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LECTURE NOTES
MG – 4111
HYDRO-ELECTROMETALLURGY
Semester I, 2010/2011
DR. M. Zaki Mubarok
Department of Metallurgical Engineering,
Faculty of Mining and Petroleum Engineering (FTTM)-ITB
Course Outline
I.
II.
Introduction to Hydrometallurgy
Thermodynamic and Kinetic Aspects
in Hydrometallurgy
III. Leaching and Solid-Liquid Separation
IV. Solution Purification and Metals
Recovery Methods from Pregnant
Leach Solution
Course Outline
V.
Leaching and Recovery of Metals and
Oxides Ores (Au, Ag, Zn, Al, Cu, Ni)
VI. Leaching and Recovery of Sulphide Ores
(Zn, Ni, Cu)
VII. Introduction to Electrometallurgy
VIII. Metals Production by Electrolysis in
Aqueous Solution
IX. Fused Salt Electrolysis
Literatures
1.
2.
3.
4.
5.
Havlik,T., ”Hydrometallurgy: Principles and
Applications,” CRC publisher, 2008.
Habashi, F. ”A Textbook of Hydrometallurgy”,
Metallurgie Extractive, Quebec,1993
Norman L. Weiss, “SME Mineral Processing
Handbook“, Volume II, SME, 1985
Unit Processes in Extractive Metallurgy:
Hydrometallurgy, A Modular Tutorial Course of
Montana College of Mineral Science and
Technology
Biswas, A.K. And Davenport, W.G., “Extractive
Metallurgy of Copper”, Pergamon, Oxford,
fourth edition, 2002
Literatures
6. Unit
Processes
in
Extractive
Metallurgy:
Electrometallurgy, A modular tutorial course of
Montana College of Mineral Science and
Technology
7. Yannopoulus, J.C,”The Extractive Metallurgy of
Gold”, Von Nostrand Reinhold, New York, 1991
Course Structure and
Mark Distribution
• Course Structure
– Lecture
– Tutorial
– Assignment and Lab Work
• Mark Distribution
–
–
–
–
45% Midterm Exam
45% Final Exam
5% Assignment
5% Lab Work
• Attendance: 70% minimum
CHAPTER I
INTRODUCTION TO HYDROMETALLURGY
Hydrometallurgy
Extraction, recovery and purification of metals,
through processes in aqueous solutions. Metals are
also recovered in the other forms such as oxides,
hydroxides.
Electrometallurgy
Recovery
and
purification
of
metals
through
electrolytic processes by using electrical energy.
Hydrometallurgy Scope
• Traditionally, hydrometallurgy is emphasized for
metals extraction from ores.
• Hydrometallurgical processing may be used for the
following purposes:
 Production of pure solutions from which high purity metals
can be produced by electrolysis, e.g., copper, zinc, nickel,
gold, and silver.
 Production of pure compounds which can be subsequently
used for producing the pure metals by other methods. For
example, pure alumina to produce smelter grade aluminium.
• However, hydrometallurgy principles can be applied
to a variety of areas such as metals recycling from
scrap, slag, sludge, anode slime, waste processing,
etc.
Unit Processes in Hydrometallurgy
• In general, hydrometallurgy involves 2
(two) main steps:
1. Leaching
Selective dissolution of valuable metals from
ore.
2. Recovery
Selective precipitation of the desired metals from
a pregnant-leach solution.
General outline of hydrometallurgical
processes
Ore/concentrate
Leaching agent
Oxidant
leaching
Solid-liquid separation
Solid residu to waste
Pregnant Solution
Precipitant or
electric current
Solution purification
Precipitation
Pure compound
Metals
• Commonly, solution purification is conducted prior to
metals recovery from the solution.
• Solution purification is aimed at obtaining a
concentrated solution from which valuable metals can
be precipitated in the next processes effectively
• Solution purification methods which are commonly
used are as follows:
– Adsorption by activated carbon
– Adsorption by ion exchange resins
