 500. Nondimensionalising the variables
with the above scales, the heat equation takes the form
0.65Tt + rer, = \12T + aa(T) 1\1 </J12.
where now a = 0 for T < 1 and a = 1 for T > 1.
3 Slender heated plume
Downstream of the top electrode, there is a plume in the moving anthracite which is
above the critical temperature to conduct electricity and so is heated ohmically. The
plume spreads as heat diffuses radially. Because the Peclet number is large, the plume is
slender, with radial extent (z / Pe) 1/2. It should be noted that the calciner is designed so
that this plume reaches the outer boundary at the exit, Le. all the anthracite is treated,
just.
The Study Group constructed a similarity solution for this plume, but in two-
dimensional rather than axisymmetric geometry. Let x be the cross-stream coordinate.
This is rescaled by x = Pe-1/2 X. Let X = h(z) be the edge of the plume defined by
where T = 1, so that inside the plume a = 1 and outside a = O. For long slender plumes,
the quasi-steady conduction of heat becomes at leading order
h</Jz = I,
where] is the total current flowing. At leading order the temperature equation becomes
7]2
Tz = Txx + h2 .
inside the plume X < h(z) and without the final term outside. (Here 7 = a/Pe.)
A similarity solution is sought in the form
T(X, z) = (}(O and h = HZ1/2 with € = X/Zl/2.
The function () then obeys
()" ~ C()' 7]2+ 2'" + H2
inside € < H and without the final term outside. Boundary conditions to be applied
are (}'(O) = 0 and () -+ 0 as € -+ 00. The constant H is determined by the condition
(}(H) = 1. It was left as a simple numerical task to find (}(€) and H. A series solution is
easy to obtain inside
()( ) _ e 7]2 ( 1 2 1 4 1 6 1 8)
€ - (0) + H2 -2€ + 24€ - 360€ + 6720€ ,
while outside () ex: erfc €. [Note: a similarity solution can be found for the axisymmet-
ric plume, which has the same Zl/2 sideways spread but has the temperature decaying
downstream like Z-l.]
3
4 Thermal boundary layer on top electrode
Now the electrodes are long and are themselves heated ohmically. Along the length of the
electrode, heat will diffuse radially out forming a thermal boundary layer in the anthracite
moving along the electrode. It is essential that the anthracite is pre-heated (in sufficient
quantity) to the critical temperature Tc before the end of the electrode in order that
thermal plume can conduct electricity.
Let the electrode carry current I, be ofradius a and have a length f. Let the thermal
boundary layer have a thickness t5 at the end of the electrode: 6 = (K,fIU)I/2. The time
for anthracite to move along the electrode is flU, so that the total ohmic heating in the
electrode during that time is (I I a2)2I oe x a2f x flU (dropping factors like 7r), where aE is
the electrical conductivity of the electrode. Increasing the temperature of the anthracite
to Tc from effectively 0 within the boundary layer requires heat pcTc x Ma. Equating
these two, we find that electrode must be at least as long as
£ = eCT~~Ea3ra:
We use the following values of the relevant parameters: density p = 850kgm-3,
specific heat c = 1700 J kg-l oK-I, critical temperature Tc = 1500 OK, conductivity of
the electrode aE = 105 mho m-I, radius of electrode a = 0.25 m, total current 1= 14 kA,
advection velocity U = 1.4 x 10-4 m s"! and heat conductivity K, = 1O-5m2s-1. These
produce the estimate for the length of the electrode necessary in order to reach the critical
temperature as f = 0.5 m, which agrees with the Elkem design. At this length, the thermal
boundary layer has a thickness of 0.17 m, which is similar to the radius of the electrode
in the Elkem design, and hence allows an easy transition to the thermal plume.
We note that the required length of the electrode f is proportional to the velocity
of the anthracite. Hence if the feed is increased too much, there will be insufficient
pre-heating of the anthracite to the critical conduction temperature Tc in the thermal
boundary layer.
