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Theses

1987

Development of a mathematical model to determine the Development of a mathematical model to determine the

temperature distribution in the metal layer and hearth of an temperature distribution in the metal layer and hearth of an

electrical resistance smelter electrical resistance smelter

Kurt B. Carlson

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ABSTRACT

A steady-state finite-difference heat transfer

mathematical model was derived to determine the temperature

distribution in the metal layer and the hearth of a

cylindrical electrical-resistance smelter. The temperature

distribution is required for the redesign of the refractory

materials and their positions in the hearth to insure that

the metal layer is kept molten at normal smelting

temperatures and that mechanical and structural integrity

of the hearth is maintained. An extensive literature

search revealed that no previously defined model of this

type existed and that consideration of a three-dimensional

model was beyond the scope of this work. The literature

search also verified that the metal layer and the hearth

could be modeled independently of the slag layer. The

finite-difference heat transfer model was then developed by

defining nine different types of nodes in the model and

deriving steady-state heat transfer equations for each type

of node. An algorithm was then developed for the solution

of the non linear dependent set of simultaneous equations.

Beyond the scope of this work, ninety percent of a Fortran

11

77 computer program was written and compiled employing the

algorithm. It is recommended that the computer program be

finished, debugged, and that various combinations of

refractory material types and positions be tried to

determine the optimum design of the refractory hearth.

Proper design of the hearth has the potential to improve

the quality of the metal poured from the smelter, reduce

operating costs, and increase the capacity of the smelter-

Ill

NOMENCLATURE

Ar - Area of heat transfer resistance [in2]

As - Area of heat transfer resistance for a partial

node [ iii2 ]

hi, h2 - Heigth of the face of a partial node [in]

I - Unit vector in the radial direction

J - Unit vector in the axial direction

k - Thermal conductivity[BTU-in/hr-ft2-0 F]

m- Equivalent slope of the bottom shell of the

smelter

Q - Heat flow from a typical node [BTU/hr-ft2- F]

Qc - Convective heat flow from a node to the air

surrounding the smelter [BTU/hr-ft2- F]

Qr - Radiative heat flow from a node to surfaces

surrounding the smelter [BTU/hr-ft2- F]

R - Radial direction

r- Length of node in the radial direction [in]

Ro - Outside radius of the shell [in]

T - Temperature of a node [F]

Tair - Temperature of the air surrounding the shell

[F]

Ts- Temperature of the surfaces surrounding the

shell [F]

Z - Axial direction

Zh - Maximum thickness of the refractory hearth [in]

Zs- Thickness of the silver layer [in]

-Emissivity of the shell

6- Stephan-Boltzman constant

[0.1714E-08 BTU/hr-ft2-R4 ]

G> - Debye temperature [K]

Listing of Figures and Tables

IV

Figure Description Page

1.1 Cross Section View of Electric Resistance 6

Smelter

1.2 Detail of Refractory Hearth of Electric 10

Resistance Smelter

2.1 Detail of Refractory Hearth of Electric 13

Resistance Smelter

3.1.1.1 Electric Glass Melting Furnace 25

3.1.1.2 Contour Plot of Power Intensity at the 28

Vertical Surface

3.1.1.3 Equipotent ial Lines of the Hydrodynamic 29

Vector Potential

3.1.1.4 Flow Velocity in the Y-Z Plane 30

3.1.1.5 Isothermal Plot in the X=0.1 Plane 31

3.1.1.6 Sketch of an Electroslag Remelting Unit 35

3.1.2.2 Streamline Patterns - Electromagnetic Effect 37

Only

3.1.2.3 Streamline Patterns of Electromagnetic and 37

Buoyancy Effects

3.1.2.4 Isotherms in the Slag and Metal Layers 38

3.1.2.5 Effect of Current Level on the Linear 40

Velocity in the Slag

3.1.2.6 Electroslag Welding Process 41

3.1.2.7 Streamline Flow Patterns 43

3.1.2.8 Computed Isotherms 43

3. 1.2.9 ESR Unit 46

Fiflure Description Page

3.1.2.10 Flow Profiles with Current Increased 50

By 10 and 100 Times

3.1.2.11 Flow Profiles with Current Increased 50

By 10 and 100 Times

3.1.3.1 Hearth Arrangement 53

3.1.3.2 Temperature History of the Hearth 54

3.2.1 Thermal Conductivities of Sodium Oxide - 57

Silicate Slags

3.2.2 Thermal Conductivity of Calcium Oxide - 58

Aluminum Oxide - Silicate Slags

3.2.3 Thermal Conductivity of Sodium Oxide - 58

Silicate Slags

4.1 Macroscopic Heat Balance in the Model 62

4.2 Location of Nodes 65

4.3 Summation of the Nodal Equations 67

5.1 Program Structure 72

Table Description Page

3.2.1 Thermal Conductivity of Slags Similar to 56

Those used in the Electrical Resistance

Smelter

TABLE OF CONTENTS

SECTION PAGE

Abstract i

Nomenclature iii

Listing of Figures and Tables iv

1.0 Introduction 2

2.0 Problem Description 12

3 . 0 Literature Search 22

3.1 Introduction to Mathematical Models. 22

3.1.1 Three-Dimensional Model Including

the Effects of Convective and

Electomagnetic Fields 24

3.1.2 Two-Dimensional Axisymmetric Models

Considering the Effect of

Convective and/or Electromagnetic

Fields 34

3.1.3 Model Considering All Layers as

Solids and Considering No Effects

of Convective and Electromagnetic

Fields 52

3.2 Physical Properties of Metallurgical

Slags 55

4.0 Theoretical Analysis 59

5.0 Program Structure 71

6. 0 Conclusions and Recommendations 78

7 . 0 References 79

8.0 Appendix 82

1.0 INTRODUCTION

The materials needed in today's highly technical world

must be of high quality. Gone are the days when man could

take materials from the earth and, with a minimum of effort

and technology, fashion the tools for his world. To

provide these high quality materials, the metallurgical

industry has had to devise processes to take the materials

from their native state in the earth and to refine them or

modify them to the desired purity levels. To do this there

are basically three refining processes: hydrometallurgy,

pyrometallurgy , and electrolytic processes.

Hydrometallurgical processes involve the separation of

the desired metal from gangue or impurities in an aqueous

or liquid phase by virtue of solubility or density

differences. Pyrometallurgical processes involve the

separation of the desired metal from the gangue by the

application of thermal energy to either volatilize off

impurities or products, or else to melt and/or solubilize

the materials and to separate them by solubility or density

differences. Electrolytic processes involve the

solubilization or ionization of metals or impurities from

partially refined metals and subsequent separation by

virtue of solubility and/or electomotive differentials.

Typical industrial refining operations might involve from

one to all three of these processes in varying combinations

and orders.

One of the pyrometallurgical processes, also called

fire refining, is smelting. Smelting usually involves

melting the charge materials in a refractory lined crucible

and either separating the materials by volatilizing off

(called fuming) the metals or impurities, or by separating

the materials in the liquid phase by virtue of solubility

or density differences. Fluxing and reacting chemicals are

added to reduce or oxidize the metals and/or impurities and

to form a slag bath. The slag bath acts as a matte or

sponge to absorb the impurities or product and is separated

from the metal by virtue of solubility and density

differences.

There are four types of smelters which are

distinguished by the manner in which the thermal energy is

supplied to melt or volatilize the materials. These types

are:

1. Reverbatory- the energy is supplied by a

fossil fuel burner and the heat is transferred by radiation

and convection.

2. Induction - the energy is supplied by

electrical induction into a charge inside of a crucible

lined by induction coils

3. Electrical Arc - the energy is supplied

by an electrical arc induced by a high voltage differential

between the metal layer and an electrode immersed into the

charge.

4. Electrical Resistance - the energy is

supplied by the resistance of the molten slag layer of the

charge.

The smelter under consideration in this thesis is an

electrical resistance silver smelter at Kodak Park (Eastman

Kodak Company) in Rochester, N.Y. (U.S.A.). Silver rich

ash derived from waste photographic products is smelted to

separate the silver and impurities. The silver halide in

the ash is reduced to elemental silver, which precipitates

to the bottom of the smelter by virtue of its greater

density. The metal and the metallic oxides in the ash are

solubilized in a fluid slag consisting of excess reducing

chemicals and fluxing chemicals. The charge is heated by

three graphite electrodes that are immersed into the slag

bath.

A schematic of the smelter is shown in figure 1.1. The

smelter is a cylindrical steel vessel lined with refractory

in the hearth and on the upper sidewalls. The unit is

covered by a refractory brick cover through which the three

electrodes are placed, the feed is introduced, and the

exhaust gases are drawn out. The three graphite electrodes

are lowered into the slag layer and the slag is heated by

virtue of its resistance to the flow of electrical current

through it. The silver, at the bottom of the slag layer,

is heated by conductive heat transfer from the slag layer.

