# THE DEVELOPMENT OF A MATHEMATICAL MODEL

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```							    THE DEVELOPMENT OF A MATHEMATICAL MODEL
OF SULFUR DIFFUSION IN THE STEEL AND SLAG PHASES
IN ITS TRANSITION FROM THE STEEL INTO THE SLAG
1
Ariel University Center of Samaria, Ariel, Israel,
2
Ben-Gurion University of Negev, Beer-Sheva, Israel

Abstract. In this paper, a deterministic mathematical model of sulfur diffusion in
metal and slag melts is presented. The model was implemented in C# language. We
have computed the distribution of sulfur in the metal and slag phases and the
concentration changes of sulfur in the volume of the metal and slag phases, both
functions of space and time. The computed results agree well with the experimental
ones.

INTRODUCTION
The new age of quality steels is distinguished by sharply higher standards on
the cleanliness of metal with respect to harmful impurities and inclusions. One of the
most important problems related to this development is the creation of an efficient and
economical technology for desulfurizing iron and steel [1-4].
Traditionally, this problem is solved by using Edisonian (trial and error)
technological experiments. The development of computer technology and its
accessibility have made it possible to solve problems by mathematical models [5-7].
There are numerous mathematical models which describe the desulfurization
process, for example [5, 6, 8-15]. However, most of them are empirical, statistical or
“black box type” input-output models [5, 12] or models that take into consideration
the processes which occur only in the metal or the oxide phase e.g. [10, 11].
Interesting results were obtained from the model [15]. It considers rates of transfer of
elements through the metal-oxide interface and mutual influence of the chemical
reactions taking place on this interface. However, transport of elements in the bulk of
the metal and oxide phases is considered without going into details.
A model for sulfur transfer from the metal to the slag phase by means of a
heterogeneous reaction and sulfur diffusion in the melts based on the thermodynamics
and kinetics analysis is presented in this paper. The model was implemented was
implemented in C# language.

THE MODELING METHOD
Sulfur transfer from the liquid metal to the slag phase is the activation step of
the heterogeneous reaction that takes place on the metal-slag interface [16]. The
reaction includes the following stages Fig. 1:
1. Sulfur transport to the reaction zone (metal-slag interface) in the metal phase.
2. Sulfur transfer from the melt metal to the slag phase.
3. Sulfur anions transport from the reaction zone (metal-slag interface) in the slag
phase.
Development of the mathematical model of the desulfurization process
involves a thorough study of the above mentioned steps of this process:
Let us consider each of these steps in details, the steps 1 and 3 are described together.

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Fig. 1. Sulfur transfer to/from the melt metal-slag reaction zone.

A MATHEMATICAL MODEL OF SULFUR DIFFUSION IN THE
MOVELESS METAL AND SLAG MELTS

Sulfur diffusion to/from the reaction zone Fig. 1 described by the 2nd Fick‟s
law. We assume that both the data and the geometry do not depend on the variables y,
z. Then the problem can be identified with an one-dimensional problem, posed in the
domain [0, l] Fig. 2. This domain is divided into two subdomains by the metal-slag
interface located on x=h, (0<h<l), where x denotes the space variable.

Fig. 2. Concentration profile of sulfur in the metal and slag phases.

In this way, the mathematical equations, which describe the transport of the element
in the metal and slag melts, are
[ S ]           2[S ]
 D[ S ]          in [0, h]                                             (1)
t             x 2
(S )            2 (S )
 D( S )           in [h, l],                                           (2)
t              x 2
where:
[S] and (S) are the sulfur concentration in the metal phase and slag phases,
respectively.
D[ S ] and D( S ) are the diffusion coefficient of sulfur in the metal and slag phases,
           
cm 2  sec1 , respectively.

Sulfur transfer across metal-slag boundary

Sulfur transfer from the metal to the slag phase occurs by reaction that is taken
place at the metal-slag interface. Since the slag is an ionic solution [16], the reaction is
electrochemical by nature and can be written as follow

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[ S ]  2e  ( S 2 ) .                                                          (3)
In the derivation of the interface transport term, it is useful to assume that the
chemical reaction is infinitely fast and that a local thermodynamic equilibrium exists
at the interface between metal and slag phases [17]. The sulfur distribution coefficient
L between liquid metal and slag phases is determined by means of this equilibrium,
namely
(S ) r
L            ,                                                                  (4)
[ S ]l
where ( S) r and [ S ]l are respectively the right and the left limit values of sulfur
concentration at the interface.
The system of equations (1, 2) together with the interface condition Eq. (4) is the basis
for mathematical model of the desulfurization process.