– Solvent extraction (using organic solvents)
– Precipitation with metals (cementation)
Solution purification
• Solution purifications by adsorption with activated
carbon, ion exchange resins (IX) and solvent
extraction (SX) have the same unit operations,
namely:
– Loading, and
– Elution
• In the elution step, the adsorbers are usually
regenerated for another process cycle.
Hydrometallurgy development
 Hydrometallurgy
is developed after pyrometallurgy.
Metals smelting has been practiced since thousands
years ago.
 Hydrometallurgy
was developed after the people
discovered acid and base solutions. However, modern
hydrometallurgy development is commonly associated
with the invention of Bayer Process for bauxite leaching
and cyanidation for gold extraction at the end of 19th
century (1887).
 One of important highlights of hydrometallurgy
development is uranium extraction (Manhattan Project)
aimed at nuclear weapon production in second world
war (1940‘s).
Important milestones in the development of
hydro-electrometallurgy
• Cementation & Aqua Regia Use - 8th Century
• Cyanidation - 1887
• Bayer Process - 1887
• Hall-Heroult Process - 1886, 1888
• Copper Electrowinning - 1912
• Zinc Electrolytic Process - 1916
• Manhattan Project (IX/SX) - 1940’s
• Biooxidation of Sulphide Concentrates - 1960’s
• Pressure Leaching
– Sherrit Gordon Nickel Process - 1954
– Pressure Acid Leaching of Ni Laterites - 1955
• Large Scale Copper SX/EW - 1960’s
Important milestones in the development of
hydro-electrometallurgy
• Carbon in Pulp (CIP)/Carbon in Leach (CIL)
for Gold Recovery - 1980’s
• Pressure Oxidation of Zinc Sulphides - 1981
• Two-Stage Zinc Pressure Leach - 1993
• Atmospheric Leaching of Zinc Sulphides
– Albion (1993), Outokumpu (1999)
Recent Developments:
• Skorpion Project (Anglo American) – 2003 (Zn from ZnS)
• Hydrozinc (TeckCominco) - 2004
• Inco’s Goro and Voisey Bay Projects - 2007
• Leaching of Chalcopyrite (CuFeS2) Ores
 Hydrocopper (Outokumpu)  Cu from sulfidic ores
• Atmospheric leaching of nickel laterite ore: 2008?
Hydrometallurgy vs. Pyrometallurgy
Hydrometalurgy
Pyrometallurgy
Treat high grade
ore?
Less economic
More economic
Treat low grade ore?
Possible with
selective leaching
Unsuitable
Treat sulphide ore
No SO2; otherwise
So or SO42- are
generated
SO2 generated (can
be converted to
H2SO4)
Separate similar
metal, such as Ni
and Co
Possible with certain
method
Not possible
Pollutant
Waste water,
solid/slurry residues
Gases and dust
Reaction rates
Slower
Rapid
Hydrometallurgy vs. Pyrometallurgy
Hydrometalurgy
Pyrometallurgy
Scale of operation?
Possibly economic to Unconomic at smale
be done at small
scale operation
scale operation and
expansion is easier
Capital cost
Generally lower than
pyrometallurgy
Higher
Energy cost
Lower
Higher
Materials Handling
Slurry Easy to be
Pumped and
Transported
Handle Molten
Metal, Slag,
Matte
Residues
Residues – Fine
and Less Stable
Slags – Coarse
and Stable
CHAPTER II
Thermodynamic and Kinetic
Aspects in Hydrometallurgy
Spontaneous Reaction, Equilibrium State
• As has been learned in basic engineering courses,
chemical reaction will spontaneously occur when the
Gibbs free (G) < 0.
G = Go + RT ln K
• G = 0  process is in equilibrium state
–
–
–
–
Go = standard Gibbs free energy
R = ideal gas constant = 8,314 J/K.mol
T = absolute temperature of the system (K)
K = equilibrium constant
• Standard Gibbs free energy is determined at:
– Gaseous components partial pressure = 1 atm
– Temperature = 25 oC (298 K)
– Ions activity = 1
Equilibrium Constant
• For reaction:
aA + bB  cC + dD

