The Southern African Institute of Mining and Metallurgy
Piotr M. Kapias and Michał Pazdziorek
Lead and Zinc 2008
THERMODYNAMIC AND EXPERIMENTAL ASPECTS OF
THALLIUM EXTRACTION FROM LIQUID LEAD
Piotr M. Kapias, Michał Pazdziorek
The Silesian University of Technology, Katowice, Poland
Keywords: extraction, lead refining, Pb-Tl solutions, PbCl2-TlCl solutions.
Abstract.
The theoretical part of the paper presents an analysis of the conditions of Tl
extraction from Pb using different extractants and determines the activity coefficients in
liquid solutions of Pb-Tl and PbCl2-TlCl. The dependence of the solution activity
coefficients upon the composition and temperature is presented in the form of
Krupkowski and Krupkowski-Fitzner equations.
An analysis and evaluation were made of the influence of the methods of PbCl2
introduction into a Pb-Tl solution upon the efficiency of the Tl extraction process under
thermodynamic equilibrium conditions. The analysis made it possible to formulate
relevant equations describing the dynamics of the process of Tl extraction from Pb
under stirring conditions as well as under the extractant injection into the liquid Pb-Tl
solution conditions.
In the part concerning experimental work, results are given for tests on the Tl
extraction process from Pb-Tl liquid solutions of initial Tl content of approximately
0.05% mass using PbCl2 or, alternatively, Cl2(g) conducted in laboratorial scale. The
tests made it possible to establish the influence of time, temperature, mass and the
method of introducing the extractants into the liquid upon the level of Tl extraction
from a Pb-Tl solution.
Labels used in the text.
wTlm
- Tl content in metallic phase, % mass
- Tl content in chloride phase, % mass
wTls
- Tl content in metallic phase in initial state, % mass
wpTlm
- Tl content in metallic phase in equilibrium conditions, % mass
woTlm
o
mm
- initial mass in metallic phase
- mass in metallic phase
mm
- lead chloride mass fed to the system
mPbCl2
mPbCl2-TlCl - mass in chloride phase
MPb
- lead atomic mass, g/mole
MTl
- thallium atomic mass, g/mole
K
- Nernst distribution coefficient
- mole friction of solution component
Xi
- mole friction of PbCl2 in saturated solution
XoPbCl2
- mole friction of TlCl in saturated solution
XoTlCl
T
- temperature, K
R
- gas constant, J/(mole K)
k
- kinetic equation constant, 1/(%.min.)
γi
- solution component activity coefficient
ε
- extraction rate, %
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The Southern African Institute of Mining and Metallurgy
Piotr M. Kapias and Michał Pazdziorek
1.
Lead and Zinc 2008
Introduction
Thallium is an admixture occurring in raw lead whose content level in refined lead is
not directly specified by standards. In some cases, low concentration of Tl in refined Pb
is tolerated, because it does not have a significant influence on the properties of most
products made of lead and its alloys. On the other hand, the presence of Tl in lead is
treated as a way of safe utilization of this extremely toxic element.
The issue of thallium extraction from lead occurs when the content level of Tl in raw
lead disables the production of refined lead, which meets the purity level specified by
standards.
2.
The goal and scope of work
The paper presented is part of a more extensive research project conducted with the
aim to develop the technology of Tl extraction from lead process as well as to treat the
semi-products of the extraction process further.
The goal of this work is to identify the basic technological parameters of the process
of thallium extraction from lead using lead chloride PbCl2 and chlorine.
The scope of work included:
• Thermodynamic analysis of the Pb-Tl│PbCl2-TlCl system.
• Laboratory performance of experiments, which allow the evaluation of process
kinetics and the verification of calculation results, achieved through thermodynamic
analysis.
3.
Thermodynamic basis of the process of Tl extraction from lead
The basic goal of the thermodynamic analysis was to specify equilibrium conditions
in four-component, two-phase thermodynamic system of Pb-Tl│PbCl2-TlCl.
3.1.
Activity coefficients in liquid Pb-Tl solutions
Specification of equilibrium conditions in the Pb-Tl│PbCl2-TlCl system requires
information about activity coefficients of the components of both phases.
The functions which define the dependence of activity coefficients in liquid Pb-Tl
solutions upon composition and temperature were determined based on data provided
by Hultgren [1] in the form of Krupkowski-Fitzner equations [2].