Solution method

For solving Eqs. (1, 2) we have used the following initial, boundary and
interface conditions:
The initial conditions (see Fig. 2)
[ S ]( x,0)  C1        0  x  h,                                               (5)
( S )( x,0)  C 2       h xl,                                                  (6)
where C1 and C 2 are the uniform initial concentration in metal and slag phases
respectively.
The boundary conditions (for t  0 )
 S ( x, t ) 
              0                                                               (7)
 x 0,l
The interface conditions
Since the heterogeneous reaction Eq. (3) occurs at the metal-slag interface, it provides
new flux boundary condition. The concentration of sulfur at the metal-slag boundary
x  h is calculated from condition Eq. (4) and the mass conservation. Let us denote
by m( S ) r and m[ S ]l the right and left limit values of sulfur mass at the interface
respectively. Then, introduce m , the sulfur mass transferred from the metal to the
slag. Finally, let M Oxr (respectively M Mel ) be the right limit value of the slag mass
(respectively the left limit values of the metal mass) at the interface. Then Eq. (4) can
be rewritten as follow
m( S )r  m      M Mel
L                             ,                                                 (8)
M Oxr       m[ S ]l  m
form which we deduce that
LM Oxr m[ S ]l  M Mel m( S )r
m                                  ,                                            (9)
M Mel  M Oxr L
and consequently
m[ s ]l  m
[ S ]l               ,                                                          (10)
M Mel
m( s )r  m
(S )r                   .                                                      (11)
M Oxr

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Discretization of the equations
Equations (1, 2) have been numerically solved by a finite-differences approach
 2 2 
using a time-explicit scheme. The second derivative  2 , 2  is approximated by
   
nd
a 2 order centered scheme. The discrete version of Eqs.(1, 2) is given below
[ S ]in1  [S ]in
                      
D[ S ]
           [S ]in1  2[S ]in  [S ]in1 , 1  i  k        (12)
t            (xMe )2
(S )n1  ( S )n
                         
D( S )
m           m
           (S )n 1  2(S )m  (S )n 1 , k 1  i  p
n
(13)
t            (xOx )2
m                     m

where k and p-k are the number of nodes along the x-axis in the metal and slag phase
respectively, t is a constant time step, x is a constant mesh size,
[ S ]in  [ S ]( ix, nt ) . The same is for (S ) .

Numerical results
The C# code has been written to carry out the solution of Eqs.(1, 2). In order
to calculate sulfur distribution in the metal and slag phases the following data were
used:
 Metal composition: 1.0 wt % C, 0.25 wt % S, 98.75 wt % Fe;
 Slag composition: 51 wt % Al2O3 , 43wt % CaO and 6 wt %MgO ;
 Temperature: 1773K;
 Diffusion coefficient in the metal phase: D[ S ]  6.2 105[cm2 / s] [179];
     Diffusion coefficient in the slag phase: D( S )  105[cm2 / s] [172];
 Metal mass: 50gr.;
 Slag mass: 50gr.;
 Length of the metal phase along x-axis: 0.94 cm;
 Length of the slag phase along x-axis: 2.19 cm.
The following results have been depicted:
1. Sulfur concentration (wt %) changes in the metal and slag phases along
coordinate x at the given time (Fig. 3).
2. Sulfur concentration (wt %) changes in the bulk of the slag and metal phases
with respect to the time (Fig. 4).

Fig. 3. Sulfur concentration changes with x-coordinate after 30 min from the
beginning of the process.

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Fig. 4. Sulfur concentration changes with respect to the time till the 800th minute.

Experimental verification

To verify the reliability of the computed sulfur distributions in the metal and
slag phases, we have compared them with experimental data. The changes of sulfur
concentration were controlled by chemical analysis. The contents of liquid metal and
slag phases were similar to those which were used in calculation. To ensure the fast
achieving of equilibrium the metal and the oxide samples were grounded and mixed.
Thus prepared mixtures were held in alumina crucibles at 1773K. Duration of the
experiments was 15-30 minutes and small samples of the metal and the slag were
taken each 5 minutes. The results of the analyses were plotted as [ S ]  t , ( S )  t (Fig.
5).

