Download Lixiviacion del Molibdeno contenido en una escoria con Acido

Modelling of Leaching of Molybdenum in
Slag
Jean-Paul Veas
University of Concepción
Stockholm, April 2005
1
Abstract
Leaching experiments of molybdenum from slag were carried out in laboratory at the
Concepción University. These experiments were modelled and the results compared with
the experimental data. Several reactions occur during the dissolution of the slag by the
acid. For sake of simplicity, the reaction of sulphuric acid with oxide of Iron (III) was
considered as representative of the system. The model assumed that the dissolution rate
varies with the pH of the solution and the surface area of the particles, which decrease
with time. Some simple cases are solved by applying analytical solutions. For more
complicate situation, numerical solutions were employed by using the software
MATLAB.
Heap leaching of slag is also modelled. The heap is irrigated with sulphuric acid. In this
case, the variation of the metal concentration in the solution, of the proton concentration,
and the concentration of metal in the slag no reacted are considered. The differential
equations were solved by using the software FEMLAB.
The experimental showed that the particle size has a large impact on the dissolution rate
of the slag. However, the impact of the pH seems to be very small. A reasonable
agreement is obtained between the experimental results and the simulations.
2
Table of Contents
ABSTRACT ....................................................................................................................... 2
CONTENTS....................................................................................................................... 3
INTRODUCTION............................................................................................................. 4
CHARACTERISATION OF THE SLAG ...................................................................... 5
EXPERIMENTAL PART ................................................................................................ 6
MODELLING USED IN THE SIMULATIONS ........................................................... 8
MODELLING OF THE LABORATORY EXPERIMENTS ............................................................ 8
Leaching model with constant reaction rate ............................................................... 8
Reaction rate in function of particle size and acid concentration ............................... 9
Model to be solved by numerical methods ............................................................... 12
DISSOLUTION OF SLAG IN A HEAP IRRIGATES WITH SULPHURIC ACID.............................. 13
RESULTS AND DISCUSSION OF THE EXPERIMENTS. ...................................... 14
EXPERIMENT A .............................................................................................................. 14
EXPERIMENT B .............................................................................................................. 19
SIMULATION OF HEAP LEACHING FOR SLAG. ................................................ 21
CONCLUSIONS. ............................................................................................................ 24
REFERENCES ................................................................................................................ 24
APPENDIX ...................................................................................................................... 25
3
Introduction
Chile is the major producer of molybdenum in the world (Reference: Statistics of Copper
and Other Minerals 1994-2003). The main mineral ores is the molybdenite. In those
deposits the molybdenite is, in general, associated to copper ores. Different processes are
used to separate the molybdenum of the cooper and other metals.
When the minerals are separated and refined by pyrometallurgy, slag is generated. The
slag formed contents still interesting percentages of some metals (e.g., copper,
molybdenum, irons, etc), which may do its recovery economically attractive. Therefore, it
is important to study the properties of the slag and investigate the separation methods that
could be applied. This could give a commercial value to the slag.
At the department of Metallurgy of University of Concepción, the recovery of metals
from the slag has been studied for several years. At present, a pilot plant is working to
extract address the extraction of metals from the slag and increase its commercial value.
Slag originated at the northern of Chile is used in the plant.
This present the results of simulations performed of experiments carried out in the plant
of University of Concepción. Dynamical model are developed and different programs
(tools) are used (e.g., Matlab, Phreeqc, Femlab).
4
Characterisation of the slag
Some properties of the slag are present below (Ref: Rory-report). Table 1 shows the
composition of a slag from Caletones (slow cooling). The density of the slag is shown in
Table 2.
Table 1: Slag composition
Elements
Concentration, %
Cu
Fe
Mo
Si
1.1
43.4
0.3
15.2
Table 2: Slag Density
Components
Density
Units
Slag
5000
kg/m3
Some data used in this report are also shown. Table 3 shows the particle size used in the
simulations equivalent to the respective size expressed as mesh N (Perry and Green,
1984). The densities of the sulphuric acid solution at ambient temperature (Perry and
Green, 1997) are also shown, Table 4.
Table 3. Total surface area of the solid
Assumed radius,
Total Surface Area in 200 g
mm
of slag, m2
Mesh N
100%-200
100%-75
-35+100
-10+20
0.037
0.100
0.2845
1.2605
3.24
1.20
0.42
0.095
Table 4: Sulphuric acid solution densities ambient temperature
Sulphuric acid solution (%)
Densities (kg/m3)
1.25
2.5
5.0
10
15
1009
1016
1032
1066
1103
5
Experimental part
The experimental part was carried out in the pilot plant at the University of Concepción
(Ref: Rory-report). A short summary of the experimental conditions and the results are
shown.
Two sets of experiments were performed. In the first set (A), the influence of the size of
the particle in the mineral recovery was addressed. The second one (B) shows the
influence of different concentration of acid at the beginning on the recovery of mineral.
The experimental conditions used in the experiments are shown in Table 5. The
experimental results, iron recovery as a function of time, are shown in Figures 1 and 2.
Table 5: Experimental conditions used in the experiments
Data
18
Temperature, C
500
Agitation, rpm
1
Liquid volume, (l)
5
Ratio (liquid/solid), l/kg
Experiment A
Volume of solution, l
Concentration of acid, g/l
Radius size, m
1.0
25
0.000037 – 0.00126
Experiment B
Radius size, m
Concentration of acid, g/l
0.000037
12.5 - 150
6
Figure 1. Iron recovery for different particles sizes (Experiment A) with an acid
concentration of 0.26 M.
Figure 2. Iron recovery for different acid concentration (Experiment B) with a particle
radius 0.037 mm.
7
Modelling used in the simulations
Regarding the reaction of the sulphuric acid with the slag, there are several minerals in
the slag that may react with the sulphuric acid. But, due to the high concentration of Fe in
the slag, only the reactions between the sulphuric acid and iron are considered. The
dissolution of the other minerals (e.g., molybdenum) is calculated assuming the
dissolution of the minerals takes place in congruent form.
There are also several iron minerals that may react with the sulphuric acid, but for sake of
simplicity, only the reaction with Fe2O3 is considered
2
Fe2O3  3H2SO4  2Fe3  3SO4  3H2O
In the simulations, models of different degree of complexity are used. In the first set of
modelling, the batch experiments carried out in the laboratory are modelled. Two cases
are modelled:


Reaction rate constant. This model may be used at the initial period, when the
initial conditions do not have changed too much.
Reaction rate is a function of the particle size and the acid concentration. This
model could be applied over long times.
In the second set of simulations a hypothetical heap with slag is modelled, where the acid
is irrigated on the top the heap.
Modelling of the laboratory experiments
In order to simplify the simulations some assumptions are done. The following
assumptions were done,
 The composition of the slag is uniform.
 The dissolution occurs on the surface of the particle.
 The congruent dissolution of the minerals takes place. The metals are dissolved in
proportion to its composition.
 The solution with the minerals are well-stirred.
 All the particles have the same size.
Leaching model with constant reaction rate
In this case the reaction rate is assumed to be constant. This model may be applied at the
initial period when the particle size (i.e., the specific surface) and the acid concentration
in the solution do not have changed too much. The results also indicate the maximum
reaction rate, since with reaction rate decreases with the time due mainly to the reduction
of the concentration of the acid. The differential equation describing the concentration of
the metal ion in the solution may be written as,
8
VL
dC
 R A , 0  constant
dt
(1)
Where VL is the volume of the solution (m3), C the metal concentration (mol/m3), t the
time (s) and R A , 0 the reaction rate (mol/s), which is constant and calculated for the initial
conditions.
The reaction rate is constant with time and its value is calculated using the conditions
(particle size and acid concentration) at the start of the experiment. The dependence of
the reaction rate with the concentration of the acid is poorly known. In this simulation is
assumed that the reaction rate is proportional to the concentration of protons in the
solution.
 