C   D
K
 A  B 
a
a
c a
a a
d
b
a
A = activity of A = γ[ A] →[ A] = A concentration
 = activity coefficient of
component A
For gaseous component of X→ a X = γ p X,
in which p X = parsial pressure of X
Nernst Equation
• Hydro-electrometallurgical processes often involve
electrochemical reactions.
• For electrochemical reaction
G = -nFE,
Go = -nFEo, therefore
RT
E E 
ln K
nF
o
Nernst Equation
In which,
E = potential for reduction-oxidation reaction
Eo =standard potential for reduction-oxidation reaction
n = number of electron involved in the electrochemical reaction,
F = Faraday constant = 96485 Coulomb/mole of electron
• Spontaneous process  E > 0  G < 0
Chemical reactions usually
perform in leaching processes
• Dissolution by acid
– Example: ZnO(s) + 2H+ → Zn2+(aq) + H2O(l)
• Dissolution by base
– Example: Al2O3(s) + 2OH- → 2AlO2-(aq) + H2O(l)
• Dissolution by complex ion formation
 Example: CuO(s) + 2NH4+(aq) + 2NH3(aq) →
Cu(NH3)42+(aq) + H2O(l)
Chemical reactions usually
perform in leaching processes
• Dissolution by oxidation
– Ex: CuS(s)+ 2Fe3+ → Cu2+(aq) + 2Fe2+ + So(s)
Other oxidators: O2, ClO-, ClO3-, MnO4-, HNO3,
H2O2, Cl2
• Dissolution by reduction mechanism
– Ex: MnO2(s) + SO2(aq) → Mn2+(aq) + SO42-(aq)
Correlation of free energy (G)
and heat (enthalphy = H)
G = H - TS
Go = Ho - TSo
∆Ho = Standard enthalpy (kJ/mol)
∆Go = Standard entropy (kJ/mol)
∆Go (reaction) = ∆Go (products) - ∆Go (reactants)
∆Ho (reaction) = ∆Ho (products) - ∆Ho (reactants)
∆So (reaction) = ∆So (products) - ∆So (reactants)
ΔHT = ΔH298 +
T
∫Cp dT
298
Cp = heat capacity at constant pressure (J/molK)
Where possible, processes are designed to be autothermal →
maintain constant temperature by the heat given by the reaction
Calc. example 1
• Find K for each reaction using
a) Standard free energy data
b) Standard electrode potential data
Calc. Example 2
a) What is the electrode potential of the
Ni2+/Ni reaction in sulphate solution at 25°C
at a Ni2+ concentration of 0.005 M
(assumption: activity of Ni2+ is equal to its
molar concentration)
b) At what pH is H2 at 10 atm at equilibrium
with this solution and pure nickel?
Ni2+ + 2 e = Ni
E° = -0.26 V
2H+ + 2 e = H2
E° = 0.00 V
Pourbaix Diagram
• Pourbaix Diagram = Potensial (Eh) – pH
Diagram.
• The diagram represents thermodynamic
equilibrium of metal, ions, hydroxides (or,
oxides) in aqueous solution at certain
temperature (isothermal).
• The boundary of stability regions of metal,
ion, hydroxides (or oxides) are equilibrium
lines.
• Does not reflect reaction kinetics.
Pourbaix Diagram
• Three possible types of equilibrium lines:
– Horizontal
– Vertical
– Slope
• Variations in ion activities are plotted as
contours/dashed lines
• Horizontal Line: for equilibrium reactions
that are independent of pH.
Horizontal Line
• Example:
Fe3+ + e = Fe2+
Eo = 0.77 V
R = 8.314 J/Kmol, T = 298 K, F = 96500 C/mol e-, n = 1 mol eRT aFe 2 +
o
E =E ln
nF a Fe3 +
If all ion concentrations are assumed to be equal to
their molar concentrations  10-6 M.
3+
[
]
RT
Fe
o
E =E ln
nF [Fe 2+ ]
8,314 x 298 [Fe 2+ ]
E = 0,77 ln
[Fe3+ ]
2x96500
[Fe2+ ]
E = 0,77 - 0,0592 log
[Fe3+ ]
E = 0,77
Vertical Line
• Reactions do not involve electron → n = 0, no
potensial , the equilibrium depends only on pH.
• Example:
Fe2O3 + 6H+ = 2Fe3+ + 3H2O
K = [Fe3+]2/[H-]6
pH = - lo ga H+ ≈ - log[H+ ] = 14 - pOH
-
pOH = - lo ga (OH)- ≈ - log[OH]
For certain Fe3+ concentration we can determine
the equilibrium pH for the above reaction.
Slope Line
• For reactions that depend both on potensial
(Eh) dan pH.
• Example:
If all ion concentrations are assumed to be equal to
their molar concentrations  10-6 M.
Water stability region (dotted lines)
• Upper boundary line
 At pO2 = 1 atm
• Lower boundary line
 At pH2 = 1 atm
Eh-pH diagram of Fe-H2O system at 25°C
Eh-pH Diagram of Zn-H2O System at 25 oC.
Eh-pH Diagram of Cu-H2O System at 25 oC.
Application of Eh – pH diagram in
hydrometallurgy
• Predicting potential leaching behaviour
for certain mineral system
• Predicting the possibility of metals ion
precipitation at the purification of
pregnant-leach solution
Application of Eh – pH diagram in hydrometallurgy
Fe(OH)3 or Fe2O3 can be precipitated from Fe3+ at lower
pH than the precipitation of Zn2+ to Zn(OH)2 or ZnO.
Fe2+ have to be oxidized to Fe3+ to gain lower pH value for
Fe(OH)3 precipitation.
Pourbaix Diagram can be constructed at various
temperature for more than two systems
Eh-pH diagram of Zn-S-H2O system at 25oC
Diagram Pourbaix in Presence of Complex Ion
• Example: Au-H2O system with the presence of
cyanide (CN-) ion (case of gold cyanidation leaching)
• Equilibrium of Au3+/Au
Au 3  3e  Au
G 0  G 0 f (Au )  G 0 f (Au 3 )
 433 KJ / mol
Standard reduction potential for Reaction 1:
0
3