1154
(1 − X Tl )2.86 + 987 (1 − X Tl )3.86
ln γ Tl = −
(1)
T
T
1154
(1 − X Tl )2.86 − 1.537(1 − X Tl )1.86 + 0.537 +
ln γ Pb = −
T
(2)
987
3.86
2.86
(1 − X Tl ) − 1.35(1 − X Tl ) + 0.350
+
T
The dependence of activity coefficients in liquid Pb-Tl solution upon solution
composition is shown on Fig. 1.
[
[
]
]
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The Southern African Institute of Mining and Metallurgy
Piotr M. Kapias and Michał Pazdziorek
Lead and Zinc 2008
1
γ Tl, γ Pb
0.9
0.8
gPb
γ Pb
gTl
γ Tl
0.7
0
0.2
0.4
0.6
0.8
1
XTl
Fig. 1. Activity coefficients of components in Pb-Tl solutions
T = 773K
3.2.
Activity coefficients in liquid PbCl2-TlCl solutions
The dependence of component activity coefficients upon composition and
temperature in liquid PbCl2-TlCl solution was determined in the form of Krupkowski
equations [3], based on phase equilibriums in PbCl2-TlCl system [4] presented in Fig.
2.
⎛ 160.1
⎞
1.709
ln γ TlCl = −⎜
+ 1.247 ⎟(1 − X TlCl )
(3)
⎝ T
⎠
⎛ 160.1
⎞
1.709
0.709
(4)
ln γ PbCl2 = −⎜
+ 1.247 ⎟ (1 − X TlCl )
− 2.41(1 − X TlCl )
+ 1.41
⎝ T
⎠
[
]
500
Pb2TlCl5
460
PbTl3Cl
0
Temperature, C
480
440
Pb 4Tl11Cl19
430oC
PbTl2Cl
419oC
420
0.74
396oC
400
377oC
380
374oC
0.14
0.43
360
0.0
0.2
0.4
0.6
0.8
X PbCl2
Fig. 2. Phase equilibrium diagram of PbCl2 – TlCl system
Page 19
1.0
The Southern African Institute of Mining and Metallurgy
Piotr M. Kapias and Michał Pazdziorek
4.
Lead and Zinc 2008
Thermodynamic equilibrium in the Pb-Tl│PbCl2-TlCl system
It was assumed that the thermodynamic Pb-Tl│PbCl2-TlCl system is in isobaricisothermal conditions, while the chemical equilibrium condition is determined by the
equilibrium of the reaction:
2[Tl] + (PbCl 2 ) = [Pb] + 2(TlCl )
(5)
where the substances which form the chloride solutions are in parenthesis, while the
metallic phase components are in square brackets.
The system equilibrium condition in the extraction process was described by the
Nernst’s distribution coefficient:
w
K = Tls
w Tlm
In general, distribution coefficient “K” depends on temperature and the composition
of the phases present in the system.
(6)
Fig. 3 shows the distribution coefficients of the Pb-Tl | PbCl2-TlCl system,
calculated with the assumption that the chemical equilibrium condition in the system is
represented by the equilibrium of reaction (5), while the phase component activity
coefficients are described by equations: (1), (2), (3) and (4).
600
K = wTls/wTlm
T = 773 K
400
T = 823 K
200
0
0.00
0.01
0.02
0.03
0.04
0.05
0.06
Tl content in the metallic phase, % mass
Fig. 3. Tl distribution coefficient in the Pb-Tl│PbCl2-TlCl system, depending upon
the composition of metallic solution and temperature
Using the statistical analysis methods, it was determined that the dependence of
distribution coefficient upon the Tl content in the metallic phase could be shown in the
form of a 3rd level product:
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The Southern African Institute of Mining and Metallurgy
Piotr M. Kapias and Michał Pazdziorek
3
Lead and Zinc 2008
2
K = A ⋅ w Tlm + B ⋅ w Tlm + C ⋅ w Tlm + D
(7)
Table 1 presents the coefficients appearing in the equation (7) for a few values of
temperature.
Table 1. Coefficients in the equation (7)
Coefficient
A.10-6
B.10-5
C.10-3
D.10-2
5.
Temperature, K
773
823
-1.052
-0.519
1.372
0.751
- 8.287
-5.375
4.904
3.956
Process modeling of thallium extraction from lead.
It was assumed, that the process of Tl extraction from lead could be accomplished in
practice using two methods: stirring or injection.
The stirring method was defined as a method of balancing the two-phase system,
consisting of liquid Pb-Tl solution and specified mass of PbCl2, which resulted in the
equilibrium condition.