R A, 0  k  A T , 0    H 
(2)
0
Where A T , 0 is the total surface area of the particles at the start of the experiment (m2), 
 
is the composition of metal in the slag, H 0 is the acid concentration at the experiment
start (mol/m3), and k is a reaction rate constant (m/s).
The Equation (1) may be integrated directly using that the solution was initially free of
solute (metal ion)
Ct  0  0
IC :
(3)
Then the integration yields
C
R A,0  t
VL

 
k  A T,0    H 
VL
0
t
(4)
The total surface area of the particles at the start of the experiment, A T , 0 , is determined
as
A T,0 
3
rp,0
VSolid,0
(5)
Reaction rate in function of particle size and acid concentration
In this case the reaction rate is considered to be function of the proton concentration and
the particle size. The latter determines the contact surface between the acid solution and
the material in the particle. The differential equation describing the concentration in the
solution may then be written as,
9
VL
 
dC
 R A  k AT   H
dt
(6)
AT is the total surface area of the particles (m2), which decreases with time. It is
determined by the particle radius, rp (m) and the number of particles in the solid, N p . It
is assumed that the number of particles is constant during the experiment. The radius of
the particles decreases with time due to the dissolution reaction.
AT  4  rp2  Np
(7)
The number of particles is determined by the volume of the solid, VSolid, 0 (m3) and the
radius of the particles, rp , 0 (m) at the start of the experiment.
Np 
VSolid,0
4
  rp3,0
3
(8)
Introducing Equation (8) in (7) and reorganizing, we total surface area during the
experiment is obtained.
 rp
VSolid, 0
A T  4  r 

r
4
  rp3,0  p , 0
3
2
p
2

  A T,0


(9)
Finally, the ratio between the radii may be expressed as the ratio between the mass of
solid during the experiment, msolid (mol) and the mass at the start, mSolid,0 (mol)
2
 m
3
A T   Solid   A T,0
 m Solid,0 
(10)
Now, a relation between solid mass and the concentration of the sulphuric acid is needed.
At the start of the experiment the mass of solid equals its initial value and no metal is
dissolved in the solution. At the end of the experiment the solid has been completely
dissolved and the concentration of the solution will be Cend
At t  0
mSolid  mSolid, 0
and
C0
At t  
mSolid  0
and
C  CEnd
Since that this relationship is linear, we can write
10
(11)
mSolid 
C 

 1 
mSolid,0  CEnd 
(12)
Introducing Equation (12) in Equation (10), the total surface area is expressed as a
function of the concentration of the metal ion in the solution.

C 

A T  1 
 C End 
2/3
 A T ,0
(13)
Regarding the concentration of protons in the solution, this decreases when the
concentration of metal ions in the solution increases.
H  H 


0
 f *C
(14)
Where f is the stoichiometric factor for the dissolution reaction
Slag  f  H  Mn 
(15)
Introducing Equations (13) and (14) in Equation (6), the differential equation for the
concentration of the metals ions in the solution is obtained,

dC
C 

VL
 k  A T    H   k  1 
dt
C
End 

 
2/3

f C 

A T , 0    H  0 1 


H
0 

 
 
(16)
The analytical integration of this equation is apparently not possible. Equation (16) is
solved for two special cases. One is the case where the concentration of the acid is
assumed to be constant and the other where the particle size is assumed to be constant.
The first case, constant acid concentration, could be applicable when the consumption of
acid is not important (small stoichiometric coefficient f) compared with the variation of
the particle radius or when acid is continuously added. The equation for the concentration
of the metal ion in the solution is then
 
dC
VL
 k  AT    H
dt

C 


0  k  1 
 C End 
2/3
 
A T,0    H 
0
(17)
This equation may be directly integrated using that the concentration of the solution is
zero at initial time (C(t=0) = 0) as initial condition
11
3
 
A T,0 
 


C  C End  1  1  k 
 H 0 t 
 
3  C End VL
 

 
(18)
The second case, constant particle size, could be applicable when the consumption of
acid is important (large stoichiometric coefficient f) compared with the variation of the
particle radius. The differential equation for the concentration of the metal ion in the
solution is then
VL
 