G
433

10
E0 

nF
3  96500
 1.5 V
(1)
• Equilibrium reaction of O2/H2O
O 2 + 4H + + 4e = 2H 2 O
(2)
Eo = 1.23 V.
Au 3+ / Au = 1.5
O 2 / H2O = 1.23
E
Au 3 + / Au
> EO2 / H2O
Therefore, Au3+ ions are not stable in water and readily
reduced to Au by oxidation of H2O to O2 (the opposite
of Reaction 2). In the other word, gold can not be
oxidized (dissolved) in water only with the presence O2.
Potensial – pH diagram of Au–H2O system without
the presence of complexing agent
With the presence of CN-, Au3+ forms STABLE
COMPLEX of “aurocyanide“ (Au(CN)2-) and the
potential-pH diagram for Au changes significantly as
follow:
Eh-pH Diagram of Au-CNH2O system at 25 oC for [Au]
= 10-4 M and [CN-] = 10-3 M
• By the presence of cyanide ions,
Au+ + e = Au
E = 1.69 – 0.0591 log [Au+]
Au+ + 2CN- = Au(CN)2(K = 2 x 1038)
Au(CN)2- + e = Au + 2CN- ...........................
(3)
E  1.69  0.0591log K  0.0591log
 CN
a

 2
/ a AuCN 2


In comparison to the first reaction that has Eo of 1.69 V,
Reaction (3) has much lower Eo at -0.57 V.
Dissolution of Au is limited by the following equilibrium of
Reaction (3).
• During cyanidation leaching, dissolved oxygen is
required to oxidize Au prior to the formation of stable
complex of Au(CN)2-.
 Interactions in Electrolyte Solution
Two types of interactions in electrolyte:
- Ion-ion interaction, and
- ion-solvent interaction
Knowledge of interaction in electrolyte solution is
important because the interactions affect solvation
effects, diffusion, conductivity, ionic strength and
activity coefficients of ions in solution.
Interactions in electrolyte solution influence the
transport properties of ions in solution.
 Ionic Strength and Activity Coefficient
- Ionic strength (I), expresses the ionic concentration
that includes the effects of ionic charge.
- Ionic strength (I) is defined as follow:
1
2
I   ci z i
2 i
in which ci = concentration of ion i in molar (mol/L)
and zi = the charge of ion i.
- It is found that activity coefficient, electrical
conductivity and the rates of ionic reactions are all
the functions of ionic strength.
Ionic Strength for unit concentration in molal
- Remember, molality = moles of solute in 1 kg
solvent. Molality can be converted to molality by the
following correlation:
ci
mi =
0.001ρ - ∑c iMi
in which Mi = the molar mass of each solute in
kg/mol (not in g/mol), ci = molarity of solute i, and 
is the density of the solution in kg/m3 (=g/L)
- In dilute solutions, ci  0.001mio (in which o =
density of pure solvent).
Ionic Strength for unit concentration in molal
- Therefore for dilute solution,
1
1
2
∑
I=
c i z i = ∑0.001m i ρ o z i 2
2 i
2
0.001ρo
∑m z 2
I=
i i
2
If the solvent is water at 25oC (density  1000 kg/m3),
then:
1
Similar form with ionic
2
∑
I ≈ mi zi
strength in molarity
2i
- Molar activity coefficient can be converted to molal
activity coefficient by the following correlation:)
 
