Mei C., Zhou J., Peng X. Simulation and Optimization of Furnaces and Kilns for Nonferrous Metallurgical Engineering

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6 Simulation and Optimization of Electric Smelting Furnace 

⎛⎞ ⎛⎞⎛⎞

⎜⎟ ⎜⎟⎜⎟

⎝⎠ ⎝⎠⎝⎠

∂∂∂ ∂

∂∂∂

++++=

∂∂∂

∂

∂∂

∂

TTT

rλλλρcV p

rzz

(6.1)

where T is the temperature;

λ is the thermal conductivity; ρ is the volumetric

density; c is the specific heat; and V is average descent velocity of the electrode.

222

ϕϕϕ

⎡⎤

⎛⎞ ⎛⎞⎛⎞

⎢⎥

⎜⎟ ⎜⎟⎜⎟

⎢⎥

⎝⎠ ⎝⎠⎝⎠

⎣⎦

∂∂∂

=++

∂∂∂

(6.2)

where σ is the electric conductivity, and

is the electric potential.

The equation for electric potential distribution is expressed as:

ϕϕϕ

σσσ

⎛ ⎞ ⎛⎞⎛⎞

⎜ ⎟ ⎜⎟⎜⎟

⎝ ⎠ ⎝⎠⎝⎠

∂∂∂

++=

∂∂∂

(6.3)

The boundary condition is given as follows:

At the top of the analytical region, i.e. surface of the electrode paste:

∂

;

()

λα

∂

=−

∂

On the two vertical surfaces of the analytical region (cross section 1 and 4 in

direction in Fig. 6.2):

∂

; 0=

∂

Along the central line of the electrode: 0=

∂

; 0=

∂

The boundary on the outer cylinder is very complicated and divided into several

parts which have different boundary conditions. The boundary condition at the

brass clamp is:

()

ϕϕ

∂

=−

∂

;

()

λα

∂

=−

∂

For other electrode surfaces upon the furnace roof:

∂

∂r

;

()

λ TT

∂

=−

∂

For electrode surfaces under the furnace roof and upon the slag level:

∂

∂r

;

()

345

ααα

∂

=−+−+−

∂

λ TT TT TT

For the exterior surface of the electrode under the slag level:

()iz

∂

TT =

For the bottom of the electrode in the slag:

()ir

∂

TT =

where

is electric potential of contact clamp, set to be zero;T

is ambient

temperature; T

is temperature of cooling water,T

is gas temperature in the

furnace; T

is temperature of inner surface of the hearth and the charge in the

furnace; T

is temperature of the slag surface;T

is temperature at the interface of

the electrode and the slag, i.e. temperature of the arc; R is contact resistance

between the electrode and clamp;

is comprehensive heat transfer coefficient

  Jiemin Zhou and Ping Zhou

between the electrode and the ambient;

is comprehensive heat transfer

coefficient between the electrode and the cooling water ;

is convection heat

transfer coefficient between the electrode and the furnace gas;

is radiation

heat transfer coefficient between the electrode and the furnace chamber ;

radiation heat transfer coefficient between the electrode and the slag ;

()

iz is

current density on the side surface of the electrode in the slag;

()

ir is current

density on the end surface of the electrode.

The distributions of

()

iz and

()

ir are shown in reference (struskij, 1982).

Obviously it is impossible to solve this problem by using analytical method, so

numerical method is adopted. Considering the shape of the electrodes, the finite

difference method is used to accelerate the calculation. The computational grids

are shown in Fig. 6.2.

6.2.2



Simulation software

The calculation software consists of main program, 13 subroutines and 2 external

functions, and the plotting software includes 3 data processing programs, 6

drawing programs and 15 subroutines.

The electric power distribution must be known in advance when to calculate

the temperature field, because the Joule heat originated from electricity is the

main heat source in the electrode during the baking. However, as the electrical

conductivity of the electrode material varies greatly with temperature, the

thermal conductivity is also a function of temperature, thus iteration has to be

used in the calculation, and the precision of solution depends on the iteration

times. Proper acceptable error must be selected to ensure a desirable precision of

calculation result as well as a reasonable computation time. To solve linear

equations, direct solution method is adopted in every circulation of iteration.

Coefficient matrix of the equation set is generally very big, from hundreds to

thousands orders. In the development of the software, characteristics such as

symmetry, positive definition as well as rarefaction of the coefficient matrix are

fully employed, and the one-dimensional memory technics and the root-squaring

method for linear equations set are adopted, to solve the large scale linear equations in

a direct way at a relatively higher speed.

To show the simulation result more illustratively, a plotting software was

developed, by which various pictures can be conveniently drawn in the printer

using FORTRAN language.