The injection method was defined as a method of injecting gaseous chlorine, or solid
chlorides like PbCl2 or ZnCl2 into liquid Pb-Tl solution. The injection process was
treated as a sum of indefinite number of subsequent stages carried out using the stirring
method, with the participation of indefinitely small extractant mass. At the same time, it
was assumed, that the initial system condition of each elementary stage corresponds to
the final system condition of the directly preceding stage.
When preparing the balance of Tl mass in the system and using the distribution
coefficient, it can be proved that in the case of stirring method, the dependence between
Tl concentration in Pb after the extraction process and the extractant mass is described
by the equation:
1
(8)
w Tlm = w oTlm
m PbCl2
1+
K
m om
It was established, that in the case the extraction process is carried out using the
injection method, the composition change of the Pb-Tl solution in the extraction process
is generally described by the differential equation:
dm PbCl2
1 dw Tlm
(9)
=−
K w Tlm
m om
If the distribution coefficient “K" can be treated as constant and independent of
metallic solution composition within the analyzed scope, the following dependence is a
result of integrating equation (9) and ordering:
⎛ m PbCl 2 ⎞
⎟
w Tlm = w oTlm exp⎜ K
(10)
o
⎜
⎟
m
m
⎝
⎠
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The Southern African Institute of Mining and Metallurgy
Piotr M. Kapias and Michał Pazdziorek
Lead and Zinc 2008
Tl content in the metalic solution , % mas.
In this study, the differential equation (9) was solved taking into account the
dependence of distribution coefficient upon the metallic phase composition. The Runge
-Kutta’s numerical method was used [5]. The calculation was made for two different
temperatures of the extraction process, taking into account the equation (7), which
defined the dependence of distribution coefficient upon the metallic phase composition.
The calculation results are shown on Fig. 4.
0.06
T = 773K (stirring)
0.05
T = 823K (stirring)
T = 773K (injection)
0.04
T = 823K (iniection)
0.03
T = 823K Expt. data (stirring)
0.02
0.01
0.00
0.0
0.5
1.0
1.5
2.0
2.5
Mass of PbCl2, Mg/100 Mg metalic solution
Fig. 4. The influence of the extraction process method on the final Tl content in Pb
For comparative purposes, this figure also shows function (8), which defines the
relation between the equilibrium concentration of Tl in metallic phase and the mass of
PbCl2 fed using the stirring method.
6.
Experimental part of the work
6.1.
Apparatus
The diagrams of the apparatus used for Tl extraction from Pb using PbCl2 and Cl2
(g) are shown on Fig. 5 and Fig. 6 respectively.
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The Southern African Institute of Mining and Metallurgy
Piotr M. Kapias and Michał Pazdziorek
Lead and Zinc 2008
5
3
4
2
Fig. 5. Diagram of the apparatus used for Tl extraction from lead using PbCl2 and
the stirring method.
1
2
3
4
5
-
Furnace with control system
Alundum crucible with the sample
Stirrer
Ni/Ni-Cr thermocouple
Ventilation hood
6
7
5
4
Ar
8
3
Cl2
9
2
10
1
Fig. 6. Diagram of the apparatus used for Tl extraction from lead using chlorine and
the injection method.
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The Southern African Institute of Mining and Metallurgy
Piotr M. Kapias and Michał Pazdziorek
1
2
3
4
5
6
7
8
9
10
6.2.
-
Lead and Zinc 2008
Alloy steel crucible with a lead sample
Steel retort
Cover
Ni/Ni-Cr thermocouple
Ceramic lance
Valve
Rotameter
Exhaust
Fan
Gas cleaning system
Methodology
All the experiments were conducted using lead samples of chemical composition
shown in Table 2. The sample’s mass was approx. 5 kg.
Table 2. Chemical composition of raw lead (mass %)
Pb
99.6529
Sb
As
Bi
Sn
Cu
Ag
Zn
Fe
Ni
Cd
Tl
0.0106 0.0004 0.2489 0.0008 0.0347 0.0028 0.0003 0.0007 0.0004 0.0002 0.0473
The stirring method:
The process of Tl extraction from Pb was conducted in alumina crucibles
(dimensions: φ 100 x 140 mm). The following procedure was used during the
experiments:
• The lead sample and a specified amount of PbCl2 were placed using the alumina
crucible in electric resistance heat furnace.
• The furnace was heated to required temperature.
