�������� ����� ��

A shrinking particle - shrinking core model for leaching of a zinc ore contain-ing silica

Vida Safari, Gilnaz Arzpeyma, Fereshteh Rashchi, Navid Mostoufi

PII: S0301-7516(09)00142-2DOI: doi: 10.1016/j.minpro.2009.06.003Reference: MINPRO 2179

To appear in: International Journal of Mineral Processing

Received date: 10 January 2009Revised date: 24 May 2009Accepted date: 11 June 2009

Please cite this article as: Safari, Vida, Arzpeyma, Gilnaz, Rashchi, Fereshteh, Mostoufi,Navid, A shrinking particle - shrinking core model for leaching of a zinc ore containing sil-ica, International Journal of Mineral Processing (2009), doi: 10.1016/j.minpro.2009.06.003

This is a PDF file of an unedited manuscript that has been accepted for publication.As a service to our customers we are providing this early version of the manuscript.The manuscript will undergo copyediting, typesetting, and review of the resulting proofbefore it is published in its final form. Please note that during the production processerrors may be discovered which could affect the content, and all legal disclaimers thatapply to the journal pertain.

ACC

EPTE

D M

ANU

SCR

IPT

ACCEPTED MANUSCRIPT

1

A shrinking particle - shrinking core model for leaching of a zinc

ore containing silica

Vida Safari1, Gilnaz Arzpeyma1, Fereshteh Rashchi2, Navid Mostoufi 1,*

1 Department of Chemical Engineering, 2 Department of Metallurgy and Materials Engineering,

University of Tehran, PO Box 11155/4563, Tehran, Iran

Abstract

A new mathematical model was developed for leaching of zinc ores containing silicates such

as hemimorphite which produce a gel during leaching with sulfuric acid. This model is based

on the shrinking core model in which the particle size and the reacting core shrink

simultaneously. It was shown that the actual dissolution time of the ore particles is longer

than the time corresponding to the dissolution of chemical zinc oxide itself. It was suggested

that because of the existence of silicates in the ore, a gelatinous layer was formed around the

reacting core. Since the gel product is soft, it breaks apart when the particles collide and as a

result, the particles shrink. However, a thin gelatinous layer always covers the reacting core

which increases the mass transfer resistance and increases the leaching time. This model was

applied to leaching of a zinc-rich tailing containing hemimorphite and the thickness of the

gelatinous layer as well as the diffusion coefficient in this layer was determined.

Keywords: Leaching; Kinetics; Zinc silicate; Shrinking core model; gelatinous silica layer

1. Introduction

The main source of zinc metal production is zinc sulfide ore. Currently, depletion of these

sulfide ores has brought more emphasis on zinc extraction from oxides, silicates or even

* Corresponding author, Phone: (98-21)6696-7797; Fax: (98-21) 6640-1024; E-mail address: mostoufi@ut.ac.ir

ACC

EPTE

D M

ANU

SCR

IPT

ACCEPTED MANUSCRIPT

2

secondary sources. Zinc extraction is performed mainly by hydrometallurgical methods. In

the hydrometallurgical process, the ore is first leached by a solvent and then before electro-

winning a purification process is used to prepare the solution for eletrolysis.

During the leaching process of zinc oxidized ore, soluble zinc sulfate forms which stays in

solution. In this process, the lead compounds form lead sulfate precipitates which transfer to

the leaching filter cake during the solid/liquid separation. Leaching of the ore at pH ca. 2,

transforms the silicate compounds of the ore to colloidal silica, i.e., a gel (Matthew and

Elsner, 1997).

Process kinetics and optimum operating conditions have been studied for the leaching of zinc

silicate ore tailings. Monhemius and Terry (1983) investigated the influence of different

parameters on the kinetic of acid dissolution of both natural and synthetic willemite

(Zn2SiO4) and hemimorphite (Zn4Si2O7(OH)2.H2O). Specific rate constants were estimated

for leaching of both willemite and hemimorphite in different acidic media. They found that

the dissolution was mixed chemical/diffusion controlled in hemimorphite and chemically

controlled in the case of willemite. Abdel-Aal (2000) investigated the kinetics of sulfuric

acid leaching of low- grade zinc silicate ore. In their study, diffusion through the product

layer was determined as the rate controlling step. Espiari et al. (2006) verified extraction of

zinc from tailings of lead flotation plant. They studied the effect of different parameters on

kinetics of zinc dissolution and found that the rate determining step is the physico-chemical

desorption process. Souza et al. (2007) studied the effect of particle size, temperature and

initial acid concentration on leaching of zinc silicate ores. They concluded that the grain

model with porous diffusion control is the rate controlling step.