The smelting process is a continuous operation where

premixed silver rich ash and fluxing and reacting chemicals

are added through the top of the smelter. When the smelter

becomes full, the slag is superheated to complete the

reduction reaction and to thin the viscosity of the slag in

order to allow as much silver to precipitate to the bottom

of the smelter as possible. A refractory plug is removed

from the slag taphole and the slag is poured out of the

smelter into an indexing casting machine. Enough slag is

left in the smelter to allow the electrodes to be immersed

into the bath and the feeding process is resumed. This

process is repeated a number of times until enough silver

TWO OF THREE GRAPHITE

ELECTRODES CONCENTRIC

TO SKELTER CENTERLINE

REFRACTORY

LINED

COVER

STEEL

SHELL

REFRACTOR*

SIDEV/ALL

SILVER

TAPHOLE

AND SFOUT

REFRACTORY LINED

HEARTH

FIGURE 1.1

CROSS SECTIOK VIEW OF ELECTRICAL RESISTANCE SHELTER

has accumulated at the bottom of the smelter to justify a

silver pour- After the last slag pour, the remaining slag

is superheated to bring the silver to a temperature where

it is fluid enough to allow it to be underpoured from

beneath the slag and not solidify in the ladle on the

casting system. The smelter is tilted up to pool the

silver in the corner of the refractory crucible, a

refractory plug is removed, and the pour is started. At

the end of the silver pour, the feeding cycle is restarted

and the process is repeated.

The principle area of concern of this thesis is the

temperature profile of the silver at the bottom of the

smelter. The temperature of the silver layer is of primary

concern due to:

1. If the silver is molten and fluid then

mass transfer of impurities can take place out of the

silver into the slag layer across the slag/silver

interface. If the silver is solid or crusted over then

mass transfer cannot take place.

2. When the slag is superheated, before a

silver pour, the following occurs:

A. the solubility of impurities in the

slag and silver layers is altered potentially resulting in

impurities precipitating from the slag layer into the

silver layer.

8

B. the slag viscosity is reduced

allowing for greater penetration of the slag into the

refractory and subsequent greater solubilization of the

refractory into the slag.

C. smelter capacity is lost as the unit

is not fed during this time.

D. additional costs are incurred for

the electrical energy.

3. Overheating of the slag and silver layers

can result in damage to the refractory sidewalls, the

hearth and the molds on the casting machine.

Presently, no method exists to give a direct indication

of the temperature of the silver layer. When superheating

the slag layer prior to a silver pour the operators must

use their judgement based upon the following factors; the

amount of residual slag in the smelter, the thickness of

the refractory sidewalls, history of the smelt (downtime,

power and feed levels, etc.), exhaust temperature, and slag

top temperature. This often ends in abortive attempts to

pour the silver which is not fluid or molten or too hot to

be properly handled in the casting equipment.

Heat is transferred from the slag to the silver layer,

which loses heat through the refractory hearth to the steel

shell, where it is lost to the ambient environment by

radiation and convection. The amount of heat lost by the

silver is greatest at the junction of the refractory

sidewalls and the hearth where the conductive path to the

steel shell is shortest. If enough heat is lost from the

silver at this point a skull of solidified material may

exist at the outer radius of the silver layer that prevents

it from being poured out of the smelter. This skull may

exist even though the silver inside of the skull may be

molten.

The layout of the refractory at the bottom of the

smelter is shown in figure 1.2. The hearth is composed of

three different materials:

1. Cured refractory brick - the brick is laid in

a stadium design where the length of the brick is shortened

at the outer radius of the dished head to match the

curvature of the head.

2. Plastic-fill refractory- this formable

refractory material is used to fill in the space between

the brick and shell.

3. Chrome-fill refractory- this powdered

refractory will allow for thermal expansion of the

refractory brick and plastic.

t

SKELTER

STEEL

SHELL

REFRACTORY

3LDEWALL

CURED

REFRACTORY

BRICK

PLASTIC

REFRACTORY

FIGURE \.g_

DETAIL OF REFRACTORY HEARTH OF ELECTRICAL RESISTANCE SMELTER

11

The type of refractory materials, their thickness, and

their relative positions in the hearth can be varied to

modify the temperature of the silver layer. Increasing the

thickness of the materials will reduce the heat loss but

will also reduce the capacity of the smelter. Installing

refractory materials with lower thermal conductivities will

decrease the heat loss from the shell to the enviroment.

This will result in greater penetration of the molten

silver into the pores and joints between the bricks.

Insulating materials below the brick layer may drive the

temperature profiles to the point where the silver may

become molten at the bottom of the refractory bricks. If

this happens the silver will pool, float the lower density

refractory bricks, and a catastrophic failure of the hearth

will result. The benefits of decreasing the heat losses

therefore have to be weighed versus the effect of the

problems listed above.

The intent of this thesis is to derive a model to

mathematically analyze the temperature profile of the

silver layer and the refractory hearth to determine the

temperature profile of the silver in the present hearth

design. The effect of varying the thickness of different

refractories, with different thermal properties, in varying

geometries could then be fed into the mathematical model to

determine the optimum materials and arrangements to be

employed to in order to keep the silver molten and fluid at

normal smeltingtemperatures.

12

2.0 DETAILED PROBLEM DESCRIPTION

The details of the layout of materials in the hearth in

the electrical resistance smelter are shown in figure 2.1.

The hearth is composed of refractory materials. The one

inch thick carbon steel smelter shell is lined with a

chromium oxide granular refractory that serves as a filler

layer and as a layer to accommodate thermal expansion of

the other refractory materials in the hearth. Just above

the chrome-fill refractory is a layer of 70* aluminum oxide

plastic refractory. The plastic is installed in a wet clay

form that is rammed into place with pneumatic hammers and

is formed to receive the rectangular cured refractory

bricks that sit above it. The purpose of the plastic

refractory is to fill the space between this dished shaped

shell and the rectangular bricks. The cured bricks are

made of 85* aluminum oxide and 10* chromium oxide and are

fitted into the hearth in a stadium course design to match

the decreasing profile of the shell bottom and to form a

flat horizontal surface on which the refractory sidewalls

are built up and on which the melt is held.

t

SKELTER

STEEL

SHELL

REFRACTORY

3LDEWALL

CURED

REFRACTORY

BRICK

PLASTIC

REFRACTORY

FIGURE Z.

DETAIL OF REFRACTORY HEARTH OF ELECTRICAL RESISTANCE SHELTER

14

The problem that arises is that at the normal smelting

temperatures of 1800-2200 degrees Fahrenheit (slag top

temperature) the silver is not molten or fluid. It is

desirable to have the silver molten because mass transfer

of impurities can take place between the silver and slag

layers. Heavy metals such as antimony, lead, copper, and

metallic sulfides can exist in a layer between the slag and

the silver layer. Due to their density being less than the

silver they may float on the silver and not be solubilized

to any large extent. These metals and metallic sulfides can

become trapped in the solid or semi-solid silver layer as

silver precipitates from the slag. During the short heatup

time before a silver pour enough time may not elapse to

allow the impurities to separate from the molten silver-

If the silver were always kept molten then equilibrium

would be reached.

Also, the slag must be superheated to drive thermal

energy down into the silver layer to completely melt it.

Superheating the slag results in higher operating costs for

electrical energy needed to melt the silver and a loss in

capacity while the slag is superheated. In addition,

superheating the slag results in a modification of the

solubilities in the slag and silver layers potentially

15

providing for a decrease in the quality of the silver by

allowing impurities to be desolubilized in the slag layer

and either precipitate into or become solubilized in the

silver layer. Superheating the slag layer also reduces the

viscosity of the slag thus allowing greater penetration of

the slag into the refractory sidewalls. This results in

greater solubilization of the refractory into the slag and

reduces the life of the refractory lining.

Due to the heat loss from the sidewalls and the smaller

thickness of the refractory hearth at the outer radius of

the smelter, the potential exists that the silver is molten

at the center and solidified near the sidewalls. When the

operator attempts to tap the smelter the silver cannot be

poured out from the smelter due to this skull or membrane

at the periphery of the silver layer. This causes abortive

attempts to tap the smelter which results in a loss of

capacity. Mechanical means of mixing the silver layer,

such as gas injection, to melt this skull are not feasible

due to the spatial constraints around the smelter.

There are two methods of keeping the silver molten at

normal operating temperatures. One is to reformulate the

composition of the slag so that it has similar viscosity

and solubility parameters to those of the existing slag, at

a temperature high enough to keep the silver at the bottom

16

of the smelter molten. The other is to provide enough

thermal insulation in the hearth and lower sidewalls to

raise the temperature of the silver above its melting

point. The second method is comparatively easy to do and

does not involve considerable expense while the first

method would encompass a detailed, expensive program.