5 Lumped models
In order to understand the temporal development, and possible instability, of non-axi-
symmetric differences between one side of the calciner and the other, the Study Group
investigated a simple lumped-parameter model. The full temperature field was replaced
by four temperatures: Tl and T2 the temperatures in the thermal plume on the left and
right sides respectively, and T3 and T4 the temperatures in the thermal boundary layers
again on the left and right sides respectively. These temperatures were taken as being
typical of their respective region and to be only a function of time. The temperature of
the electrode was taken to be T.
The heat balance for the left plume region of length L and cross-sectional area A is
pcALT1 + pcUA(T1 - T3) = k1(T2 - Td + k2(T3 - Td + k3(T - Td + nI,
with coefficients k; for the heat transfer between different regions, which could be esti-
mated from the studies above, and with ohmic heating n1. The equation for the right
4
plume regions is similar with T2 replacing T1 and T4 replacing T3. The heat balance for
the left thermal boundary layer is
and a similar equation for the right thermal boundary layer with T2 replacing T1 and T4
replacing T3.
There was much discussion about the form to be adopted for the ohmic heating.
In the model eventually adopted, most of the voltage drop V was taken to occur in the
thermal plume, and so
A 201 = L 0"(T1)V ,
and a similar expression in the right thermal plume with T2 replacing T1. The alterna-
tive model which was eventually rejected had most of the heating occurring near to the
electrode
01 = 'Y(T1, T)V2,
with the function '1' still to be determined somehow.
More sophisticated models with the cross-sectional area A varying were not pursued.
6 Reduced model
In order to understand the possibility of non-axisymmetric states, the above lumped-
parameter model was further simplified by deleting the thermal boundary layers, leaving
the two-variable model
1'1 = -UTI + >'(T2 - T1) + 'Y0"(T1)
1'2 = -UT2 + >'(T1 - T2) + 'Y0"(T2)
Recall that O"(T) = 0 for T < Te and O"(T) = 1 for T > Tr: It is important to realise now
that 0" rises very sharply from 0 to 1 in a neighbourhood of Tc, and so we may consider
it to take on arbitrary values between 0 and 1 for T = Te.
If U > '1'ITe, the only solution is the symmetric solution T1 = T2 = 0, i.e. too cold
for any ohmic heating.
If U < '1'ITe, there are three symmetric solutions
T1 = T2 = 0, Te and 'YIU,
corresponding to no heating (0" = 0), partial heating (0" = UTel'Y) and full heating
(0" = 1), respectively.
If '1'(>.+ U)ITe(2)' + U) > U > 'Y>'ITc(2)' + U), there are also two asymmetric
solutions with full heating in one side O"(Td = 1 and no heating in the other 0"(T2) = 0
'1'(>.+ U)
T, = U (2)' + U) ,
and a similar solution with T, and T2 interchanged.
5
If ,ITe > U > ,)..ITe(2)"+U), there are two asymmetric solutions with full heating
on one side a(Td = 1 and partial heating on the other T2 = Te
T _ ,+)"Te
1- )..+U '
T - T. . h _ U(2)" + U)Te - )..,
2 - e WIt ae - ()..+ Uh '
and a similar solution with T, and T2 interchanged.
Finally if U < ,(),,+U)ITc(2)"+U), there are two asymmetric solutions with partial
heating on one side T, = Te and no heating on the other a(T2) = °
. U(2)" + U)Te
'T, = Te with ae = ()..+ Uh '
and a similar solution with Tl and T2 interchanged.
A simple phase-plane analysis find that the solutions with some partial heating are
all unstable (saddle points or unstable nodes). At high flow rates U > ,ITe the only
stable solution is the cold symmetric state T, = T2 = 0, i.e. the anthracite fails to reach
the critical temperature before it passes out of the calciner and so in this simplified model
there is no heating anywhere. At lower flow rates U < ,ITe, in addition to this cold stable
symmetric state there is the symmetric state Tl = T2 = ,IU, in which ohmic heating
can exceed the critical temperature before the material exits the calciner. In the range
,()..+U)ITe(2)" + U) > U > ,)..ITe(2)" +U) there are also two stable asymmetric states,
one side with ohmic heating and one without. In these asymmetric states, the ohmic
heating on the one side is sufficient for the anthracite to reach the critical temperature
on that side, but via heat transfer between the two sides is insufficient for the other side.