The calculation result can be expressed as:

a) The isopotential line and isothermal line of the vertical cut planes in the

electrodes.

b) The electric current field and heat flux field of the vertical cut planes in the

6 Simulation and Optimization of Electric Smelting Furnace 

electrodes.

c) The stereo distribution graph of the electric potential, temperature and

electric power of the vertical cut planes in the electrodes.

d) The stereo distribution graph of the electric potential, temperature and

electric power of the cross sections in the electrodes.

e) The table of the heat balance of the electrodes.

Besides, the needed information can also be printed out as tables.

The simulation software can be applied in following situations:

a) Optimal design of the electrode configuration. Some optional design

schemes, such as different electrode diameter, different electrode shell material

and thickness, different fin dimension and number, different brass clamp height

and spacing, etc., can be calculated beforehand when the self-baking electrode

for a new electric furnace is designed or the current electrode is improved. In

this way, the electrode configuration corresponding to the most reasonable

temperature distribution can be selected, and this configuration is the best design

scheme.

b) Selection of the most proper operation conditions. By importing various

electrode operation parameters into the computer, such as electrode current,

descent speed of the electrode, height of the paste surface, insert depth of the

electrode into the slag, flow rate of the cooling water (or air), etc., the

influences of the parameters on the temperature distribution during the baking

can be examined, then the most reasonable operation condition can be

obtained.

c) The influence of the distance between the clamp and the furnace roof, the

height of the furnace chamber, the nudity area of the melt slag, the variety

(standard or inclosed) and property of the electrode paste and some other factors

on the baking process of the electrode can be investigated, so that proper measures

can be taken to improve the electrode baking condition, to strengthen the electrode

intensity and to decrease the energy consumption.

6.2.3



Analysis of the computational result and the baking process

Fig.6.3 to Fig. 6.7 show the simulation results of the self-baking electrode of the

No. 2 furnace in Jinchuan Nonferrous Metal Company.

Fig. 6.3 (a) is the electric potential distribution on the electrode vertical cut

plane 1 (division of the vertical plane and cross section is shown in Fig. 6. 2), in

which the electric potential is drawn as height of the plane. The electric potential

above the clamp is almost constant, but it rises gradually from the lower margin of

the clamp to the bottom of the electrode. The maximal electric potential drop (V

)

is about 3.5V.

  Jiemin Zhou and Ping Zhou

Fig. 6. 3 (b) is the three-dimensional temperature distribution on the electrode

vertical plane 1. The acute temperature change at the slag surface and near the top

of the furnace can be seen very clearly from this figure, and it can also be seen that

the central temperature of the electrode is higher than the surface temperature in

the furnace chamber. In this figure, the 300

ć and 500ć isothermal lines and

their position on the electrode have been drawn. Although the behavior of the

paste varies with the composition, some assumptions can be made based on

general conditions. That is, when temperature of the paste reaches 300

ć, the state

of the paste will change from fluid into plastic and the possibility of the paste

flowing out of the cracks in the casing caused by arc is diminished greatly; when

the temperature exceeds 500

ć, the paste binder polymerizes and the risk of soft

breakage is decreased. Even if the electrode breaks under the 500

ć isothermal

line, the electrode paste will not leak into the furnace, so such accidents can be

easily dealt with (“Ferroalloy Production” composition group 1975). Under the

present baking condition, the 300

ć isothermal line of the No.2 electric furnace

electrode is located between the brass clamp and the top of the furnace, while the

500

ć isothermal line is near the top of the furnace.

Fig. 6. 3 Electric potential and temperature distribution on the vertical

section of the self-baking electrode

(a) Electric potential distribution;(b) Temperature distribution

Fig. 6.4 (a) shows the current field in the electrode plane section 1. It can be

seen clearly the current’s magnitude and direction in the electrode. Because the

electrode around the clamp has not been sintered yet, the electrode paste is almost

electrically insulated. When current flows from the brass clamp into the electrode,

it will all flow down along the electrode shell and fin. After the electrode entered

the furnace, electric conductivity of the electrode paste will become stronger and

stronger as the temperature increases. When the electrode shell is burned out, flow

direction of the current will move towards the inner past of the electrode

remarkably. Finally, the current will be distributed uniformly in the baked

electrode.

6 Simulation and Optimization of Electric Smelting Furnace 

Fig. 6.4 Current field, heat flow field, equipotential line and isothermal line on the vertical

cross section of self-baking electrode

(a) Current field; (b) Heat flow field;(c) Equipotential line; (d) Isothermal line

Fig. 6.4 (b) is the heat flow field in the plane section 1. In this figure, it can be

seen that heat is transferred from the lower side to the upper side of the electrode.