Modifying the hearth design does, however, involve

making sure that the integrity of the hearth is

maintained. Catastrophic results can occur if the hearth

fails and allows the hot metal and slag to come into

contact with the steel shell. The steel shell would lose

its strength if heated to normal smelting temperatures and

could mechanically fail. This could present a serious

problem to personnel and adjoining equipment.

In a hearth such as the one under consideration the 85*

aluminum oxide(alumina)- 10* chromium oxide (chromia) fire

cured brick is employed because it is relatively insoluble

to the slag and the silver and will provide a mechanically

sound hearth. The problem, however, arises at the joints

of the bricks where a mortar is used to join and fasten the

bricks. These joints are the weakest portion of the hearth

and will allow much easier penetration and

17

solubilization of the refractory by the slag and silver.

At normal smelting temperatures silver and slag (primarily

silver because it occupies the hearth the bulk of the time)

penetrate the semi-porous firebrick and the joints until

they reach the solidification point. This point is where

the temperature of the refractory material is the same as

the melting temperature of the silver or slag. The highly

conductive silver reduces the net thermal conductivity of

the brick further lowering the solidification position.

In the design of a refractory hearth it is of the

utmost importance to control the position of the

solidification point. In the case of the hearth under

consideration, the solidification point must be above the

firebrick/plastic refractory interface. If the

solidification point is below the interface then molten

silver will exist at the interface. Due to the density of

the silver being so much greater than the density of the

refractory (650 versus 180 pounds per cubic foot,

respectively) the molten silver can float the refractory

brick out of the hearth. The full weight and temperature

of the silver layer will then be against the plastic

refractory. The thermal resistivity of the plastic and

chrome-fill refractories will allow the steel shell to rise

to a temperature where it will soften, lose its strength,

and catastrophically fail.

18

There are four potential design techniques for the

hearth under consideration. The first is to minimize the

thermal resistivity of the materials underneath the bricks

so that the solidification point is above the bottom of the

bricks. While this technique provides for the maximum

safety in the hearth it also has the disadvantage of

maximizing heat loss through the bottom of the hearth by

decreasing the total thermal resistivity of the hearth.

The second method is to install materials whose thermal

conductivities are lower than the existing bricks at the

top of the hearth to raise the solidification point and

minimize heat loss. This will minimize the flow of thermal

energy from the silver, but can lead to a catastrophic

failure. The refractory bricks that are used to hold the

slag and silver at the top of the hearth are high density

bricks that are relatively non-porous and prevent the

silver from penetrating through the brick. The high

density brick, however, has a relatively high thermal

conductivity that maximizes heat loss. If an insulating

refractory brick were used at the top of the hearth it

would lead to a quick failure of the hearth. The

insulating refractory materials have a relatively low

density and high porosity. The silver and slag can easily

19

penetrate the insulating brick and either rapidly

solubilize it or greatly increase its net thermal

conductivity and quickly bring the solidification point to

the refractory materials interface.

The third technique is to minimize heat losses from the

shell to the environment by providing an insulating layer

below the brick layer- This, however, will cause the

solididfication point to move downwards towards the

interface. In the event that the solidification point

descends below the bricks and the molten silver is allowed

to come into contact with the insulating material it will

quickly penetrate it for the reasons listed in the

preceding paragraph. This can lead to a catastrophic

failure of the hearth. In the preceeding explanation of

the potential failure of the existing hearth when the

higher thermal conductivity refractory brick failed there

was still a fair amount of plastic refractory left in the

hearth that would keep the molten silver away from the

steel shell. The failure of the steel shell would take

some period of time and the smelter could be potentially

shutdown and cooled before the refractory failed. In this

example the silver would penetrate the insulating material

so quickly that the failure of the steel shell

20

would occur before the smelter could be cooled down. For

this reason, a backup safety lining is installed below the

insulating layer that is of sufficient strength and thermal

properties that in the event of a failure of the brick and

insulating layers the backup lining will protect the steel

shell against the slag and silver until the smelter can be

shutdown. Thermocouples can be buried in the backup lining

that would detect the rise in temperature of the backup

lining, when the primary lining fails, and warn the

operator that the smelter must be immediately shut down.

This technique has the disadvantage of increasing the

hearth thickness which reduces the capacity of the

smelter. The advantage is that the loss of energy from the

bottom of the hearth is minimized.

The fourth technique is to install a plastic monolithic

lining at the bottom of the hearth. The formable plastic

refractory is pneumatically rammed into the bottom of the

hearth and then cured in the smelter. The resultant

monolithic hearth then has no joints. Insulating materials

and a backup lining can be added below the plastic

refractory depending upon the thickness of the plastic and

the location of the solidification point. The disadvantage

of this design is the capital cost for the equipment to

cure

21

the plastic in the hearth. Also, If the plastic is not

cured correctly then moisture will remain in the plastic

hearth. Due to the extremely high vapor pressure of water

at normal smelting temperatures, moisture poses a serious

personnel safety and equipment availability concern if it

exists below the slag and silver. The advantage is a

mechanically integral hearth with no joints for easy slag

or silver penetration.

The potential therefore exists, as can be seen from the

previous discussion, to redesign the hearth of the electric

smelter in order to keep the silver layer molten and fluid

at normal smelting temperatures while maintaining the

integrity of the hearth.

22

3.0 LITERATURE SEARCH

3.0 Introduction to the Chapter

An extensive literature search was conducted to

determine if any previously defined models could be

employed for the smelter under consideration. No useable

models were found but similar models revealed important

information for the development of the new model. A

three-dimensional model of a rectangular glass melting

furnace, which is simpler than the smelter under

consideration, revealed that the complexity of a

threedimensional model was well beyond the scope of this

work. The model took over seventy minutes to run on a CDC

6500 computer. Five two-dimensional models were found in

the literature of electrical resistance smelters or melting

processes. The models revealed that a simulation of the

electromagnetic and convective induced flow currents in the

slag and metal layer was also well beyond the scope of this

work. Due to the presence of the three concentric

electrodes in the smelter under consideration it is not

felt that a two-dimensional model of these parameters would

be accurate.

23

The two-dimensional models, however, also revealed

that, in the radial direction of the cylindrical smelter,

the temperature of the slag/metal interface is essentially

constant. This allows the slag and metal layers to be

modeled independently of each other. Since this work is

not concerned with modeling the slag layer, the silver

layer and the refractory hearth will be modeled assuming

the slag/silver interface to be at a constant temperature

throughout

3.1 Introduction to Mathematical Models

The mathematical models are of the temperature and the

convective and electromagnet ically induced flow profiles in

rectangular and cylindrical smelters. One of the models is

of a three-dimensional rectangular glass melter and from

the complexity of the model it was determined that a

three-dimensional model of the non-axisymmetric smelter

under consideration would be far beyond the scope of this

thesis. The two-dimensional axisymmetric models revealed

that the temperature of the slag/silver interface is

constant in the radial direction. This allows the silver

and slag layers to be modeled independently.

24

3.1.1 Three Dimensional Model Including the Effect of

Convective and Electromagnetic Fields

A. Abstract

Chen andGoodson1 formulated a mathematical model of

the three-dimensional temperature and convective flow

profiles of an electric resistance glass furnace. The

model involves the numerical solution of the energy,

momentum transfer and the heat generation equations. The

article gives the assumptions, the defining equations and

boundary conditions for the cubic furnace. Results showing

plots of the temperature and convective flow profiles in

the furnace are shown. The authors conclude that the

results agree quantitatively with characteristics of an

operating furnace, that the three dimensional model is

absolutely needed, and that the major limitation on the

model is the large amount of computer time needed for a

solution. The major limiting assumption appears to be that

the viscosity was assumed to be constant with respect to

temperature.

25

B. Furnace Descripti on

The furnace considered in the model is an electric

glass melting furnace as shown in figure 3.1.1.1

/ TMeltmir end Channel

Cilass level

1 orehearth

Figure 3.1.1.1 - Electric Glass Melting Furnace

The melting end, where the raw materials are fed, is the

primary concern of this model. The melting box is a 2.12

meter cube where heat is generated by the flow of

electricity from electrodes, in the refractory lined walls,

through the glass.

C. Model Assumptions

The following major assumptions are considered in the

model:

1. Glass is a Newtonian fluid (experimentally

verified) .

26

2. Flow is laminar.

3. Viscous heat dissipation can be neglected.

4. The Boussineq approximation is valid (the maximum

temperature variation is small compared to the reciprocal

of the cubic thermal expansion coefficient).

5. The electrical resistivity of the glass is assumed

to be constant.

6. Glass is optically thick; therefore, there is no

radiative heat transfer in the glass.

7. All fluid properties are constant except that the

buoyancy effect due to density difference is accounted for.