Clearly one way to eliminate the asymmetric state would be to decrease the flow rate
until U < ,)..ITe(2)" + U).
Future work should incorporate the electrode into the two-variable theory, and to
return to the four-variable theory of section 5 which could be solved numerically.
7 Control
The Study Group considered the control of the two-variable model of section 6 through
monitoring the total current I = A( a(T1) +a(T2)) VI L and in response adjusting the flow
U of the anthracite. It is to be expected that some control mechanisms U = f (1) might
be unstable: the time-delay resulting from the non-zero residence time (not in the model
of section 6) may require responding to the time-integral of the current rather than the
instantaneous value in order to avoid control instabilities.
The simplest control strategy is 'bang-bang', in which either the flow runs at its
normal constant value, U = U say, if both sides of the thermal plume are being ohmically
heated, I > 2Aa(Te+) VI L, or the flow is shut down totally, U = 0, if both sides are not
above the critical temperature for ohmic heating.
Under bang-bang control, if the initial state has ohmic heating on both sides, T, > Te
and T2 > Te, then the system will evolve to the stable symmetric state with ohmic
heating, T, = T2 = ,IU, so long as the flow is sufficiently slow for this state to exist,
U < ,IT e. This follows immediate from the governing equations in this regime
TI + T2 = -U(Tl + T2) + 2" and i,- T2 = -(U + 2)")(T1 - T2).
6
If on the other hand the initial state has no ohmic heating on both sides, the system
would evolve under bang-bang control to a symmetric state at the average of the initial
temperatures, and no ohmic heating. Again this immediately follows from the governing
equations in this regime
The initial condition of one side ohmically heated and the other not, as would occur
in the stable asymmetric states, is more complicated. If one side is significantly above
the critical temperature and the other side is only just below, then the cold side will
warm up to the critical temperature and the system will then evolve into the stable
symmetric state with both sides heated ohmically. If one side is only just above the
critical temperature and the other side is significantly below, then the hot side may cool
down to the critical temperature, with the system then evolving into a symmetric state
with no ohmic heating, or if the rate of heating is high enough the cold side may warm up
to the critical temperature. To distinguish between these different cases, we consider the
governing equations for the left side being heated T, > Te and the right side not T2 < Te
dD 2),
-=--D+1dS ,
with solution D = .i.+ (D _~) -2(8-80»';-y
2), 0 2), e ,
where Do and So are the initial values.
We now need to write down the boundaries of this regime of one sided heated and
the other not in terms of the new variables D and S. The constraint that Tl > Te and
o < T2 < Te gives D > 0 and S > Te at least. The right side will warm up to the critical
temperature while the left side is still hot if according to the above solution T2 = Te
while T, > Te, i.e. D = S - 2Te in S > 2Te. The left side will cool down to the critical
temperature while the right remains cold if T, = Te while T2 < Tn, i.e. D = 2Te - S in
Te < S < 2Te.
From the solution D(S) it is clear that if the rate of heating is sufficiently large
, > 2)'Te all the trajectories will move away from the boundary D = 2Te - S in
Te < S < 2Te and so the system must go to the symmetric heated state.
If, < 2)'Te, we have to contemplate solution trajectories intersecting this boundary.
The last trajectory to intersect will be tangent, and so has slope dD IdS equal to -1 at
the point of intersection. We thus find the point of intersection as
D = ,I)., and S = 2Te - ,I)..
We note that this intersection does not occur in the domain S > Te (i.e. T, > Te) if the
heating is too high, > )'Te. For the case of weak heating, < )'Te, we can follow the
7
trajectory back to S = Te and so deliminate all the initial conditions which would cause
the whole system to cool down to below the critical temperature
We conclude that bang-bang control will drive the system to the stable symmetric
solution with ohmic heating if initially at least one side is ohmically heated, and if under
dangerous conditions the heating rate is sufficiently strong 'Y> ATe.
8 Contributors
Chris Budd, Jon Chapman, John King, Andrew Lacey, Dale Larson, Mark Peletier, Colin
Please, David Riley, Adam Wheeler, Gerard Wood.
8


Temperature variations and control of a Calciner

Temperature variations and control of a Calciner

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