ACC

EPTE

D M

ANU

SCR

IPT

ACCEPTED MANUSCRIPT

3

Some researchers have reported the possible existence of internal diffusion resistance. For

instance, Pecina et al. (2007) have reported formation of a sulfur layer in zinc sulfide

leaching with high concentration of sulfuric acid solution containing hydrogen peroxide and

indicated that this layer reduces leaching efficiency. Mulak et al. (2005) found that in

leaching of spent nickel oxide catalyst with sulfuric acid, an aluminium-rich layer surrounds

the unreacted core of the particle and grows inward as the particle reacts. In leaching process

of sphalerite containing a lower concentration of iron, the decrease in zinc dissolution rate (as

compared to a greater iron-containing sphalerite mineral) was attributed to formation and

growth of a polysulfide surface layer during the initial rapid leach period (Weisener et al.,

2004).

Based on the above evidences, it can be concluded that when a core shrinks, an internal

resistance layer, either a reaction product layer or a gel film forms around the core and results

in a decrease in the extraction yield in the leaching process. However, in many cases, the size

of the particle (including unreacted core and the layer) decreases with time. In other words,

although the product layer forms around the core, it shrinks as the core shrinks. In all the

previous studies, existence of this gelatinous layer around the particle was neglected during

the kinetics calculations and its effect on the kinetics of leaching was not taken into account.

In the present study, a mathematical model has been developed based on the shrinking core

model in which the resistance of the gelatinous product film is also considered. This kinetic

model consists of three steps: external diffusion in the liquid, internal diffusion in the

gelatinous product film and chemical reaction on the surface of the core.

2. Model Development

ACC

EPTE

D M

ANU

SCR

IPT

ACCEPTED MANUSCRIPT

4

Zinc silicate ore, as hemimorphite, reacts with sulfuric acid according to the following

reaction:

4 2 7 2 2 2 4 4 2 6 2( ) . 4 4 ( ) 3Zn Si O OH H O H SO ZnSO Si O OH H O+ → + + (1)

Zinc sulfate is soluble in water. However, disilicic acid, Si2O(OH)6, which apparently

polymerizes to produce polysilicic acid, forms a gelatinous phase at specific acidic pH which

remains around the particles surface and leads to a decrease in the extraction yield.

The most common models considered in leaching are illustrated in Fig. 1. Fig. 1a corresponds

to the case when the reaction takes place on the exposed surface of the particle and the

product completely dissolves in the liquid. This shrinking particle model (SPM) has been

used by researchers such as Espiari et al. (2006), Aydoğan et al. (2006) and Velardo et al.

(2002). If the product does not dissolve in the liquid, the particle size would not change but

the reacting core shrinks inside the particle. This situation is shown in Fig. 1b and the model

is called “Shrinking Core-Constant Particle Size”. It has been used by researchers such as

Liu et al. (2006), Liddell (2005) and Szubert et al. (2006). Fig 1c demonstrates schematic of

a model called “Shrinking Core-Shrinking Particle”. In this case as the reaction proceeds, the

unreacted core of particle shrinks while a gelatinous silica layer forms around the core.

However, since this layer is soft, it breaks apart when the particles collide. Nevertheless, a

thin layer of silica remains around the core. The silica layer creates a resistance during acid

transfer from the solution to the surface of the core. In the present work, the last model was

considered as the base kinetic model since the silica product does not dissolve completely in

the acid solution. Therefore, the internal diffusion through the gel film should be taken into

account.

2.1. Assumptions

ACC

EPTE

D M

ANU

SCR

IPT

ACCEPTED MANUSCRIPT

5

The assumptions of the model are as follows:

• The particles are spherical.

• During the process, the particle shrinks uniformly, thus, it maintains its spherical

shape.

• In the absence of adequate information, the thickness of the silica layer around the

core was assumed to be constant during the leaching process.

• Although there exist many reactions in the leaching of zinc by sulfuric acid from the

ore, for the sake of simplicity, the main reaction considered in this work is dissolution

of zinc oxide in acid. In other words, the main source of zinc in the leaching was

assumed to be zinc oxide.

• Hemimorphite particles were considered as the source of silica responsible for the gel

formation.

• Other substances present in the ore do not have any significant effect on the kinetics.

• The temperature remains constant during the process.

• The particle and the gelatinous layer are both non-porous. Thus, mass transfer occurs

through molecular/ion diffusion in these phases.