8. The furnace is at steady state with no glass

throughput.

9. The complications caused by melting and varying

input batch blanket thickness at the top surface has been

neglected.

D . Equational Setup and Method of Solution

The conservation equations of mass, energy, momentum,

and electric charge were formulated in terms of the system

variables. The equations were then recast in vorticity

transport form and expressed in terms of the nondimensional

Prandtl, Grashof, and Nusselt numbers and the furnace

aspect ratio as the only physical parameters. The method

27

is given by Aziz and Heliums2. The resulting form yields

differential and integral equations of the following form

in three dimensional coordinates: voltage, energy,

vorticity, hydrodynamic vector potential, velocity and

power. The boundary and initial conditions for the

equations are given by the authors.

The method of solution is an alternating direction

implicit finite-difference method3 that is applied to the

voltage, energy, vorticity and hydrodynamic vector

potential equations. An approximate solution is obtained

over an 11 point grid system in the glass in a finite

time. The voltage equation is first solved to determine

the energy input to the glass. The assumption of constant

electrical resistivity allows the voltage equation to be

uncoupled from the energy equation. Knowing the voltage

distribution, the power equation is solved to yield the

power intensity. An algorithm is solved over a finite time

differential until steady state is reached. The values of

the temperatures, vorticities, hydrodynamic vector

potential, and the velocities are fed in at their initial

values. The algorithm solves the energy equation for the

temperature field and the vorticity transport equation for

the velocities at all interior grid points, the

28

hydrodynamic vector potentials are solved knowing the

vorticities, and the velocities and boundary vorticities

are then calculated. The boundary and initial conditions

are given in the Appendix at the back of the article.

E . Results of the Program

The outputs of the llxllxll grid point simulation are

given in terms of graphic plots of the power intensity,

hydrodynamic vector potential, velocity, and temperature

fields at differing planes throughout the melter. Figure

3.1.1.2 shows the contour plot of power intensity at one of

the electrode surfaces that are at two opposite walls. The

pattern of intense power input at the edge of the

electrodes agrees well with wear patterns on industrial

smelters .

-i 1 1 1 1 1 1 1f-

Figure 3.1.1.2 - Contour Plot of Power Intensity

at the Vertical Surface

29

Figure 3.1.1.3 shows equipotential lines of the

hydrodynamic vector potential at the plane 0.1 normalized

units distance from and parallel to the electrodes. The

core or roll is expected due to the electric current taking

the path of least resistance through a central core.

Figure 3.1.1.3 - Equipotential Lines of the

Hydrodynamic Vector Potential

Figure 3.1.1.4 shows a plot of the vector sum of the y

and z component velocities at the plane 0.1 normalized

units distance from and parallel to the electrodes. A

picture of the flow profile can be obtained from this plot

30

\ \ t

1t \ \ t

//ft

t ft t

/ t 1 f

ft t t

' > t f

y

* n

Figure 3.1.1.4 - Flow Velocity in the Y-Z Plane

Figure 3.1.1.5 shows a plot of the isotherms in the

plane 0.1 normalized units from and parallel to the

electrode surface.

31

I .\M)( |'5/,M (

I 5Wr(

-I5()C

400C

: i ( \j:

Figure 3.1.1.5 - Isothermal Plot in the X=0.1 Plane

The authors state that the verification of the output

data was not made due to a lack of data on operational

furnaces but the gross quantities such as temperatures and

power consumption agree well with industrial estimates and

measurements .

The authors list the factors that affect the accuracy

of the results. The first is the estimation of parameters

and physical properties such as constant viscosity, no

system throughput, and no insulating layer at the top of

the melt due to unmelted glass are deemed to be the most

32

critical assumptions. Chen4 has done work on the effect

of the physical properties on free convection but no work

has been done to date on industrial furnaces. Data is

available for the effect of temperature on the heat

capacity, thermal conductivity, and viscosity of the slag

employed in the smelter under consideration in this

thesis. No data is available on the effect of temperature

on the electrical resistivity of the slag. The second

factor affecting the accuracy is the assumption that the

formulation of the conservation equations and the boundary

conditions were assumed to be less critical than the

previous assumptions. The third is that the spatial

discretion error in the numerical solution, due to the

complexity of the convective flow profile, is assumed to be

small compared to the first assumption.

F. Conclusions

The conclusions of Chen and Goodson are:

1. The results of the model for flow and

temperature profiles agree qualitatively with operating

industrial furnaces.

33

2. For a model with simple electrode placement,

such as the melter, the three dimensional model is

justified. They state that for a complex electrode

placement the method offers even more justification. The

smelter under consideration in this thesis offers a complex

arrangement of three cylindrical electrodes placed

symmetrically in a cylindrical crucible.

3. The large amount of computer time

(approximately 70 minutes on a CDC 6500 computer) needed is

caused by the large Prandtl numbers seen in slow viscous

flow (1 to 10 poise). Chen and Goodson give a detailed

explanation of the problem in the Appendix of their

article. The Prandtl number for the glass slags

investigated by Chen and Goodson was 1000. Since this is

within the Prandtl number range for the smelter under

consideration in this thesis the problem would be

encountered if the slag layer was modelled.

From this article it was determined that a

three-dimensional model of the smelter under consideration

would be well beyond the scope of this thesis project. The

smelter is much more complex than the smelter in this

article because of the three cylindrical electrodes placed

symmetrically into the cylindrical vessel. The problems

that Chen and Goodson had with large computation times

would occur due to the large Prandtl number for the slag

under consideration.

34

3.1.2 Two-Dimensional Axisymmetric Models Considering

the Effect of Convective and/or Electromagnetic Fields

A. Mathematical Model of an Electroslag Refining

Unit

Dilawari and Szekely5 developed a mathematical

formulation to represent the electromagnetic force field,

fluid flow and heat transfer in an electroslag refining

unit. The output includes plots of velocity and

temperature fields in a cylindrical axisymmetric system.

An electroslag refining unit is used to melt or

dissolve an electrode made of a mixture of slag and metal

to separate the two phases. The model used is shown in

figure 3.1.2.1

35

- COOLING MTCM

POWEP

SLWU

OR

RtCTrtCR

->CT1. ROOl.

-JOUOTCD MOOT

Figure 3.1.2.1 - Sketch of an Electroslag

Remelting Unit

The major differences between this model and the

smelter under consideration are first that this model is

axisymmetric. The ESR unit is also water cooled which will

set up higher temperature and lower flow gradients than the

smelter under consideration in this thesis which has a low

thermal conductivity refractory liner cooled by the ambient

air- The feed is introduced as ash feed and the power is

supplied by graphite electrodes. The smelter power is

conducted between the three electrodes and heats the slag

bath by the resistance of the slag layer.

36

The mathematical formulation includes the following

conservation equations set up in the vorticity transport

form:

1. Fluid flow - the equation of continuity and

the equation of motion with the effect of buoyancy and

electromagnetic force fields included

2. Electromagnetic force field - Maxwell's

equation

3. Heat transfer - the convective heat balance

with allowance for eddy transfer, heat generation, and

transport of thermal energy by droplets descending through

the slag layer is solved.

The authors present the detailed formulation of the

equations and the boundary equations with references to

validate the assumptions.

The results are derived in terms of the flow patterns

obtained for the system considering electromagnetic effects

only and also considering electromagnetic and bouyancy

effects together, temperature profile snd isotherms, slag

velocity versus current level, and overall process

parameters such as melting rates.

37

Figures 3.1.2.2 and 3.1.2.3 show the computed

streamline patterns for electromagnetic and electromagnetic

and buoyant effects, respectively, with a current level of

45 KA (rms).

RADIAL DISTANCE (cm)

Figure 3.1.2.2 -

Streamline Patterns

Electromagnetic Effects only

RADIAL DISTANCE (cm!

Figure 3.1.2.3 - Streamline Patterns of

Electromagnetic and Buoyancy Effects

38

It can be seen that considering the effect of buoyancy in

figure 3.1.2.3 shows the formation of a secondary flow

loop. However, the magnitude of the flow caused by the

buoyancy effect is approximately one-tenth of the magnitude

of the elect romagnet ical ly induced flow loop in figure

3.1.2.2.

Figure 3.1.2.4 shows the computed isotherms for the

case considering only electromagnetic effects with a

current level of 45 KA (rms).

RADIAL OISTANCE (cml

Figure 3.1.2.4 - Isotherms in the Slag

and Metal Layers

The large temperature differential at the outer wall is

caused by the water cooled jacket on the ESR unit. It is

expected that this differential would be less in the

refractory lined smelter- Of interest is the fairly

39

constant temperature of the slag-metal interface in the

slag region. The gradient at the wall would not be as high

for the smelter under consideration for the reason listed

above. For the model, an assumption of constant slag

temperature at the slag-metal interface would be justified

by this result.