  1  fH C 
dC
 k  AT    H   k  AT ,0    H 
dt
0


0

(19)
This equation may also be directly integrated
H 
C

0
f
k  f  A T,0   

1  Exp (
t ) 
V
L


(20)
Model to be solved by numerical methods
Equation (16) may be however integrated numerically using e.g., Runge-Kutta. However,
in practical cases, the dependence between the variables and parameters may be more
complicated and difficult to visualise. In order to give transparency to the equations and
allow that modifications and improvements may easily be made, the Equation (16) is
writen using three differential equations: for the concentration of the metal ion in the
solution, concentration of acid in the solution, and for the mass of metal that has not
reacted.
Equation for the concentration of the metal ion in the solution
VL
dC
 RA
dt
(21)
Equation for the concentration of the acid in the solution
VL
dH
 f  R A
dt
(22)
Equation for the amount the metal that has not reacted
dm
 R A
dt
(23)
12
Where H is the concentration of protons (mol/m3) in the solution, m is the mass of metal
(mol) to be dissolved. The reaction rate R A , varies with the particle size (specific surface
area) and the pH of the solution,
 
R A  k  AT    H
(24)
There are several tools that may be used to solve these differential equations. The
equations were integrated by using the Runge-Kutta method. Software MATLAB was
used.
Dissolution of slag in a heap irrigates with sulphuric acid
In this case, the slag is deposited forming a heap of a given height. Solution of sulphuric
acid is irrigated on the top of the heap. For sake of simplicity, it is assumed that the
height of the heap is practically constant. This means that when the metals are dissolved,
the size of the particle is maintained constant; no gangue is consumed by the acid. This
assumption will be discussed later. In this case, we have a variation of the concentration
of the acid in the solution and the particle size. They vary in time and along the bed.
A balance of material is written for the concentration of the metal ion in the solution, for
the concentration of protons in the solution and for the amount of material that has not
reacted. These equations are:
Variation of the concentration of the metal ions (iron):
W
C
C   2C 
q
  D  2   k  AT  H
t
x 
x 
 
(25)
Variation of the concentration of sulphuric acid:
H
H   2H 
W
q
  D  2   f  k  AT  H
t
x 
x 
 
(26)
Variation of material:
 
m
 k  A T  H 
t
(27)
These equations were solved by using the Software FEMLAB
13
Results and Discussion of the Experiments
In the laboratory two set of experiments were done
 Experiment A, the influence of the particle size on the dissolution rate is studied.
 Experiment B, the reaction rate as a function of the pH of the solution is
addressed.
Experiment A
This experiment shows the influence of size of the particle in the mineral recovery. Four
different particles sizes are used (See Table 3).
Case A.1: These simulations are for constant reaction rate, i.e., the dissolution rate
independent of the particle size and pH. This case is only applicable at the initial period
(very short time) when the variations in particle size and pH may be neglected.
According Equation (4), the recovery may be determined as
 

C
1 k  A T,0    H
Re cov 


C End C End
VL
0
t
(28)
Where C End is the concentration of the solution when the totally of the mineral has been
dissolved in the solution. The experimental data is fit using Equation (28). Therefore the
value of the constant k may be determined. The experimental recovery and those obtained
using Equation (28) are shown in Figure 3. The fitting is carried out using the
experimental points for very short time (less than 10 min).
The calculations were done for one of the metal ions. But, since congruent dissolution is
assumed, the values for the other ions may be calculated directly from the metal
composition. The theoretical final concentration, C End , and the specific reaction rate
constant, k. for the different metal are shown in Table 6.
Table 6: Specific reaction rate constant
Elements
Cend concentration,
Reaction rate,
(mol/l)
(m/s)
Cu
Fe
Mo
0.03
1.55
0.006
2.0E-08
8.0E-07
5.5E-09
14
Fig. 3: Experimental data is compared with the analytical solution obtained when the acid
concentration and particle size are assumed to be constant.
The values obtained for the specific reaction rate constant are rather uncertain due to that
the fitting is not sufficiently good and that not data is found for very short time. The first
data point is for about 6 minutes, which may be too long for the smallest particle size. For
small particle size may be expected that an important fraction of the acid was consumed
in a very short time.
15
Case A.2: In these simulations, the reaction rate changes with the particle size (Total
surface area) and the pH of the solution. As discussed above, the analytical solution is not
possible. Therefore, two special cases are solved: a) the pH of the solution is maintained
constant and b) the particle radius is maintained constant.
For the first case, when the pH of the solution is assumed to be constant, the recovery
may be calculated from Equation (17) as
3
 