• The stirrer was switched on when required temperature was reached. The rotation
speed of the stirrer was ca. 120 rpm.
The injection method:
The process of Tl extraction from Pb using chlorine was conducted in alloy steel
crucibles dimensions: φ 100 x 140 mm. The following procedure was used during the
experiments:
• The lead sample was placed in the crucible and then in the retort in the heat furnace.
• Chlorine was introduced into the system using a ceramic lance with a nozzle with
two cylindric openings of 1mm diameter at the end. While heating up the system to
the desired temperature, argon was fed into the system through the nozzle situated
over the surface of the sample.
• Once the desired temperature had been reached, the inflow of argon was switched
off and an inflow of a controlled amount of chlorine was switched on.
• Once the inflow of chlorine had been switched on, the lance was put approximately 5
cm into the lead sample. The amount of chlorine introduced into the sample was ca.
5.5 dcm3/h. During the experiment the gases from the retort were evacuated using a
pump and utilized in a system of washers.
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The Southern African Institute of Mining and Metallurgy
Piotr M. Kapias and Michał Pazdziorek
Lead and Zinc 2008
During the process of Tl extraction, samples of lead of approximately 30 g were
taken from the crucible by means of a special scoop through the opening in the
apparatus cover (not shown in Fig. 5 and Fig. 6) and then cast into a steel mould. The
Tl content in the samples was determined using the atomic absorption spectrometry
method.
6.3.
The results of the experiments
The results of the experiments are shown on figures below. Fig. 7 and Fig. 8 show
the influence of time and extractant mass on the Tl content in Pb in the process of Tl
extraction using PbCl2 and the stirring method.
Tl content in Pb, ppm
500
400
PbCl2
% m ass
of refined lead
300
200
5.0%
2.5%
100
2.5%
0.50%
0
0
5
10
15
20
25
30
Extraction process time, min
Fig. 7. The influence of PbCl2 mass on the kinetics of Tl extraction from Pb
Tl content in Pb, ppm
500
PbCl2
% m ass
of refined lead
400
2.5%
300
2.5%
0.50%
200
0.50%
100
0
0
10
20
30
40
50
60
Extraction process time, min
Fig. 8. The influence of PbCl2 mass on the kinetics of Tl extraction from Pb in the
two-stage process
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The Southern African Institute of Mining and Metallurgy
Piotr M. Kapias and Michał Pazdziorek
Lead and Zinc 2008
Fig. 9 and Fig. 10 show the influence of time on the level of thallium extraction from
lead in one-stage and two-stage stirring processes, respectively. The level of thallium
extraction from lead was defined as the ratio of difference between the initial and final
thallium content in lead and the initial content of Tl in Pb (in %):
p
w Tlm
− w Tlm
ε = 100 ⋅
(12)
p
w Tlm
Tl extraction rate, %
100
80
PbCl2
% m ass
of refined lead
60
5.0%
40
2.5%
0.5%
20
0
10
20
30
Extraction process time, min.
Fig. 9. The influence of PbCl2 mass and time on the level of Tl extraction from Pb
in the one-stage process
Tl extraction rate, %
100
PbCl2
% m ass
of refined lead
80
5.0%
60
2 x 2.5%
2 x 0.5%
40
20
0
0
10
20
30
40
50
60
Extraction process time, min.
Fig. 10. The influence of PbCl2 mass and time on the level for Tl extraction in the
two-stage process
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The Southern African Institute of Mining and Metallurgy
Piotr M. Kapias and Michał Pazdziorek
Lead and Zinc 2008
Fig. 11 and Fig. 12 show the influence of the amount of the introduced chlorine and
temperature on the Tl content in Pb as well as the Tl extraction degree using gaseous
chlorine and the injection method. For the purpose of comparing the two methods and
extractants used, the amount of gaseous chlorine introduced into the sample has been
given as hypothetical, equivalent PbCl2 mass in the figures.
Tl content in Pb, ppm
500
400
T =793 K
T = 723 K
300
T = 673 K
200
100
0
0
1
2
3
4
PbCl2 as a equivalent of gaseous chlorine, % mass
of refined lead
Fig. 11. The influence of the amount of the gaseous chlorine on the content of
thallium in the lead
Thallium extraction rate, %
100.0
80.0
60.0
T =793 K
40.0
T = 723 K
T = 673 K
20.0
0.0
0
1
2
3
4
PbCl2 as a equivalent of gaseous chlorine, % mass
of refined lead
Fig. 12. The influence of gaseous chlorine on the Tl extraction degree.