The authors also present additional plots that show:

1. With decreasing current level the dominance of

the electromagnetically driven fluid flow over the bouyancy

induced flow is reduced

2. The ratio of effective thermal conductivity to

molecular conductivity is much greater than one and the

flow is turbulent

3. The effect of droplet formation in the slag

pool is also demonstrated. This is not of interest, due to

the graphite electrodes used in the smelter, but could be

of potential interest if the ash and chemical feed to the

smelter is ever modeled.

Figure 3.1.2.5 shows the effect of current on the

velocity in the slag layer.

40

o a*

ic

CURRENT (KA)

Fig. 13The effect of current on the maximum value of the

linear velocity in the slag phase; ( ) nonisothermal condi

tions, ( ) isothermal conditions. Other property values used

Figure 3.1.2.5 - Effect of Current Level

on the Linear Velocity in the Slag

In the event that the power input to the slag layer is ever

modeled, this paper brings up some useful observations. Of

primary importance would be the temperature and flow

profile in the delta region between the electrodes. A

tremendous amount of power is fed into this small region.

B. Model of the Electroslag Welding Process

Dilawari, Szekely, andEager5 took the computer model

developed for the electroslag refining unit in the previous

article6and adapted it to the case of electroslag

welding. The authors set forth the two dimensional

41

mathematical model employing Maxwell's equations, the

turbulent Navier-Stokes equations, and the convective heat

balance equation.

Electroslag welding, which is a similar process to

electroslag remelting, is used to produce relatively

defect-free welding joints at fast deposition rates. The

authors set forth to understand the process in terms of the

relationship of the size of the weld pool, the thickness of

the weld joint, the system geometry, current, voltage, and

flux conductivity to obtain fundamental understanding of

the process. A model of the electroslag welding process is

as shown in figure 3.1.2.6.

electrodeguide

electrode tube

SLAG POOL

METAL POOL

PLATE 2

COMPLETED WELD

PLATE I

Figure 3.1.2.6 - Electroslag Welding Process

42

The derivation of the governing equations are given in

both cylindrical and rectangular coordinates. The

equations are formulated separately for the slag pool, the

metal pool, electrode, cooling jacket, and the solidified

weld. The critical assumptions are: steady state

conditions, all physical properties are independent of

temperature (except density), flow is laminar, and the

process is axisymmetric. The primary objective was to

assess the effect of the electromagnetic and buoyancy force

fields on the fluid flow and temperature distribution.

This analysis is only of the cylindrical case and the

results for the rectangular system will not be discussed.

The primary difference between this model and the model in

the previous article is the distance from the electrode to

the container wall. The article by Dilawari and Szekely5

used an electrode to crucible diameter ratio of 5 to 7,

while this article has only a ratio of 1 to 10. This study

would be thought to more closely simulate the delta region

between the electrodes where the electrodes are close to

one another while this article would simulate the space

between the electode and the refractory crucible.

Maps of the computed streamline patterns of the fluid

flow and isotherms for the cylindrical system are shown in

figures 3.1.2.7 and 3.1.2.8.

43

Slog-metol^,

interfoce

'35O 0.3 0.6 0.9 1.2 IS

RADIAL DISTANCE (cm)

Figure 3.1.2.7 - Streamline

Flow Pattern

Slag-metal L-iinterface

0 0.3 06 09 12

RADIAL DISTANCE (cm)

Figure 3.1.2.8 - Computed

Isotherms

As in the previous article, the authors conclude that

the electromagnetic forces are the dominant force for fluid

flow. The lower velocities in this model can be attributed

to the smaller electrode diameter to crucible diameter

ratio and the lower current applied to the system (376 amps

versus 45,000 amps, respectively). Figure 3.1.2.7 would

show the maximum velocity of the slag to be in the range of

0.02 kg/sec.

44

Of interest in the given streamlined flow patten figure

is the depth of electrode penetration in the slag and how

close it is to the metal layer. In this model, the

electrode bottom is two-thirds of the way through the slag

layer, and its minimal effect on the flow in the metal

layer can be seen. The maximum velocity in the metal layer

is one-fifth of that in the slag layer. With the water

cooled casing it is felt that the flow in the metal layer

is controlled more by buoyancy forces than by

electromagnetic forces due to the higher thermal

conductivity of the metal. The location of the maximum flow

streamline being next to the container wall would tend to

support this.

In the smelter under consideration the electrode

penetrates only one-fourth to one-half the way through the

slag layer and where in the model the electrode sits only

0.5 cm above the metal layer it sits 18 to 36 inches above

the metal layer in the smelter- For this reason it would

be thought that the effect of electromagnetic forces on

fluid flow in the silver layer in the smelter would be

minimal. Also, since the smelter investigated in this

thesis is not water jacketed but is jacketed by a low

thermal conductivity refractory the buoyancy induced

currents would be expected to be orders of magnitude lower

than those shown in the model.

45

Of interest in the isothermal plot is the temperature

profile of the slag and metal layers in the vicinity of the

slag/metal interface. Again the plots show fairly

isothermal behavior along the interface. The

non-isothermal behavior under the electrode and at the wall

would not be thought to exist in the smelter for the

reasons noted in the previous paragraph.

The authors state that the vast majority of the heat

flow to the metal layer is by the descending metal droplets

from the consumable electrodes. In the smelter under

consideration droplets of silver would be precipitating

from the feed ash but the electrodes are made of graphite.

During the heatup time for slag and silver pours there is

no silver rich ash being fed to the smelter so the flow of

thermal energy to the silver layer is by conduction and

convection only. This would drastically reduce the rate of

heat flow to the silver layer by this theory.

Fluid Flow in the Electroslag Remelting Unit7

This paper is concerned with the effect of fluid flow and

droplet formation in the Electroslag Remelting Process

(ESR) . The process is the same as in article (5) in this

46

section. The paper is based upon visual observations of a

transparent model fabricated by the author as shown in

figure 3.1.2.9.

consumable

electrode:

r^ "r

MOLD

LIQUID SLAG

LIQUID METAL

SOLIDIFICATION

FRONT

INGOT

ASBESTOS SEAL

GRAPHITE CHILL

WATER COOLED COPPER BASE

Figure 3.1.2.9 - ESR Unit

The bulk of- the work is concerned with the effect of

droplet formation on the ESR process. Since the smelter

under consideration employs non-metallic electrodes, that

part of the work will not be considered in this analysis.

47

If the feed of silver rich ash to the smelter is modeled,

the effect of silver droplets on heat transfer and fluid

flow needs to be reviewed in this article.

The author experimented with consumable and

non-consumable electrodes and found the results differed

sharply. The observation with consumable electrodes had

agreed qualitatively with observations by Dilawari. et.

al.5'6. The results from the non-consumable graphite

electrodes showed that a hot spot was formed around the

electrodes. In the ESR process the heat around the

electrode is dissipated by the melting of the electrode.

The hot spot around the graphite electrode results in the

formation of convection currents in opposition to the

electromagnetic currents. A turbulent region existed

around the electrodes while the remaining slag layer

remained largely quiescent.

The author proposes equation 3.1.2.1 to determine a

qualitative value of the degree of mixing in the slag

layer. The equation is derived from a pressure balance of

a point just below the mixed region and in the cooler

quiescent region.

48

X = k * (1**3)* (1/A1 -1/A2)**2 Equation 3.1.2.1

where:

X = depth of the stirred region

k = thermal conductivity of the slag

Al= electrode area

A2= area of the metal layer at the bottom of the

crucible

I = current

if X is less than the diameter of the electrode, the

turbulent region is localized around the electrode; and if

X is equal to or greater than the diameter of the

electrode, then the slag layer is essentially uniform

throughout the volume.

The smelter under consideration is somewhat different

than this model, with three electrodes concentrically

mounted in the crucible. A mean electrode diameter

formulation of the three electrodes could allow this

formulation to be used.

Due to the metal layer being opaque, the author was not

able to make observations of flow and temperature profiles

in the metal layer- He does, however, make several

observations on the entry of the metal droplets into the

pool that would be of interest if the slag and liquid metal

pool were simulated.

49

The author notes the existence of a cool boundary layer

at the slag/metal interface and that the boundary seems to

exist as a rigid boundary. The slag and metal layers

appeared to move independently of each other. The presence

of this boundary layer would support the assumption of the

interface in the smelter being considered isothermal in the

radial direction.

D. Temperature and Flow Profiles in an ESR Unit

This paper presents the modelling of temperature and

velocity fields for an ESR unit. The Navier-Stokes

equations and heat transfer equations are solved for the

cylindrical axisymmetric model. A detailed derivation of

the equations and boundaries are given in the article.