Fig. 5. Experimental data of sulfur concentration changes.

The comparison of the experimental results with the calculations demonstrated
the identical character of the concentration dependencies on time (Fig. 7.15, 7.16 and
7.18). However, the computed time for achieving the equilibrium is significantly
longer than its experimental value. This can be due to the fact that the natural
convection flows of liquid metal and slag phases were not taken into consideration.
A mathematical model, which takes into account convection in the melts, is
presented below.

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A MATHEMATICAL MODEL OF SULFUR TRANSFER IN A MOVING
METAL AND SLAG MELTS

This process comprises two mechanisms: the molecular diffusion and the
 mol 
convective transport. Thus the flux of element j  2  consists of two components,
 cm 
one due to diffusion j D and the second due to convective flow jC .
j  jD  jC .                                                                     (14)
Thus, sulfur transport in the metal or sulfur ions transport in the slag phase, is
described as follows
[ S ]           [ S ]           2[S ]
 u[ S ]         D[ S ]          In [0, h],                               (15)
t              x              x 2
(S )            ( S )           2 (S )
 u( S )         D( S )           in [h, l],                              (16)
t              x                x 2
where u[S ] and u ( S ) are the velocities of the movement of the liquid metal and slag
phases respectively, [cm/s ].

Numerical solution

In order to describe the convection (8.12), (8.13), the one-dimensional domain
[0, l] is divided into two layers (“upper” and “lower”). More precisely, the nodes
along the x-axis are doubled with two degrees of freedom at the same coordinate xi . In
addition, it is assumed that the velocity of the metal and slag phases in the “upper”
layer is positive (u ) and the velocity of these phases in the “lower” one is
negative (u ) . Thus, these equations can be written
[ S ]U           [ S ]U            2 [ S ]U
 u[ S ]           D[ S ]             ,                                (17)
t               x                x 2
[ S ]L            [ S ]L            2 [ S ]L
 u[ S ]           D[ S ]              ,                              (18)
t                  x                 x 2
 ( S )U           ( S )U            2 ( S )U
 u( S )            D( S )               ,                             (19)
t                x                  x 2
(S ) L             (S ) L             2 (S )L
 u( S )            D( S )                ,                           (20)
t                  x                 x 2
where [ S ]U and [ S ] L are sulfur concentration in the “upper” and “lower” layers
respectively in the metal phase, ( S)U and ( S) L are sulfur concentration in the “upper”
and “lower” layers respectively in the slag phase.
To solve these equations, a reference frames which move with the velocity  u were
chosen. Then the new reference frames are
 Me  x  u[ S ]t ,                                                              (21)
 Sl  x  u ( S ) t ,                                                            (22)
 Me  x  u[ S ]t ,                                                              (23)
 Sl  x  u ( S ) t ,                                                            (24)

1-150
where  Me and  Me are the moving reference frames in the “upper” and “lower” layers
respectively in the metal phase,
 Sl and  Sl are the moving reference frames in the “upper” and “lower” layers
respectively in the slag phase,
u[ S ]t and u ( S ) t are the distance traveled by the metal and slag phases respectively in
time t.
Consequently, sulfur concentration can be transformed into
[ S ] ( x, t )   ( Me , t ) ,
U
(25)
( S )U ( x, t )   ( Sl , t ) ,                                                    (26)
[ S ] ( x, t )   (  Me , t ) ,
L
(27)
( S ) ( x, t )   (  Sl , t ) .
L
(28)
Substituting equations (25 - 28) into (17-20) respectively leads to
[S ]U            [S ]              2 [ S ]            2
 u[ S ]           D[ S ]               D[ S ] 2 ,                       (29)
t               x               x 2      t         Me
[S ]L           [ S ]           2 [ S ]             2
 u[ S ]         D[ S ]               D[ S ] 2 ,                          (30)
t              x              x 2      t         Me
(S )U          ( S )           2 ( S )            2
 u( S )         D( S )               D( S ) 2 ,                           (31)
t             x              x 2      t         Sl
( S ) L            ( S )               2 ( S )                2
 u( S )           D( S )                     D( S ) 2 .         (32)
t                  x                 x 2      t            Sl
To complete the model, one has to supplement to Eqs. (29 - 32) together with
Eq. (7) equations that describe the movement of the melts bottom-up and top-down.
We have chosen Eq. (33).
 ( , t )0   (  , t )0 ,  (  , t ) h   ( , t ) h ,
l                  l