A T,0 
 
C


Re cov 
  1  1  k 
 H 0 t 
C End  
3  C End VL
 

 
(28)
The results for the four different particle sizes are shown in Figure 4 and compared with
the experimental data.
Fig. 4: Experimental data is compared with the analytical solution obtained when the acid
concentration is assumed to be constant and the particle size decreases with time.
For long reaction time, the recovery will be approximate to 1.0, since acid is available
during all the time. Only for short times, an acceptable agreement is obtained with the
experimental data, since only a small fraction of the acid has been consumed. When the
acid in the experiments decreases, large differences are observed between the simulations
and the experiments. This is very clear for the smallest particle size, where the acid is
rapidly consumed. For the largest particle size, the agreement is sufficient good for a
longer time.
16
For the second case, when the particle size is assumed to be constant, the recovery may
be calculated from Equation (18) as
 
H 0
C
Re cov 

C End C End  f
k  f  A T,0 

 1  Exp (
t ) 
VL


(29)
The results for the four different particle sizes are shown in Figure 5 and compared with
the experimental data.
Fig. 5: Experimental data is compared with the analytical solution obtained when the
particle size is assumed to be constant and the acid concentration decreases with
time.
The patterns of the curves are similar, in particular for the case with the smallest particle
size. However, the asymptotic values is some higher in the simulations. This may be due
to that the stoichiometric coefficient for the reaction is some higher than the assumed
value of 1.5 Another reason may be that part of the acid is consumed by the gangue.
These results indicate that a larger amount of acid is required.
Case A3. Finally, Equation (15) was integrated numerically, to consider the two factors
(pH of the solution and particle size) simultaneously. The results are shown in Figure 6.
They are similar to the results shown in Figure 5, where constant particle size is assumed.
This is due to the in these simulations/experiments the concentration initial of the acid
reaches only to dissolve about 10 - 11 % of the initial amount the metals in the slag. This
17
means that when the acid is totally consumed, the particle radius has decreased only a
few tens percent.
Fig. 6: Experimental data is compared with the analytical solution obtained when the
particle size and acid concentration vary with time.
As discussed above, the model may also be written in function of three differential
equations, for the metal and acid concentration in the solution, and the amount of metal
that has not reacted. The differential equations were solved by using a solver from the
Software MATLAB.
Of course, the results obtained in this way are identical with those results obtained by
numerical integration of Equation (16).
As a partial summary of the results obtained, it is observed that
1. - To increase the recovery of minerals, the i9nitial concentration of acid would be
increased.
2. - The calculations were done for iron, but similar calculations may be done for the
other metals (molybdenum or copper).
18
Experiment B
This experiment shows the impact of the acid concentration at the start of the recovery of
mineral and the ratio liquid/solid on the recovery of metals. Three different situations are
studied. In the first case, a high acid concentration is used with a large ratio liquid/solid
(1.5 M and L/S=10). In the second case, a small concentration of acid is used with the
same ratio liquid/solid (0.25 M, L/S=10). In the third case, the ratio liquid/solid was
decreased to 5 (0.25 M, L/S=5). The experiments were carried out using the same particle
size, the smallest particle size (100% below mesh 200).
The experimental results show that the relative reaction rate is decreased with increasing
ratio L/S. The equilibrium is rapidly reached for the case with the smallest ratio L/S, 5.
However, no large differences are found in the relative reaction rate when the
concentration of the acid is increased. It seems that the acid concentration has a
negligible impact in the dissolution rate of the slag.
The acid concentration is very important in determining the amount of slag that is
possible to dissolve for one given situation.
In summary the initial acid concentration determine the location of the equilibrium, but
has little influence on the kinetics of the reaction. A higher acid concentration means a
larger metal recovery. It is not recommendable to work with concentrated acid since does
not increased the reaction, but more difficult to handle and can be dangerous.
These large differences are shown in Figure 7 and 8. Figure 7 compares the experimental
data at very short times with the analytical solution for the case where the acid
concentration and particle sizes are assumed to be constant during the experiment.
Figure 8 shown the numerical solution when the particle size and the acid concentration
vary with time. The simulations show that for the first case (1.5 M and L/S=10), the
totality of the slag is dissolved, since the acid is in excess in the solution. In the other two
cases the recoveries about 20 and 10 % of the metal in the slag. This is due to that the
shortage of acid in the solution, The acid is totally consumed when only a small fraction
of the slag has been dissolved
19
25
Metal recovery, %
20
15
10
5
1.50 M; L/S=10
0.25 M, L/S=10
0.25 M, L/S=5
0
0
2
4
6
8
10
12
14
time, min