The calculated flow lines show the effect of electrode

penetration and current levels on the system. These

outputs are shown in figures 3.1.2.10 and 3.1.2.11.

50

Figure 3.1.2.10 - Flow Profiles With Current Level

Increased by 10 and 100 Times

Figure 3.1.2.11 - Flow Profiles With Current Level

Increased by 10 and 100 Times

51

From figure 3.1.2.10 the effect of raising the current

levels by a factor of ten for the second plot and a factor

of 100 for the third plot is shown. The effect of raising

the currents level is to compress and accelerate the flow

core. Figure 3.1.2.11 shows the effect of increasing the

electrode penetration from 9 to 40 mm. It can be seen

that, compared to figure 3.1.2.10, the effect is to

compress the flow regime and accelerate the flow core. The

flow core is also shifted to the right towards the

container wall as the current level and electrode

penetration are increased.

The smelter under consideration allows for variable

penetration of the electrodes into the slag bath at

variable voltage and current levels. If the heat transfer

in the slag layer is ever simulated this article would be

useful .

E. Metal Layer Growth in an ESR Unit9

This paper is concerned with the growth of the metal

ingot in the bottom of the ESR unit. Liquid metal is

introduced into the slag layer and the solidification and

52

segregation of the metal in the ingot are studied to

determine the effect of process parameters on metal ingot

quality. This article would not be of direct use in this

simulation, but would be useful if the effect of the silver

precipitating through the slag layer was studied.

3.1.3 Models Considering All Layers as Solids and

Considering No Effect of Convective and Electromagnetic

Fields

This mathematical model is of a three electrode

circular furnace. This model by Kaiser andDowning10

predicts the hearth temperature distribution resulting from

different operations. A sketch of the hearth arrangement

is shown in figure 3.1.3.1.

53

Figure 3.1.3.1 - Hearth Arrangement

For this model the temperature of the metal/hearth

interface is assumed to be constant in the radial

direction. The effect of the heat loss from the bottom of

the hearth by radiation and convection is considered in the

model, which sets up a gridwork in the hearth and uses a

forward-differencing technique to achieve the solution.

The model starts out by assuming values for the temperature

distribution and iterates the solution of the equations

until steady-state is achieved to yield the correct values.

54

The effect of temperature perturbations is then

simulated by making the appropriate changes in the

variables and plotting the transient temperature

distribution as a function of time. One important

parameter was the effect of losing the underhearth cooling

fan. If the fan is lost, the steelwork (grillage) under

the furnace can overheat and potentially undergo permanent

or catastrophic damage. A plot of the effect of losing the

fan and leaving the furnace energized is shown below. The

plot reveals that the steel temperature rises above the

maximum permissible temperature.

80C

l- 60C "

Z/TZTIZ/C/'Z/:/

20C :

Maximum Allowably C/ _

Interface Temperature

Maximum Allowable

Steel Temperature Range

Carbon/Refractory Interface @ C

Top of Grillage 9 C

YYY y yyyyy.

Wall @ Hearthllne

-i_

12 16 20 24 28 32 36

Time After Upset, Hr.

Fig.7Temperature history following a lo o( forced convection with furnace energized

48

Figure 3.1.3.2 - Temperature History Of Hearth

55

Plots are given for different variable pertubations with

different materials lining the hearth.

3.2.0. Physical Properties of Metallurgical Slags

This section of the thesis will summarize results of

physical properties of metallurgical slags similar to the

slags employed in the smelter under consideration.

A. Thermal Conductivity and Heat Capacity 1 i

Kishimoto, et . al. measured the thermal conductivity and

specific heat of metallurgical slags by the laser flash

method at temperatures from room temperature to 1750

degrees Kelvin. The slags were calcium

oxide-silica-alumina and sodium oxide-silica base

materials.

The data for thermal conductivity is shown in table

3.2.1. It can be seen that at low temperatures the thermal

conductivityincreases linearly with increasing temperature

while at high temperatures the thermal conductivity

56

TABLE 3.2.1 - THERMAL CONDUCTIVITY OF SLAGS SIMILAR

TO THOSE USED IN THE ELECTRICAL RESISTANCE SMELTER

Composition (6K) Thermal Conductivity(W/m-_Kl

40% Calcium Oxide 800 0. 883+2 . 87*10E-4*T 0.260+730/T

40% Silica

40% Alumina

45% Calcium Oxide 800 0. 828+3 . 03*10E-4*T 0.196+747/T

40% Silica

15% Alumina

50% Calcium Oxide 830 0 . 757+2 . 74*10E-4*T 0.156+737/T

35% Silica

15% Alumina

29.0% Sodium Oxide 650 1 . 11+1 . 0*10E-4*T 0.114+792/T

71.0% Silica

38.6% Sodium Oxide 650 1. 08+1. 0*10E-4*T 0.089+734/T

61.4% Silica

57

decreases proportionally with temperature. The thermal

conductivity also increases with decreasing basicity.

The authors also reference three articles which contain

the density of the slags in the liquid state by Kammel12,

Bockris13, and Haggins14. The author of this thesis

was not able to obtain any of these articles.

B . Thermal Conductivities15

The thermal conductivities of sodium oxide and calcium

oxide based slags were measured by the hot wire method from

room temperature to 1500 degrees centigrade. The results

are given in graphical form in figures 3.2.1, 3.2.2, and

500 1000

Temperature CO

1500

Figure 3.2.1 - Thermal Conductivities of

Sodium Oxide-Silicate Slags

58

30-

2.5-

20-

* 15

E

1.0

0.5

& o a)Ca0-A0SO2-20AI203

25CaO-60SiOj-15AI,03& a 55Ca0-45SiO2

& * 50CaO-50AI203

Fused SO?

Sakurayaetal. / %.^ \\

m.p.t x

500 1000

Temperature (*C)

1500

Figure 3.2.2 Thermal Conductivity of Calcium

Oxide-Aluminum Oxide-Silicate Slags

50Na20-50Spj

200 400 600 800

Temperature (*C)

1000 1200 1400

Figure 3.2.3 - Thermal Conductivity of

Sodium Oxide-Silicate Slags

59

4.0 THEORETICAL ANALYSIS

A finite-difference analysis model was formulated to

determine the flow of thermal energy through the hearth of

the electrical resistance smelter. The goal of the model

was to develop the temperature distribution in the hearth

and the total heat loss from the smelter to the environment

employing varying refractory materials and geometries in

the hearth. The optimum selection of refractory materials

and placement could then be determined to keep the silver

layer at its highest temperature and reduce heat loss while

maintaining mechanical and structural integrity in the

hearth.

In order to reduce the complexity of the model, it was

decided that the slag layer would not be modeled. The

model would therefore consist of the silver layer, the

lower refractory sidewalls, and the refractory hearth. The

assumption of not modeling the slag layer is felt to be

valid due to the work of Dilawari andSzekely5'7

and

Dilawari, et. al.6,referred to in the literature search

that shows the metal/slag interface temperature to be

constant from the axial centerline to the outer radius of a

60

cylindrical smelter. The effect of the large radial and

axial temperature differentials in the slag layer therefore

does not have an effect at the metal/slag interface. This

interface temperature could then be fed into the model as a

variable, but the temperature will not be derived from the

effect of heating the slag layer with the electrodes.

The work of Chen and Goodson1showed that a

threedimensional model of a simple cubic glass furnace was

well beyond the scope of this thesis. The smelter under

consideration in this thesis is more complex than the glass

furnace that was modeled due to the presence of the three

non-concentric electrodes and the need to analyze the heat

loss through the refractory. The previously mentioned

assumption of the slag/silver interface being a constant

temperature coupled with the radial symmetry in the hearth

made the decision to use a two-dimensional model valid.

It was also assumed that due to the thermal

conductivity of the silver being so much greater than that

of the refractory (4.0 versus 0.4 W/cm-K, respectively),

the convective heat transfer in the silver layer would be

negligible. This allows the silver layer to be treated as

a solid layer-

61

The model generated is therefore a two-dimensional

heat flow analysis with the refractory and silver being

considered as solids. The next step was to analyze the

heat inputs and outputs to the model as seen in figure

4.1. The only heat input is from the slag/silver

interface, which is at a constant temperature. The sole

heat loss is from the shell to the ambient enviroment by

convection and radiation. Axial heat transfer in the

refractory sidewalls at the extension of the slag/silver

interface line is assumed to be negligible in order to

simplify the model.