2.2. Kinetic Modeling

Considering all the above assumptions, the first step for developing the model is to define a

criterion indicative of the advancement of reaction versus time. The rate of reaction per unit

surface of the core can be related to the dissolution rate of zinc oxide as follows:

ZnO ZnOr

c

M dnR

S dt= − (2)

The rate of zinc oxide disappearance can be expressed as:

ACC

EPTE

D M

ANU

SCR

IPT

ACCEPTED MANUSCRIPT

6

ZnO Ore c c

ZnO

dn GS dr

dt M dt

ρ= (3)

It has been shown in several references that the rate of reaction for dissolution of zinc oxide,

Rr, is first order with respect to the concentration of the solvent (e.g., Espiari et al., 2006;

Monhemius and Terry, 1983):

Ar

ZnO

kCR

M= (4)

Therefore, the rate of shrinkage can be expressed as:

c A

Ore

dr kC

dt Gρ= − (5)

The acid concentration, CA, used in Eq. (5) should be evaluated at the surface of the core. To

find the acid concentration at reaction surface, mass balances should be written for both the

liquid film layer and the gelatinous layer. In both cases, mass transfer occurs only in radial

direction and the mass transfer equation becomes:

22

10AdCd

rr dr dr

=

(6)

The boundary conditions are:

( )0

Ac D A A g

dCr r h C C D

drδ= + − = (7)

Ac g A

dCr r D kC

dr= = (8)

Considering that the process is in quasi-steady-state conditions, the rate of reaction at the

surface of the core would be equal to the rate of mass transfer to and through the gel film:

( )0c c

D A A Ar rh C C k Cδ+

− = (9)

ACC

EPTE

D M

ANU

SCR

IPT

ACCEPTED MANUSCRIPT

7

Solving the differential Eq. (6) with boundary conditions (7) and (8) and then solving Eq. (9)

for finding CA(rc) would result in obtaining CA:

0

2

2 2

1 1

1 1( )

g

c cA A

g g

c c c D c

D

kr r rC C

D D

kr r r h rδ δ

− −

= + − + + +

(10)

Therefore, the dissolution rate of zinc oxide can be expressed as:

0

2

2

1

11

( )

cA r r A

rZnO ZnO c c c

g c D c

k C CR

M M r r r

k D r h rδ δ

=

= =

+ − + + +

(11)

or in form of the shrinkage rate:

0

2

2

1

11

( )

Ac

Ore c c c

g c D c

Cdr

dt G r r r

k D r h r

ρδ δ

= −

+ − + + +

(12)

Eq. (12) can be expressed in terms of advancement of the reaction. The conversion can be

determined based on the residual volume of particle as follows:

3

0

( )( ) 1 cr t

X tr

= −

(13)

Therefore, the final differential equation, from which the extent of dissolution of particles as

a function of time can be obtained, can be then achieved by combining Eqs. (12) and (13):

( )( )

32/3 2 2/3 20

0 021/3 1/3 1/3

0 0 0

3 1

1 (1 )1 1 1(1 ) (1 ) (1 )

A

Ore

g D

CdX

dt Gr X r X r

k D r X r X h r X

ρ

δ δ

= − − + − + − − + − +

(14)

ACC

EPTE

D M

ANU

SCR

IPT

ACCEPTED MANUSCRIPT

8

2.3. Chemical and Physical Properties

The only chemical reaction considered in this work on the surface of the core is dissolution of

zinc oxide by sulfuric acid. Reaction rate constant for dissolution of pure zinc oxide in

sulfuric acid is calculated from the following equation (Kristovnikov and Davydovskaya,

1936):

4 13634.966.028 10 expk

RT− = × −

(m/s) (15)

It is worth mentioning that in all previous investigations on kinetics of leaching of zinc with

sulfuric acid, different rate constants were reported (e.g., Espiari et al., 2006; Abdel-Aal.,

2000). However, since in all these investigations the effect of gelatinous layer was not taken

into account, the reported rate constant is in fact a combination of reaction rate constant and

dispersion coefficient of acid in the gelatinous layer. In the present study, the effect of

gelatinous layer is separated from the reaction. Therefore, only dissolution of pure zinc oxide

in sulfuric acid was considered on the surface of the ore. Of course, based on the

assumptions listed above, effect of other substances in the ore, on the dissolution kinetic of

zinc oxide was neglected in this work.