Micro arc area is located on the interface of the electrode and the slag, where

temperature is very high, and the heat transfers from the interface to the

electrode. Above the slag surface, because temperature of the furnace chamber is

much lower than that of the electrode, heat in the electrode is transferred by

radiation acutely through the shell surface, whilst convection heat transfer also

exists. The heat radiation intensity decreases with the increasing of the electrode

position. Above the top of the furnace, temperature of the workshop is 500

̚600

  Jiemin Zhou and Ping Zhou

ć lower than that of the furnace chamber, and the temperature difference

between the electrode and the ambient is also great, therefore, the heat loss from

the electrode to the environment is remarkable. Near the brass clamp, heat flows

from the electrode towards the clamp because there is a water cooling system

here.

Fig. 6.4 (c) is the isopotential line on the electrode plane section 1. The voltage

drop on the interface of the clamp and the electrode is 0.6

̚0.8V, which is about

one fifth of the total voltage drop. The isopotential line near the top of the furnace

has a relatively big slope, which means the electric conductivity of the electrode

paste changes greater there.

Fig. 6.4 (d) shows the isothermal line used to analyze the temperature

distribution in the electrode. It can be seen that the isothermal lines in the

electrode are very dense near the slag surface, which indicates an acute

variation of temperature. The electrode isothermal lines are very sparse in the

upper part of the furnace chamber, so the change of temperature is very smooth.

Meanwhile, there is a relatively big temperature gradient near the top of the

furnace because there is a dense distribution of isothermal line there. The

electrode enters the clamp at a temperature of 60

ć, and leaves the clamp at a

temperature of 200

ć. Its temperature is a little higher than 400ć when it is

near the top of the furnace. Therefore, the electrode paste above the top of the

furnace is generally in a state of fluid or plastic, and the paste will leak out if

there is a hole on the electrode casing. Between the clamp and the top of the

furnace, almost the whole weight of the electrode is born by the electrode

casing and fins and all the current passes through the casing and fins, thus, an

unsuitable operation or a bad welding of the electrode casing will easily result

in a soft-broken accident.

Fig. 6.5 shows the power density distribution of the electrode, whose four

plane sections have evident difference from each other. For the electrode

casing, at the clamp, there is a big contact electric resistance on the interface

of the clamp and the casing, so the current’s power density is very big, and it

increases with the descent of the electrode and the augmentation of the current

(see plane 1 and 2). When the electrode leaves the clamp, its power density

drops abruptly. However, with the slipping down of the electrode and the

increasing of the temperature, the electric resistance of the casing increases,

then the power density will increase continuously. With the increasing of the

temperature, the paste starts to conduct electric current, the current in the

casing decreases, the power density generally drops again. When the casing is

burned out, the difference of power density in the electrode will be very small.

Plane 4 is a section in the gap between the two clamps but not contacting

either of them, so there is no peak in the power density of the casing near

clamp. Inside the electrode, the paste under 400

ć has a poor electric

6 Simulation and Optimization of Electric Smelting Furnace 

conductivity and low power density, but the ascent of the temperature results

in an increasement of the electric conductivity as well as the power density. In

the slag, a part of current flows from the lateral surface, which makes a

decreasing power density. Plane section 1 locates on the interface of the fin

and electrode paste. With the slipping down of electrode and the ascent of

temperature, power density in the fin increases firstly, then decreases, while

the power density in the electrode paste increases smoothly. Hence the

distribution curve of power density is very complicated.

Fig. 6.5 Power density distribution on the vertical plane section of

the self-baking electrode

(a) Plane 1; (b) Plane 2; (c) Plane 4

Fig. 6.6 (a) is the stereo electric potential distribution on electrode cross

section (division of the cross sections is shown in Fig. 6.2, the area of the

sectors is one eighth of the circle). Distribution of the electric potential is

uniform on the cross sections except for (cross section 10) near the furnace

roof.

Fig. 6.6 (b) presents the power density distribution on the cross section. In the

upper portion of the electrode where the paste has a poor electric conductivity,

power density in the casing is much higher than that in the inner electrode, and the

power density in the interior fin is also much higher than that in the electrode

  Jiemin Zhou and Ping Zhou

paste.

Temperature distribution of the cross section is given in Fig. 6.6 (c). It can be

seen that the central temperature in the electrode is higher than that on the surface

above the slag (cross section 20).

Heat balance data of the electrode is given in Table 6.3. Calculation results

indicate that most of the heat (more than 80%) is from the Joule heat of current.

The heat from the melt is only a small portion, and the heat from the furnace

gas is even less. The heat that consumes to bake the electrode is only 20% of

all the energy, whilst most heat is lost through the surface of the electrode.