f
1  M s m
for salt, or
i 
fi
1  M s  m
for single ion.
in which  = total moles of ion formed during
complete dissociation, m = ionic molality and Ms =
molecular weight of solvent (kg/mol).
Activity and Activity Coefficient,
DEBYE-HUCKEL LAW
- Debye Huckel Law correlates the activity coefficient (fi , i)
with ionic strength (I).
- Forms of Debye-Huckel equations
depend on
concentration of solution and the unit concentration used.
- For dilute solution at 25 oC and I given in molar (M),
log f i  0.51159zi
2
I
log f ± = 0.51159 z + z - I
for single ion, and
for salt.
- The above equations are known as LIMITING DEBYE
HUCKEL LAW.
The limitation of LIMITING Debye-Huckel Equation
• The D-H Limiting Law is called a ”limiting” law
because it becomes increasingly accurate as the limit
of infinite dilution is approached.
• Up to concentrations of about 0.01m THE LIMITING
D-H LAW gives reasonable values, but at higher
concentrations
the
calculated
activity
coefficient
become inaccurate (high %error compared to the
values determined experimentally).
 Debye-Huckel Law for Concentrated
Solution
- For concentrated solution (> 0.01 molal), Limiting
Law D-H is modified by considering the ionic size
parameter:
log f  
 A z z
I
1  Ba I
- in which A and B are constants that depend on the
kind of solvent and temperature, a = ion size
parameter.
- For aqueous solution at 25 oC, A = 0.51159 and B =
3.2914 x 109 meter.
ACTIVITY AND MEAN ACTIVITY
- Molar activity and molar activity of a single ion i is
determined as follow:
a i  i mi
a i  fi Ci
and
- For 1 mole of M+A- salt that dissociates to + mol of
Mz+ and - mole of AzM+A-  + Mz+ + - Az-
 = + + -
ACTIVITY AND MEAN ACTIVITY
Mean molal activity coefficient can be determined by
the following correlation:

         