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The Southern African Institute of Mining and Metallurgy
Piotr M. Kapias and Michał Pazdziorek
7.
Lead and Zinc 2008
Discussion
The theoretical part of this work included the thermodynamic analysis of the fourcomponent two-phase Pb-Tl│PbCl2-TlCl system. During the analysis, the dependence
of component activity coefficients in two-component Pb-Tl solutions and pseudo twocomponent solutions PbCl2-TlCl upon composition and temperature was determined.
These dependencies were shown in the form of Krupkowski-Fitzner and Krupkowski
equations, respectively.
The relations determined between activity coefficients of solution component and its
composition and temperature were used to determine equilibrium conditions in the PbTl│PbCl2-TlCl system, depending on temperature and the PbCl2 mass to metallic phase
mass ratio. Then, on this basis, the value of Nernst’s distribution coefficient was
calculated, as well as the dependence upon phase composition and temperature. It was
found that the distribution coefficient decreases as the Tl content in Pb increases. For
example, at the temperature of 773 K, with Tl concentration in Pb of 50 ppm, the
coefficient is ca. 460, while with Tl concentration in Pb of 500 ppm, it is approximately
twice smaller – ca. 280. As the temperature increases, the distribution coefficient value
decreases. Taking the above into account, it can be stated that from this point of view
the lowest possible temperature is preferred in the process of Tl extraction from lead.
The dependence of the distribution coefficient upon metallic phase composition was
given an approximation via the 3rd level product, using the methods of statistics. The set
functions were used to determine the dependence between the mass of fed PbCl2 and
the content of Tl in Pb in equilibrium conditions.
Two alternative methods of introducing PbCl2 (or a different chlorine carrier, e.g.
Cl2(g)) to the system were used. The first – called the stirring method – was understood
as a way of balancing the entire system, comprising of the specified metallic phase
mass and a portion of PbCl2, to equilibrium condition. The second – called the injection
method – was understood as a way of feeding the same PbCl2 mass to the metallic
phase using the injection method. Mathematic models were prepared for both methods.
The testing of models led to the conclusion that the way of feeding the extractant to the
system may have a significant influence on the extraction level. The curves presenting
both methods shown on Fig. 4 indicate that during the process of Tl extraction from Pb,
containing in the initial condition 500 ppm of Tl, in the case of stirring method, the
mass of extractant (PbCl2) required to reach the final content of 100 ppm is, compared
to the injection method, approximately twice larger. Meanwhile, to reach the final Tl
content in Pb of 50 ppm, it has to be three times larger. The curves on the figure in
discussion also show the insignificant influence of temperature on the extraction
process within the analyzed scope.
During the experimental part of this work, two sets of experiments were conducted
which involved the application of raw lead samples, whose initial mass was ca. 5 kg
and contained ca. 500 ppm Tl.
The first set of experiments was conducted in constant temperature of 793, 813 and
853 K, using the stirring method. The influence of extraction process time and the
PbCl2 mass to lead ratio on the extraction process kinetics was studied.
The extraction process kinetics was described with the equation:
m PbCl2 ⎞
⎛
p
w Tlm = w oTlm + w Tlm
− w oTlm exp⎜⎜ − 100 ⋅ k
t⎟
(11)
m om ⎟⎠
⎝
(
)
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The Southern African Institute of Mining and Metallurgy
Piotr M. Kapias and Michał Pazdziorek
Lead and Zinc 2008
The values of constants WoTlm and “k” in equation (11) were determined based on
results of stirring method measurements and shown on Fig. 7.
The set of points in Fig. 9 and Fig. 10, which show the dependence between Tl
content in Pb and the extraction process time and the mass of used extractant, reveal
that in the initial process phase, the Tl content in Pb decreases rapidly, while in the final
phase it approximates asymptotically the equilibrium value. This value depends on the
PbCl2 mass fed. The results indicate that in the case of lead with ca. 500 ppm in the
initial condition, in order to achieve final Tl content in Pb of 50 ppm, it is necessary to
use PbCl2 mass, which equals ca. 2.5% of lead mass. In the case the extractant mass is
decreased to 0.5% of lead mass, the final Tl content in Pb is ca. 200 ppm. Further
decrease of Tl content in Pb can be achieved by feeding another portion(s) of PbCl2.