Fig 7. Experimental data compared with analytical solution when the particle size and
acid concentration are med to be constant.
120
Metal recovery, %
100
80
60
40
20
1.50 M; L/S=10
0.25 M, L/S=10
0.25 M, L/S=5
0
0
20
40
60
80
100
120
140
time, min
Fig 8. Experimental data compared with numerical solution when particle size and acid
concentration vary in the experiment.
20
Simulation of heap leaching for slag.
These simulations model the leaching of slag carried out in a heap when acid sulphuric is
irrigated on the top to recover the metals in the slag. These simulations correspond to a
hypothetical case; therefore, no real data are used. The main aim of this section is to
develop a model that may be applied to leaching of minerals in heap, when a solution is
irrigated on the top the heap. The other aim was the use of the software (FEMLAB),
which solves the model in a friendly way. Some data used in the simulations are shown in
Table 7.
Table 7. Data used in the simulations
Data
Entity
Temperature, C
Radium size, m
Height, m
Initial acid concentration,
mol/m3
Value
15
0.00126
1.0
500
In these simulations, the concentration of mineral, the sizes of the particles and
concentration of acid vary with time and throughout of the heap. Three balances of
material are applied to the heap with the slag: for the concentration of the iron in the
solution that flow downward in the heap, the concentration of sulphuric acid in the
solution, and the amount the slag in each location of the heap. This system of partial
differential equations is solved with the software FEMLAB. The results are directly
plotted, or may be transferred to some other software for additional calculations or
plotting.
In the simulations is observed that due to the high reaction rate of the slag with the
sulphuric acid, sharp variations in concentration are observed in the zone de reaction.
This front or reaction zone divides the system in three regions. In the upper part, where
the slag has been depleted, the lower part where the slag is intact, and the central part
where the dissolution is taking place.
Figure 9 shows the concentration of iron in the solution along the bed, for three different
times, for a short time (blue line), for an intermediate time (red line), and finally for a
long time (black line). The values in the legend may be considered as relative, since they
depend on the parameters used in the simulations.
The location of the reaction depends on the amount of metals in the slag (actually on the
consumption of acid by the slag), the concentration of the acid in the solution, and on the
irrigation rate. On the other hand, the particle size, the reaction rate constant and the
irrigation rate determine the extent of the reaction zone (slope of the curves in Figure 9 to
11.
21
It is also observed that the increase of the concentration of iron (or molybdenum, or
copper) in the solution along the bed is exactly opposite to the decrease of the
concentration of protons.
Between the blue line and black line we have the variation of Iron versus the time
throughout of the heap. At the beginning of the time and the heap we can see the
concentration of Iron is increase quickly. In the middle of the heap (red line) we can see
the same increase that before this is because the reaction has the same rate reaction in all
the time, so that, in the end of the heap (black line) has the same way.
Figure 11 shows that the variation of the concentration of iron in the heap (no reacted) is
similar to the variation of the concentration of iron in the solution. This is due to that all
the iron is dissolving at the reaction zone.
Fig. 9: Variation of acid sulphuric in the heap of slag and the time.
22
Fig. 10: Variation of concentration of the Iron in the heap of slag and the time.
Fig. 11: Variation of mass at the heap of slag and the time.
23
Conclusions.
Comparisons between experimental data obtained at laboratory scale and results of a
model are carried out. A fair agreement is obtained.
The experimental results and the simulations show that the dissolution of slag is strongly
dependent on the particle size men only slightly influenced by the pH of the solution. The
amount of acid in the solution is only important when the amount of slag that will be
dissolved is determined.
Several assumptions were made in the conceptual model, so that, some of they need to be
further evaluated. The simulations were made with the software Matlab and Femlab.
References
Femlab. www.comsol.com
Perry R.H. and Green D.W., “Perry’s Chemical Engineers’ Handbook” Page 21-15, table
21-6 U.S. Sieve Series and Tyler equivalents (ASTM-E-11-61), Sixth Edition, McGrawHill 1984.
Perry R.H. and Green D.W., “Perry’s Chemical Engineers’ Handbook” Page 2-107, table
2-101 Sulphuric acid solution, Seventh Edition, McGraw-Hill 1997.
24
APPENDIX
A schematic picture of a typical heap is shown. We choose a part located in the interior of
the heap as our study object, to avoid the boundary effects. In this element of volume we
describe the respective balances (protons, iron and mass of slag no reacted).
20 m
1010
meters
m
100-200 m
40 m
If we take a small piece at the x-direction, we have:
X
X + X
Now we can make a balance of material:
Rate of increase of mass of Fe in the volume element
Rate of flow in of mass of Fe across face at x by convection
and diffusion/dispersion
Rate of flow out of mass of Fe across face at x+x by
convection and diffusion/dispersion
Rate of production of mass of Fe by chemical reaction
25
  Fe