 (  , t ) h   ( , t ) h , ( , t )l   (  , t )l
r                r
(33)

Discretization of the equations

Equations (29 - 32) have been numerically solved by a finite-differences
approach. For the space variable x, the mesh size x Me in the metal phase, and x Sl
for the slag phase are chosen following
x Me  u[ S ] t , xOx  u ( S ) t .                                         (34)
The convection-diffusion equation can be discretized by an explicit time-scheme.
Despite its simplicity, such an implementation requires to ensure a stability condition
that is written in that case
D  t 1
 .                                                                     (35)
( x ) 2 2
This means that the time to propagate through a distance x is proportional to ( x ) 2 .
Moreover, from Eqs. (34) and (35) we get
x
t     ,                                                                      (36)
u

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(x) 2
t                .                                                                       (37)
2D
As we intend to apply the model for low values of u, it appears a contradiction.
( x ) 2
Indeed, as u tends to zero, t tends to infinity and has to remain lower than                  .
2D
For these reasons, we rather use a time-implicit discretization scheme, which
does not require any stability condition. Remark however that implicit schemes are
stable for larger time step, but tend to damp the smaller scales. Hence, t can be
chosen greater than for an explicit time-scheme, but not too large, in order to preserve
the accuracy of the modelisation.
In such a way, equations (29 - 32) can be rewritten in their discrete versions as
 i  in
n 1

                   
D[ S ]
                  in1  2 in1  in1     1  i  k 1 ,          (38)
t           (xMe )    2        1                    1

 m1  m
n         n

                   
D( S )
                  m1  2 m1  m1
n
1
n         n
1    1  m  p  1,         (39)
t           (xOx )    2

in1  in
                   
D[ S ]
                 in1  2in1  in1
1                1        1  i  k 1 ,            (40)
t          (xMe )    2

m1  m
n        n

                    
D( S )
                 m1  2m1  m1
n
1
n       n
1      1  m  p  1.            (41)
t          (xOx )    2

The final concentration of sulfur is given by
 n 1  in 1
[ S ]in 1  i                  ,                                                          (42)
2
 n 1  m1
n
( S ) n 1  m
m                        .                                                         (43)
2
Numerical results and experimental verification
The data for calculations of the sulfur distribution are the same as the ones
which were used for moveless melts. Figure 6 represents sulfur concentration changes
at 5th min from the beginning of the process.

Fig. 6. Sulfur concentration changes with x-coordinate at 5th min from the beginning
of the process.

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Figure 7 shows computed and experimental results of sulfur concentration variations
with respect to the time. Computed results are plotted by a firm line, and experimental
ones, obtained by experiment which has been mentioned above are marked by dots.

Fig. 7. Sulfur concentration changes with respect to time.

The concentration of sulfur in metal phase is decreasing during the process until the
system reaches a thermodynamic equilibrium. It will happen after about 30 min Fig.
7. As it is indicated in the Figs. 6, 7 most of the sulfur transfers from metal to the slag
in 30 minutes after the beginning of the process. It agrees with experiments

CONCLUSIONS AND PERSPECTIVES

A deterministic kinetic mathematical model of sulfur transfer from the metal
to the slag phase has been developed. To implement this model a program in C#
language has been written. The distribution of sulfur in the metal and slag phases
along the x axis at a given time and the sulfur concentration changes in the volume of
these phases with respect to the time has been computed. The numerical results agree
well with the results which were found by experiment.
This computational model enables to obtain results, for example sulfur
concentration changes along the x axis, which is very difficult, almost impossible, to
get by experiment.
Using this mathematical model allows us to predict the sulfur distribution in
the metal during the refining process and to reduce expenses in selection of optimum
process' conditions, which provide the metal with required composition.
This model can be used for quantitative analysis of the diffusion stage of the
heterogeneous reactions.

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