1/ 
Mean molality,


m  m m

  1 /  
Note that m  m
Thus, mean molal activity,
a    m


m     

  1 /  
m
Exercise: 1
1. Determine the molar activity coefficient of Ca2+ at
25oC using relevant Debye Huckel Equation in the
following solution:
a. 0.0004 mole of HCl and 0.0002 mole of CaCl2 in
one liter solution
b. 0.004 mole of HCl and 0.002 mole of CaCl2 in one
liter solution
c. 0.4 mole of HCl and 0.2 mole of CaCl2 in one liter
solution
Ion size parameter for Ca2+ = 0.4 nm.
Exercise 2:
2. The stoichiometric mean activity coefficient at 25 oC
of the sulphuric acid in a mixture of 1.5 molal
sodium sulphate (Na2SO4) + 2 molal H2SO4 is
0.1041. If the second dissociation constant, K2, for
sulphuric acid is 0.0102 and the pH of the solution is
–0.671, calculate:
a) the molal activity of H2SO4
b) the molal activity of SO42c) the molal activity of HSO4d) the mean activity of H2SO4
Exercise:3
1 gram FeCl2, 1 gram NiCl2 and 1 gram of HCl are added to 200
ml of deaerated water. Platinum electrodes are used to deliver
electrical current so that the electrolysis performs. The anodic
and cathodic current density are 1000 A/m2. The following are
the reactions and Eo (in the reduction direction) that may occur:
Fe2+ + 2e = Fe
Eo = -0,277 V
Ni2+ + 2e = Ni
Eo = -0,250 V
2H+ + 2e = H2
Eo = 0 V
Cl2 + 2e = 2ClEo = 1,359 V
a)
b)
c)
d)
Calculate molar activity coefficients of the cations and anion
contained in the solution (use the Finite Size of Debye
Huckel Limiting Law)
Calculate the activity of the cations and anion contained in
the solution
Determine the half cell potential of the above reactions
Which pair of redox (reduction –oxidation reaction) that
would occur (based on the calculation of c)
Exercise: 3 (cont.)
e) What would be the cell voltage of the reaction d
Data: Atomic weight Fe = 55.8, Ni = 58.7, Cl = 35.5, H =1
Ion size parameter in nm : Fe2+ = Ni2+ = 0.6, H+ = 0.9, Cl- = 0.3
H2 overpotential = 0.28 V
Cl2 overpotential = 0.03 V
Ohmic overpotential = 0.25 V.
Kinetics in Hydrometallurgy
• Kinetics in hydrometallurgy deals with the kinetics of
leaching, adsorption and precipitation
• Studying of leaching kinetics is done for the
establishment of the rate expression that can be used
in design, optimization and control of metallurgical
operations.
• The parameters that need to be estabished:
– Numerical value of the rate constant
– Order of reaction
– Rate determining step
– Activation energy
Leaching Kinetics
• Consider the dissolution of a metal oxide, MO, with
an acid by the following reaction:
MO(s) + 2H+(aq)  M2+(aq) + H20(aq)
• The reaction rates for this leaching system can be
given by
dC MO
1 dC H+
rR = = dt
2 dt
or
dC
rP =
M2+
dt
=
dC H2O
dt
Leaching Kinetics
• For general example if a chemical reaction involves A
and B as reactants and C and D as products, the
stoichiometric reaction can be written as follows:
k1
aA + bB ⇔cC + dD
(1)
k2
where
a, b, c, and d = stoichiometric coefficients of species
A, B, C, and D, respectively
k1, k2 = reaction coefficients in the forward and
reverse directions, respectively
Leaching Kinetics
• The rate expression of this stoichiometric reaction
can be written in a more general way:
1 dC A
1 dC B 1 dC C 1 dC D
= =
=
= k1CnA CBm - k 2CpCCDq
a dt
b dt
c dt
d dt
(2)
where
CA, CB, Cc, and CD are concentrations of species A,
B, C, and D, respectively and m, n, p, q are orders of
reaction.
Leaching Kinetics
• However, if the reaction given in Eq. 1 is irreversible,
as in most leaching systems, Eq. 2 is reduced to the
following form:
1 dC A
= k 1CnA CBm
a dt
or
dC A
= k1' CnA CBm
dt
where k1’= k1 x a.
• For this system, the rate constant, k1', and the orders of
reaction, n and m, should be determined with the aid of
leaching experimental data.
• The rate expression given in the above equations can be
further reduced if the reaction is carried out in such a
way that the concentration of A is kept constant.
• For such situations, the rate expression is reduced to:
dC A
= k 1″
CBm
dt
where k1”= k1’ x CAn. It should be noted that the rate
constant and the order of reaction are constant as