Additionally, the heat that is carried away by the cooling water is also

considerable.

Fig. 6. 6 Electric potential, power density and temperature distributions on cross section

of self-baking electrode

(a)Electric potential distribution on cross section 10; (b) Power density distribution

on cross section 15;(c)Temperature distribution on cross section 20

6 Simulation and Optimization of Electric Smelting Furnace 

The plots and charts mentioned above give a lot of information. They illustrate

the rules of electricity conduction, electricity-heat conversion, and heat transfer

from various perspectives, including electricity and heat, potential field and flow

field, microscopic and macroscopic scale, longitudinal and transverse direction,

etc. This provides fundamental knowledge for the investigation and the analysis of

electrode baking process.

Table 6.3 Heat balance of the self-baking electrode of the electric smelting furnace

Item Heat income/kW

Heat

expenditure/kW

Joule heat from the electrode resistance 46.5

Joule heat from the contact resistance

between the clamp and the electrode

10.1

Heat conducted from melt 7.9

Heat from hot gas 0.4

Heat dissipated from the electrode surface 36.4

Heat consumed in baking electrode 12.1

Heat taken away by cooling water 16.5

Error

0.1

Total heat 64.9 64.9

The simulation results indicate that, for the self-baking electrode of nickel

electric smelting furnace in Jinchuan Company under the configuration, dimension

and operation condition introduced above, the electrode above the top of the

furnace is in a serious shortage of heat, the electrode paste has little intensity and

electric conductivity, and the electrode is in a danger of paste-leaking and

soft-broken. As there is a big temperature gradient near the roof of the furnace, the

too fast baking of the electrode will influence its baking quality, which is prone to

result in a hard-broken of the electrode. In this case, the size of the casing and the

fin should be chosen to ensure that they can bear the total current in the electrode

and the weight of the electrode from the clamp to the bottom. It should also be

done to make the welded seam to be smooth and regular to ensure a high welding

quality. The way that the clamp is mounted should be improved as well to reduce

the risk of paste-leaking and soft-broken of the electrode. Additionally, the baking

condition should be reformed to obtain a smooth temperature gradient in the

electrode to improve the electrode performance.

  Jiemin Zhou and Ping Zhou

6.2.4



Optimization of self-baking electrode configuration and

operation regime

In the following text, examinations are to be made on the influence of various

parameters on the temperature distribution in the electrode through the analysis of

simulation result to get some optimization schemes that aim for reducing the

consumption of electrode and ensuring a safe operation of the electrode (Zhou,

1990; Zhou et al., 1990; Zhou et al., 1989).

a) Influence of the electrode consumption (descent velocity) and current density

on the baking process. When the electrode current density drops down to

1.25A/cm

, or even 0.2 A/cm

, the positions of the two isothermal lines (300ć

and 500

ć) change little, but central temperature of the electrode at the top of the

furnace decreases. When current density increases up to 2.5 A/cm

, the 300ć

isothermal line rises to the lower margin of the clamp, and it will continuously rise

up to the intermediate section of the clamp when current density reaches 3.0

A/cm

(Fig. 6.7 (a)). It has little influence on the temperature distribution when

electrode consumption varies from 0.15

̚0.6m/d.

b) Influence of furnace temperature and electrode material physical properties

on the temperature distribution in electrode. Fig.6.7(b) shows the temperature field

when a lot of slag is exposed and the furnace temperature is 700

ć. Under such a

condition, the two isothermal lines rise a little bit. The isothermal lines will then

fall down a bit when the slag surface is completely covered by the charge and the

hearth temperature is 500

ć. If the clamps are cooled by hot water or vaporization

and the cooling medium is 100

ć, there will be no evident temperature change on

the electrode between the clamp and the top of the furnace.

Previous researchers attempted to combine the baked electrode paste with the

crude paste, then put them into the casing to improve the electric and thermal

conductivity of the upper electrode paste

˄Cavigli, 1978˅. However, simulation

result shows that it is helpless to the electrode baking even if the electric

conductivity of the upper paste increases to 1

h10

S/m; but, when the paste’s

thermal conductivity increases to 33.2 W/(m

gK), the 300ć isothermal line rises

up to the lower margin of the clamp. The similar effect can be obtained if the

casing and the fin are made of metals with larger electric resistivity so as to double

the resistance (Fig. 6.7 (c)).

c) Influence of electrode configuration parameter and insulation measures on

the upper electrode on temperature distribution in the electrode. It is found that

joule heat will increase and the 300

ć isothermal line will rise up to reach the

inner of the clamp when the electrode diameter is 0.75m and the casing

thickness is1.5mm (Fig. 6.7(d)). When the electrode diameter increases up to