The result of such action was illustrated in Fig. 8, which shows the characteristic curve
of changes of Tl content in Pb, depending on PbCl2 mass and time in the two-stage
process, while Fig. 10 shows the extraction level in this process. The data presented in
these figures show that the level of Tl extraction from Pb, achieved by feeding the
second portion of PbCl2, is increased as the fed portions of PbCl2 is decreased. It is
worth noting that, as a result of feeding 5-times smaller amount of PbCl2, the extraction
level was only slightly lower. This is an indirect confirmation of the conclusion,
formulated on the basis of testing the extraction process models, that the injection
process (which can be treated as a sum of indefinite number of indefinitely small
portions of extractant) is more efficient than the stirring (one-stage) process.
Fig. 4 shows that the experiment results correspond well to the results of calculations
conducted via thermodynamic analysis of the stirring method. Due to the above, it can
be used in practice to determine the PbCl2 mass required to lower the Tl content in Pb
to a specified level.
The second set of experiments was conducted using gaseous chlorine as an
extractant. In order to compare the results of this series with the results in the first
measuring series, the amount of gaseous chlorine in Fig.10 and Fig. 11 has been given
as the equivalent PbCl2 mass. It is assumed that in practice the chlorine injection
process could be conducted by means of a reactor shown in Fig. 13.
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The Southern African Institute of Mining and Metallurgy
Piotr M. Kapias and Michał Pazdziorek
Lead and Zinc 2008
Chloride
or chlorides and
transportation gas
Lance
Exhaust
Lead pump
Gas cleaning system
Lead in the kottle
Fig. 13. Schematic diagram of a reactor for Tl extraction from Pb using the
extractant injection method.
A reactor of this type has been tested on industrial scale in the oxidizing lead
refining process [6]. The results of the research concerning Tl extraction from lead
using chlorine presented in Fig. 11 and Fig. 12 show that gaseous chlorine can be an
efficient means of extracting Tl from lead. At the stage of the laboratory research
presented the results were comparable. The results can certainly be further improved
while optimizing the technological parameters of the process as well as during the
construction of the reactor.
8.
Concluding remarks
1. Pb-Tl solutions in the wide range of Tl concentrations in Pb (ca. 0-30% atm.)
show thermodynamic properties approximate to ideal solutions. Solution
component activity coefficients are close to unity within this scope.
2. The Tl distribution coefficient in the Pb-Tl│PbCl2-TlCl system decreases as the
Tl content in Pb increases. For example, at the temperature of 773 K, with Tl
concentration in Pb of 50 ppm, the Tl distribution coefficient is ca. 460, while
with Tl concentration in Pb of 500 ppm, it is approximately much smaller – ca.
280.
3. The distribution coefficient value decreases as the temperature increases. Taking
the above into account, it can be stated that possibly lowest temperature is
preferred in the process of Tl extraction from lead. It was found that the
experiment results correspond well to the results of calculations conducted via
thermodynamic analysis.
4. The method of feeding extractant to the system may have a significant influence
on the extraction level. The above is a conclusion drawn from the analysis of
mathematic models of the extraction process. It was found that, in the case of
stirring method, which involves balancing the system, comprising of metallic
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The Southern African Institute of Mining and Metallurgy
Piotr M. Kapias and Michał Pazdziorek
Lead and Zinc 2008
phase and specified PbCl2 mass, to the equilibrium condition, the mass of
extractant (PbCl2) required to achieve the desired final Tl content is several
times larger, compared to PbCl2 mass fed using the injection method.
Acknowledgment. This research project Nr. N507 022 31/0629 is supported by The
State Committee for Scientific Research.
9.
Literature
1. Hultgren R..,: “Selected Values of the Thermodynamic Properties of Binary Alloys.”
American Society for Metals, Metals Park, Ohio, 1973.
2. Fitzner K.: Arch. Hutn. 28, 1983.
3. Krupkowski A.: “Laws of thermodynamics and their application in metallurgy and
metallographic.” (Polish) PWN, Kraków 1958.
4. Roth R. S., Cotts D. B. and Parker H.S.: Am. Cryst. Ass’n. Abstracts, Ser. 2, 4, 17,
(1976).
5. Collatz L.: “Numerical methods of solving differential equations.” (Polish), PWN,
Warszawa 1960, s. 72, 81.
6. Kapias P, Bednarek A., Brzezina E., Szkutnik L., Wójcik S.: „The pilot plant for lead
refining by oxygen.” (Polish), Rudy Metale R 44, Nr 8 - 9, 1999, s 385 - 390.
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