long as the temperature of the system is maintained
constant.
• Consider the dissolution of zinc in acidic medium:
Zn(s) + 2H+(aq) → Zn2+(aq) + H2(g)
• For the above reaction, the rate of disappearance of H+
ion is directly related to the rate of appearance of Zn2+
ion; thus,
dC
Zn2 +
dt
=
1 dCH+
= k′
CmZnCHn = kCHn
2 dt
• If concentration of zinc metals is assumed to be constant and CH is
further abbreviated generally as CA, then the equation can be written
as follow:
dC A
= kC nA
dt
• The order of reaction, n, can be any real number (0, 1, 2, 1.3, etc.).
• When n = 0, the reaction is referred to as “zero order” with respect
to the concentration of A.
dC A
= kC A 0
dt
CA
∫dC A
Co
A
or
= CA - C
XA =
o
A
t
= k ∫dt
0
= -kt
k
t
CoA
where CAo represents the concentration of A at t = 0, and XA
represents the fractional conversion, i.e., XA = [ (CAo — CA)/ CAo].
• If the plot of XA versus t gives a straight line, the zeroorder assumption is consistent with experimental
observations and the k value can be obtained from the
slope of the plot.
XA
k/CAo
time
• When n = 1, the reaction is first order with respect to the
concentration of A:
dC A
= kC A
dt
t
dC A
∫
= -k ∫dt = -kt
o
0
C
CA
A
CA
CA
ln o = -kt
CA
ln(1 - X A ) = - kt or X A = 1 - e -kt
ln (1 - XA)
k
time
For second order reaction,
t
dC A
∫ 2 = - k ∫dt = -kt
0
CA
Co
A
CA
1
1
= -kt
C A CoA
XA
= CoA kt
1 - XA
XA
(1 - XA)
k
CAok
time
If the second-order assumption is valid, we obtain a
straight line from a plot of XA/(1 - XA) versus t, and the
rate constant can be determined from the slope of the
plot.
Temperature Effect on the Reaction Rate
(Arrhenius Law)
Reaction rate increases markedly with increasing
temperature. It has been found empirically that
temperature affects the rate constant in the manner
shown in the following equation:
k = k oe-Ea / RT
ln k = ln k o -
Ea 1
R T
log k = log k o -
Ea
1
2.303R T
where Ea is the activation energy and k° is a constant
known as the frequency factor, frequently assumed to
be independent of temperature.
Modeling of heterogenous reaction
kinetics
• Heterogenous reaction between solid and fluid in
hydrometallurgical processes is frequently modelled with
“shrinking core“ model.
• If we select a model we must accept its rate equation, and
vice versa.
• If a model corresponds closely to what really takes place,
then its rate expression will closely predict and describe
the actual kinetics;
• If a model differ widely from reality, then its kinetic
expressions will be useless.
• Detailed of modeling and relevant kinetics equations for
various rate determining steps can be found in previous
course (Metallurgical Kinetics).
• For determination of Ea, number of experiments, at least
at three or four different temperatures are needed, with
all other variables being kept constant. The next step is
to calculate the rate constant for each temperature as
discussed previously.
• A plot of In k versus 1/T yields a straight line from which
the activation energy, Ea, can be determined
• Activation energy value can be used to predict the rate
determining step of the reaction:
• Ea = 40 – 80 kJ/mol: process is controlled by surface chemical
reaction
• Ea = 8 – 20 kJ/mol: process is controlled by diffusion to and from
the surface
Mass Transfer in Solution
• For hydrometallurgical system, mass transfer of
component i in solution frequently consists of a
molecular diffusion term, migration term, and convective
diffusion term, as indicated in the following expression:
Ni =
- Di∇Ci - ziμiFCi∇Φ + Ci V
where
Ni= flux of i, Ci = concentration of i, Di = diffusion coefficient of i
Ci = concentration gradient of i, zi = valence of the specified ion,
µi = ionic mobility, F = the Faraday constant, Ф = electrical
potential gradient, and V = net velocity of the fluid of the system
First and Second Fick’s Law of Diffusion
• If Ni consists of the molecular diffusion term only,
Ni = -Di∇Ci
(Fick's first law)
∂Ci
+ Di∇ 2Ci = 0
∂t
(Fick' s second law )
• Dimensionless Parameter for Convection Calculation
LV  LV   