The desired inputs and outputs to and from the model

were then determined. The desired inputs are:

1. the temperature of the slag/silver interface

2. the depth of the silver layer

3. the diameter of the shell

4. the slope of the bottom head of the shell

5. the thickness of the lower refractory

sidewalls

6. the position of various types of refractories

in the lower sidewalls and the hearth

7- the thermal conductivity, as a function of

temperature, of the various refractory materials and the

silver

J

ao

hi

1

0

9<J

<

0

r-

<

ID

vl

0

62

63

8. the temperature of the air surrounding the

shell

9. the temperature of the surfaces that receive

radiant energy from the shell

10. the emissivity of the shell

Employing these variable inputs would allow the model

to be used for the solution of the heat flow problem for

various shell geometries, various depths of the silver

layer, and various types, thicknesses and orientations of

the refractory materials in the lower sidewalls and the

hearth. The only input that will be a function of another

variable is the thermal conductivity of the silver and the

refractory which will be fed in as functions of the

temperatures of the materials. This is necessary due to

the large variation of the thermal conductivity of the

refractory materials with respect to temperature. If, for

instance, a solid plastic refractory were used in the

hearth, then if the top of the hearth were at 2400 degrees

Fahrenheit, then the bottom of the hearth would probably be

in the 300 to 400 degrees Fahrenheit range. For a 90

percent alumina plastic refractory, the thermal

conductivityincreases from 17 to 23 BTU-inch/square

foot-hr-degrees Fahrenheit between 2400 and 400 degrees

fahrenheit, respectively.

64

The desired outputs from the program were two. The

first is a description of the temperature field in the

hearth, and the second is the heat flow from the shell to

the environment by convection and radiation. The

temperature distribution in the hearth is needed to

determine where the freeze point for the metal exists.

The silver, lower sidewalls, and the hearth were then

broken down into a finite-difference analysis. It was

found by analyzing the model that nine distinctive types of

nodes existed in the model. These various types of nodes

are defined and shown in figure 4.2. A steady-state heat

balance was performed on each type of node to derive the

descriptive steady-state heat transfer equation for each

node in the model. The form of the steady-state heat

balance equation is as shown in in equation 4.1.

Sum of the Heat Inputs - Sum of the Heat Outputs = 0

Into the Node Out of the Node

Equation 4. 1

Where the heat flow into and out of the nodes is

defined by Fourier's equation:

Q = -k * Ar * dT Equation 4.2

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66

The derivation of the equations for the nine nodes can be

found in the Appendix of this thesis. The summary listing

of all the the equations for the nodes can be found on

figure 4.3.

The model described by the equation is for a

rectangular node of variable size where the node dimensions

would be fed into the model. The major assumptions that

were used in the derivation were that each node can only be

made of one distinctive material and that all nodes must be

of the same size.

The descriptive simultaneous equations for all of the

nodes form a rectangular matrix equation of the form shown

in equation 4.3

COEFFICIENT

TERMS

NODAL

TEMPERATURES

REMAINDER

TERMS

Equation 4.3

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68

The only unknowns are the temperature of each node and the

thermal conductivity, as a function of the node

temperature, of the material in the node. The solution is

also non-linear due to the presence of the temperature

terms raised to the fourth power that is derived from the

flow of radiant energy from the shell to the environment.

A non-linear solution of the equations would be

difficult because the derivative of each equation would

have to be taken. It was therefore decided to translate

the solution to a linear form by employing the following

algorithm. This algorithm converts the solution to linear

form and uncouples the dependency between the temperature

and the thermal conductivity.

1. Input the desired values of node size, smelter

geometry, refractorymaterial properties and position,

depth of the silver layer, and the temperature of the air

and the emissivity and the temperature of the ambient

surroundings around the smelter.

2. Formulate an initial estimate of the temperature

in the nodes by assuming a linear temperature drop from th

slag/silverinterface to the steel shell, which is assumed

to be at 350 degrees Fahrenheit. The steel shell on a

typical smelter will usuallybe in the 300 to 400 degrees

Fahrenheit range.

e

69

3. Calculate the coefficient terms in the temperature

matrix and the remainder terms using the initial estimate

of the temperature fields in the nodes to calculate the

thermal conductivity of the material in the nodes. Use the

initial estimate of the temperature field to calculate the

values of the coefficients having fourth powered

temperature terms. This transforms the fourth powered

terms into constant remainder terms. The equations are now

in linear form and dependent only upon temperature.

4. Using a linear solution method, such as Newton's

method, solve for the temperature field now that the

coefficient and remainder terms are known.

5. Compare the values of the temperature field

derived to the initial estimated values and if the

difference is not less than a predetermined tolerance

criterion then the algorithm needs to be reiterated from

steps 2 to 6 using the last calculated values of the

temperature field in place of the initially estimated

values .

6. The algorithm is terminated either by the

tolerance criterion being satisfied for all of the nodes or

by a predetermined convergence criterion not being

satisfied.

One problem that may cause the solution to diverge

during the algorithm is the presence of the nodal

temperatures raised to the fourth power. The value of the

70

remainder term associated with this quantity may be so

adversely large as to affect the solution. This may

present a real problem in the first few iterations, when

the value used for this temperature term may be

significantly different than the true value. In order to

check to see if this is happening the solution should first

be obtained negating the effect of radiation from the

shell. This could be done by inputting a value of zero for

the emissivity of the shell. The solution could then be

attempted with the actual value of the emissivity to see if

convergence could still be obtained. If it cannot, then

the solution could be repeatedly iterated slowly increasing

the value of the emissivity to its full value.

71

5.0 PROGRAM STRUCTURE

Using the algorithm for the solution of the

temperature field in the hearth that was derived in the

previous section a Fortran 77 program was written. The

program contains the desired inputs and outputs listed in

the prevoius section. The program is composed of a master

program that accesses a network of 11 interwoven

subroutines. This was done outside the scope of the thesis

and is included as extra work.

The structure of the program is shown on figure 5.1 on

the next page. An explanation of the program and

subroutines is listed below:

PROGRAM SMELTER - This master program controls the

entire program flow. It accesses the INPUT and OUTPUT

subroutines and subroutine CONTROL, which controls the

solution of the temperature distribution. Subroutine CALC

is also called to insure that the input data is in the

proper format.

SUBROUTINE INPUT - This subroutine reads all of the

data in from a DATA file and stores it in common block

form. The subroutine also prints out the data fed in.

72

FIGURE 5.1 PROGRAM STRUCTURE

SUBROUTINE <

INPUT

> PROGRAM <

SMELTER

> SUBROUTINE

OUTPUT

INPUT DATA

> SUBROUTINE <

CONTROL

> SUBROUTINE

CALC

SUBROUTINE

TINT

SUBROUTINE

LIBRARY

SUBROUTINE

COEFMAT

SUBROUTINE

REFPOS

SUBROUTINE

COEFCAL

SUBROUTINE

INTCEPT

SUBROUTINE

THERCON

73

SUBROUTINE CALC -

This subroutine analyzes the data

fed in to make sure it is in the proper format and that it

satisfies certain fundamental criteria established by

assumptions used in the development of the

model.

SUBROUTINE OUTPUT - The output of this subroutine is

the derived hearth temperature field and heat losses by

convection and radiation.

SUBROUTINE CONTROL - This subroutine controls the flow

of data to the subroutines that calculate:

1. the initial temperature assumption- TINT

2. the refractory material in each node- REFPOS

3. the value of the coefficient and remainder

matrices- COEFMAT

4. a library subroutine that solves for the

value of the temperature field - LIBRARY

Subroutine CONTROL also controls whether the solution has

been achieved to a tolerance, a convergence criterion has

been exceeded, or if the algorithm for the determination of

the temperature field must be reiterated.

74

SUBROUTINE TINT -

This subroutine calculates the

initial value of the temperature field based on an

assumption that the shell temperature is at 350 degrees

Fahrenheit and that there is a linear temperature drop

between the shell and the slag/silver interface.

SUBROUTINE REFPOS - This subroutine assigns a

refractory or metal material to each finite difference

node. This program transduces the refractory and metal

position data that was fed in into a nodal form.

SUBROUTINE LIBRARY - This subroutine employs a library

subroutine to calculate the value of the temperature field

from the derived values of the coefficient matrix and a

remainder matrix. The library subroutine is a linear

solution type.

SUBROUTINE COEFMAT - This subroutine is employed to

control the calculation of the coeffcient and remainder

matrices. This subroutine and subroutine INTCEPT (which it

calls) are used to identify each node as one of the nine

distinctive types of nodes. Subroutine COEFCAL is then

called to calculate the actual values of the coefficient

and remainder terms in the heat balance equation for that

type of node. Once called from subroutine CONTROL, this

75

subroutines controls the calculation of all the coefficient

and remainder terms for the model before returning to

subroutine CONTROL.

Subroutine INTCEPT -

This subroutine calculates the

intercept points of the nodal system and the bottom shell

or head of the smelter.

SUBROUTINE COEFCAL - This subroutine calculates the

value of the coefficient and remainder terms in the

descriptive heat transfer equation for each type of node.

The subroutine receives a node identifier from subroutine

COEFMAT that guides the subroutine to the correct

calculation for the node type under consideration.