 t

  x

n Fex x
n Fex
x  x
rFe  x
We can rewrite the balance of material as
 C Fe
=
t
Rate of increase
in moles of Fe
per volume unit
 C 
  Fe 
 x 
Net rate of addition
in moles of Fe per
volume unit by
convection in the xdirection
 J 
  Fe 
 x 
Net rate of addition
of moles of Fe per
volume unit by
diffusion in the xdirection
 R Fe
Net rate of
production of moles
of Fe per volume
unit by chemical
reaction
Data:
Total porosity
  0.4
Water porosity
 W  0.2
Slag density
 slag  5000
Content of Fe in slag, in weight
  0.45
kg
m3
Calculus
Initial volume of slag
V0slag  0.6
m 3slag
m 3 bed
Initial mass of slag
m 0slag  3000
Equations
Mass balance for Fe. From (1)
W
C Fe
 



   D  C Fe   R  u  C Fe
t
x 
x
x

If we suppose a coefficient of diffusion constant, we have
C Fe 
 2 C Fe 
C
  R  u  Fe
W
   D 
2 
t
x
x 

26
kg slag
m 3bed
Units:
W
C Fe  m 3 water   molFe  1   molFe 
 3
 3

    3
t
 m bed   m water   s   m bed, s 
 molFe 
R 3

 m bed, s 
Reaction rate, R
 
R  k  AT  H
 molFe   molFe m 3   m 2   molH  
 m 3 bed, s    m 2 , s molH     m 3 bed    m 3 

 
 
 

r 
r 
3
Vslag  V0slag
Volume of slag
p
3
0p
 m 2   m 3slag 
3
Vslag  3
 3

rp
 m slag   m bed 
Surface area
AT 
Surface area
rp 
rp 
3
A T  V0slag
 3  V0slag
3
rp
r0p 
r0p 3
3
Reaction rate
2
2

rp 
R  k  3  V0slag
 H  
3
r0p 
Radium of slag
rP  m slag
mSlag 2 / 3
R  k  3  V0slag
 H  
2/3
r0p   m0Slag 
R  Tp  m Slag 
2/3
Tp  V0slag
 H  
3 k
r0p   m 0Slag 2 / 3

If we put R into equation of balance to Fe, we have
27
1/ 3
slag
r0p 
m 0slag 1 / 3
C Fe 
 2 C Fe 
C
  Tp  m Slag 2 / 3  H    u  Fe
W
   D 
2 
t
x
x 

28