 
Di    Di 
where µ = the viscosity of the
fluid, ρ = the density of the
fluid
Dimensionless Parameter for
Convection Calculation
• The parameter LV/Di is known as the Peclet number and
can be separated into two other parameters: Lvρ/µ that
known as the Reynolds number, and µ/ρDi is the
Schmidt number.
• Peclet number is regarded as a measure of the role of
convection against diffusion,
LV
∇Ci V
convective diffusion
=
=
D i D i (∇Ci / L) molecular diffusion
• For most hydrometallurgical systems, the Schmidt number
is on the order of 1,000 because the diffusivity of ions and
kinematic viscosity of water are, respectively, on the order
of 10-5 cm2/s and 10-2 cm2/s.
• Therefore, if the Reynolds number is greater than 10-3,
the Peclet number is greater than 1, and consequently,
convective diffusion is more dominating than molecular
diffusion in such systems.
Mass Transfer Coefficients for Convective Diffusion
• For systems with large Peclet numbers, it is frequently
assumed that there is a diffusion boundary layer at some
distance from the solid surface. For such systems, it is
quite common to write the mass flux from the bulk
solution to the solid surface as follows:
Ni = km (Cb - Cs)
where
Ni = mass flux of species i
km = mass transfer coefficient, in cm/s
Cb = concentration of species i in the bulk solution,
in mol/cm3
Cs = concentration of species i at the solid surface,
in mol/cm3
• Because the units of measure of km are the same as those of
(D/), where  is the diffusion boundary layer thickness, km, is
often substituted by this ratio. Therefore,
D
N i = (Cb - Cs )
δ
• The diffusion boundary layer thickness is often estimated
by the relationship km = D/δ, provided km is known.
Mass Transfer from or to a Flat Plate.
• The mass transfer coefficient for a flat plate where fluid
is flowing over the plate at a velocity V0 has been well
documented.
• The mass transfer coefficient for such a system can be
estimated from first principles and has the following
form:
k m = 0.664D2 / 3ν -1/ 6 L-1/ 2 V01/ 2
For Re < 106
where
D = the diffusivity of the diffusing species
v = the kinematic viscosity of the fluid
L = the length of the plate
Rotating Disk.
• Although it is not a practical geometry, because the
mathematical representation of the system is exact and
follows very closely to the experimental data, a rotating
disk is frequently used to determine the mass flux and
the mass transfer coefficient.
• The mass transfer coefficient for this system is as follow
k m = 0.62D2 / 3ν -1/ 6ω1/ 2
• The equation is valid for the Reynolds number, r2ω/ν is
less than 105, where r and ω are, respectively, the radius
and the angular velocity of the disk.
Particulate System
• It has been demonstrated that the mass transfer
coefficient for particulate systems can be given by the
following equation:
2D
km =
+ 0.6Vt1/ 2 d -1/ 2 ν -1/ 6 D 2 / 3
d
where d = the diameter of the particle, Vt = the slip
velocity, which is often assumed to be the terminal
velocity of the specified particle.
• The terminal velocity of a particle can be
calculated using the following equation
depending on the Reynolds number of the
system, which is defined by dVtρ/μ, where ρ is
the density of the fluid:
2r 2 (ρs - ρ)g
Vt =
9μ
where ρs is the density of the particle. The
preceding equation is often referred to as the
Stokes' equation and is valid as long as the
Reynolds number is less than 1.
When the Reynolds number is between 1 and
700, the following equations are used:
μ A
Vt = 10
dρ
1/ 2
(
)
A = 5.0 0.66 + 0.4 log K - 5.55
4gd 3ρ(ρs - ρ)
K=
3μ2
where g is the gravitational coefficient.
Example 1:
• A cementation reaction, Zn + Cu2+ → Cu + Zn2+,
is taking place at the surface of a zinc plate of 10
cm x 10 cm area.
• Feed flowing parallel to the plate at a velocity of
1 m/s contains copper at 1 mo/dm3.
• Suppose we want to estimate the rate of
deposition assuming that the mass transfer of
Cu2+ to the zinc plate is rate determining step.
The diffusivity of Cu2+ is 7.2x10-6 cm2/s, and the
kinematic viscosity of water is 0.01 cm2/s.
1 dN Cu 2+
= k m (Cu 2b+ - Cu s2+ ) = k m Cu 2b+
S dt
where
S
Ncu2+
Cub2+
Cus2+
= the surface area of the plate
= the number of moles of Cu 2+ ion
= the concentration of Cu2+ in the bulk
= the concentration of CU2+ at the interface
km = 0.664 (7.2 x 10-6)2/3 (0.01)-1/6 (10)-1/2 (100)1/2
= 0.664 x 3.7 x 10-4 x 2.15 x 0.316 x 10
= 1.7 x 10-3 cm/s.
Therefore,
1 dN Cu 2

 1.7 10 3 1,000  1.7 mol / cm 2 .s
S dt
and
100 10
Re 
 105
0.01
Example 2:
• Consider the situation from the previous example,
except that instead of a zinc plate, zinc particles 100
µm in diameter are suspended in a 1 mol/dm3 Cu2+
solution. Suppose we want to estimate the rate of
deposition of Cu2+ (Note that the density of Zn is 7.14
g/cm3.)
For particulate,
4  981 0.01 1 7.14  1
K
 80.3
4
3  10
1/ 2
A  50.66  0.4 log 80.3  5.55  0.412
3
Therefore,
Vt  100.412  2.58 cm / s


Re  10  2  2.58 / 10  2  2.58
As a result ,
2  7.2 10 6
km 
 0.6  2.581/ 2  0.011/ 2  0.011/ 6  7.2 10 6
0.01
 1.44 10 3  7.75 10 3

 9.19 10 3 cm / s
Finally ,
1 dN Cu 2
km  
 9.19 10 3 1,000  9.19 mol / cm 2 .s
S dt

2/3