Subroutine THERCON is called to calculate the value of the

net thermal conductivity of this node and the adjacent

nodes. These values are needed for the calculation of the

coefficient and remainder terms. As each node is

identified in subroutine COEFMAT, this subroutine is called

to determine the terms for that node and then returns to

subroutine COEFMAT. This subroutine also calculates the

value of the convective and radiative heat transfer from

the nodes on the shell boundary to the environment. The

total heat flows are summed up in subroutine COEFMAT.

76

SUBROUTINE THERCON -

This subroutine calculates the

values of the thermal conductivity, as a function of node

temperature, for the node under consideration and for that

of the the adjacent nodes. The net thermal conductivity of

each pair of adjacent nodes is then calculated and

transferred back to subroutine COEFCAL. The thermal

conductivity is calculated from anth

order polynomial

equation of whose coefficients were fed into subroutine

INPUT.

All of the subroutines except CALC and OUTPUT have

been written, typed into the computer and compiled. A

listing of each of the programs and subroutines is in the

Appendix. Subroutines Input and Tint have been run and

correct output values obtained. These are listed with the

subroutines in the Appendix.

Each of the programs or subroutines is written in a

structured programming format including the following:

1. Program name

2. Author and date written

3. Purpose and function

4. Form of call statement

5. Explanation of input variables

6. Explanation of derived output variables

7. Call statement

8. Implicit, real, integer, and common statements

77

9. Body of program, including common statements at

every distinct point

10. Write statements to print out internal

calculations- the signals for these write statements are

fed in through the the input file.

The work remaining on the program is as follows:

1. Write subroutines CALC and OUTPUT

2. Link and debug the program

3. Run the program employing variations of the input

data

78

6.0 CONCLUSIONS AND RECOMMENDATIONS

From the work completed to date on deriving the nodal

equations and writing and compiling the bulk of the

program, it appears that solution of the problem by the

model is feasible if the program converges. It is

therefore recommended that another student be given the

charge to complete the model by finishing the computer

program, debugging the program, and running the program

with various combinations of input variables. The benefits

of improved silver quality, increased smelter throughput,

and decreased operating costs would appear to far outweigh

the cost of completing this thesis.

In addition, if financial and personnel resources

allow, it may be advantageous to extend the mathematical

analysis to include the slag layer- From the information

gained in the literature search, an evaluation considering

the effect of convective currents in the slag layer induced

by density and electromotive differentials would be

feasible. This analysis would, however, be extremely time

consuming and costly and the justififcations for it would

have to be derived.

79

7.0 REFERENCES

1. Chen, T.S., and Goodsen, R.E. "Computation ofThree-

Dimensional Temperature and Convective Flow Profiles for an

Electric Glass Furnace", Glass Technology. Volume 13,

Number 6, December, 1972.

2. Aziz, K and Heliums, J.D. Physics Fluids, 1967, 10

(2), 314

3. Douglas, J. Numerical Math, 1962, 4, pages 41-63

4. Chen, T.S. PhD Thesis - Purdue University, 1971

5. Dilawari, A.H., and Szekely, J. "Heat Transfer and

Flow Phenomena in Electroslag Refining", Metallurgical

Transactions B, Volume 9B, March, 1978, pages 77-87.

6. Dilawari, A.H., Szekely, J., and Eager T.W.

"Electromagnetically and Thermally Driven Flow Phenomena in

Electroslag Welding", Metallurgical Transactions B. Volume

9B, September, 1978, pages 371-381.

7. Campbell, J. "Fluid Flow and Droplet Formation in the

Electroslag Remelting Process", Journal of Metals. July,

1970, pages 23-35.

80

8. Sandler, V.Y. "Numerical Analysis of Temperature and

Velocity Fields in Slag Bath', Magnetohvdrodynamics , vl8,

n2, April-June, 1982, pages 195-200.

9. Ridder, S.D., et. al. "Steady State Segregation and

Heat Flow in an ESR", Metallurgical Transactions Bt Volume

9B, September, 1978, pages 415-425.

10. Kaiser, R.H. and Downing, J.H. "Heat Transfer Through

the Hearth of an Electric Smelting Furnace", Electric

Furnace Proceedings. 1977.

11. Kishimoto, M., Maeda, M.

, Mori, K., and Kawai, Y.

"Thermal Conductivity and Specific Heat of Metallurgical

Slags", Second International Symposium on Metallurgical

Slags and Fluxes, Publication of the Metallurgical Society

of the AIME, 1984, pages 891-904.

12. Kammel, R. and Winterhager, H. "Struktur und

Eigenschaften von Schlacken der Metalhutenprozesse V

Dichtebstimmungen und Electrische Leitfaigkeitmessungen an

Schmelzen des SystemKalk-Tonerde-Kieselaure"

, Erzmetall,

18(1), 1965, pages 9-17.

81

13. Bockris, J.M., Tomlinson, J.W., and White, J.L. "The

Structure of the Liquid Silicates: Partial Molar Volumes

and Expansivities", Trans. Faraday Society, 52(1956),

pages 299-310.

14. Haggins, M.L. and Sun, K.H. "Calculation of Density

and Optical constants of Glass From its Composition in

Weight Percentage", J. Amer. Ceramic Society, 26(1), 1943,

pages 4-11.

15. Nagata, K. and Goto, K.S. "Heat Conductivity and Mean

Free Path of Phonons in Metallurgical Slags', Second

International Symposium on Metallurgical Slags and Fluxes,

Publication of the Metallurgical Society of the AIME, 1984,

pages 875-889.

82

8.0 APPENDIX

8.1.0 Derivation of Nodal Equations

8.2.0 Thermal Conductivity of Silver

8.3.0 Listing of Computer Programs

8.3.1 Program Smelter

8.3.2 Subroutine Input

8.3.3 Subroutine Tint

8.3.4 Subroutine Refpos

8.3.5 Subroutine Control

8.3.6 Subroutine Coefmat

8.3.7 Subroutine Intcept

8.3.8 Subroutine Coefcal

8.3.9 Subroutine Thereon

83

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108

8.2.0 Thermal Conductivity of Silver

109

Table 1

Therwal Conductivity of Silver

(Temperature, 7\ K; Thermal Conductivity, t. ' cm"1

K"')

SolidLiquid

T k r k T k T k

0 0 123 2 4 36 1235 08 1.75*3273 1.85*

1

2

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4 147 223.2 4.30 1400 1.83*3673 1.74*

5 172 250 4.29 1473.2 186*

3800 1.70*

6 187 273 2 4.29 1500 1.87*3873 1 68*

7 193 298 2 4.29 1573.2 1.89*4000 1.63*

8 190 300 4.29 1600 1.90*4073 1.61*

9 181 323.2 4.28 1673.2 1.92*4273 1.54*

10 168 350 4.27 1700 1.93*4500 1.45'

11 154 373 2 4 26 1773.2 1.94"4773 1 33*

12 139 400 4.25 1800 1 95*5000 1.23*

13 124 473.2 4 20 1873 2 1 96*5273 111*

14 109 500 4.19 1900 1.96*5500 1 01*

15 96 573.2 4 14 1973.2 1 97*5773 0.875*

16 85 600 4 12 2000 1.97*6000 0.764*

18 66 673.2 4 07 2073.2 1.98*6273 0.628*

20 51 700 4 04 2173.2 1.98*6500 0 514*

25 29.5 773.2 3.99 2200 1.98*6773 0.373*

30 19.3 800 3.96 2273.2 1.99*7000 0 251*

35 13.7 873 2 3.90 2400 1.98*7273 0.104*

40 10.5 900 3.88 2473.2 1.98*

45 8.4 973.2 3 82 2600 1.97*

50 7.0 1000 3.79 2673.2 1.96*

60 5.30 1073.23.73*

2800 1.95*

70 4.82 1100 3.70*2873.2 1.94*

80 4.64 1173.2 3 63*3000 1.91*

90 4.51 1200 3.61* 3073 1.90*

100 4.44 1235.083.58*

3200 1.87*

fThe values are for well-annealed high-purity silver, and those below 150 K are applicable only to a specimen having residual electrical

resistivity of 0.000621 pft cm. The values for molten silver are provisional.

'Extrapolated or estimated.

J. Phys, Chen. Ref. Data, Vol, 3, Suppl. 1, 1974, p. 607

110

8.3.0 Listing of Computer Programs

8.3.1 Program Smelter

8.3.2 Subroutine Input

8.3.3 Subroutine Tint

8.3.4 Subroutine Refpos

8.3.5 Subroutine Control

8.3.6 Subroutine Coefmat

8.3.7 Subroutine Intcept

8.3.8 Subroutine Coefcal

8.3.9 Subroutine Thereon

Ill

8.3.1 Program Smelter

112

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134

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139

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