The mass transfer coefficient was calculated from (Ranz and Marshall, 1952):

1/21/3

1/3 1/22 0.6 Re 2 0.6 p pd uSh Sc

D

µρ µ

= + = +

(16)

Evaluation of mass transfer coefficient from Eq. (16) requires estimation of viscosity, density

and diffusion coefficient of the acid solution. These properties are not simple functions of

their compositions. There can be found several equations for viscosity, density and diffusion

coefficient of the mixture in this process in various literatures. In the present study, the

following correlations of Guerra et al. (2006) were used:

ACC

EPTE

D M

ANU

SCR

IPT

ACCEPTED MANUSCRIPT

9

[ ] [ ]3 5 2 8 3

11 44 2 4

(0.4332344 4.998831 10 2.174276 10 4.216447 10

3.072309 10 ) exp(0.6182 0.1801

T T T

T ZnSO H SO

µ − − −

−

= − × + × − ×+ × × +

(17)

[ ] 2 22 4

3

1153.82 66748 181.436 158.354[ ]

396.312[ ] 0.55

H SO Zn Fe

Fe T

ρ + +

+

= + + +

+ − (18)

[ ] [ ]( ) [ ][ ] [ ] [ ]( )

4

0.5

4 2 4 4

1.5 102 4 4 2 4

[8.083 7.496 0.296 4.105

3.924 0.739 1.615 ] 10

ZnSOD ZnSO H SO ZnSO

H SO ZnSO H SO −

= − + +

+ − + × (19)

3. Results and discussion

Performance of the proposed model was examined using the experimental data reported by

Espiari et al. (2006) for leaching of zinc from a zinc-rich oxide silicate tailing with sulfuric

acid. Their XRF analysis results showed that the sample contained 37% zinc oxide and

23.7% silica. The leaching data reported by Espiari et al. (2006) are available as zinc

recovery vs. time at various temperatures. Eq. (14) was solved for different operating

conditions reported by Espiari et al. (2006). In this equation, thickness and diffusion

coefficient of the gel layer were considered as fitting parameters and their values at different

temperatures were determined by fitting the equation to the experimental data.

Fig. 2 illustrates a sample solution of the model as well as the corresponding experimental

data. This figure shows the conversion of the ore as a function of time in the batch system. It

can be seen in Fig. 2 that the model fits satisfactorily to the experimental data (Espiari et al.,

2006). Prediction of the conversion when the effect of the gel film was neglected (only zinc

oxide reaction with acid and liquid film resistance was considered) is also shown in the same

figure. As it can be seen, the particles dissolved very fast if the effect of gel film formation is

neglected. However, the trend of the experimental data suggests that the leaching process is

not as fast as dissolution of zinc oxide in sulfuric acid. Slower reaction can be justified by

ACC

EPTE

D M

ANU

SCR

IPT

ACCEPTED MANUSCRIPT

10

adding the resistance of the gel film which is modeled in the present study. Therefore, it is

not possible to neglect the effect of the gel film because its effect on the leaching is

substantial.

Espiari et al. (2006) reported conversion vs. time at various temperatures. Using these

experimental data it was found that the thickness of the layer is almost constant in the range

of temperature considered in this work and its average was determined to be 1.2 µm with

standard deviation of 0.8 µm. However, diffusion coefficient is a strong function of

temperature. The Arrhenius plot of diffusion coefficient of the gel against temperature is

shown in Fig. 3 from which temperature dependency of diffusion coefficient was found to be:

28167.83ln 12.92gD

RT= − − (20)

There are three terms in the denominator of Eq. (14) which correspond to resistances due to

chemical reaction, diffusion through gel film and mass transfer in the liquid film,

respectively. Resistance of the liquid film is negligible as compared to the other two

resistances for the operating conditions considered in this work. Therefore, in order to

investigate the effect of temperature on the leaching rate, only chemical reaction and

diffusion through the gel were considered. Fig. 4 demonstrates the reaction and internal

diffusion resistances as a function of temperature. It can be seen in this figure that both of the

resistances decrease by increasing the temperature. At low temperatures, reaction resistance

is negligible and kinetic mechanism would be reduced to gel diffusion control. Increasing the

temperature has a significant effect on both reaction and mass transfer rates. At high

temperatures these two rates are of the same order of magnitude as shown in figure 4.

Therefore, at high temperature, the process is controlled by both chemical reaction and

diffusion though the gelatinous layer.

ACC

EPTE

D M

ANU

SCR

IPT

ACCEPTED MANUSCRIPT

11

4. Conclusion

A mathematical model was developed for leaching of zinc from a zinc ore containing silica.

It was shown that considering only the chemical reaction of zinc dissolution is not sufficient

for estimating the leaching time of the ore. A thin gelatinous layer was considered to cover

the reacting core of the ore in order to correct the model prediction. Adding the mass transfer

resistance of this layer to the model considerably improved the predictions. It was shown that

the thickness of the gel is almost constant but the diffusion coefficient of the gel decreases

with temperature. At low temperatures, the reaction rate is significantly lower than the rate

of mass transfer through this layer which alters the mechanism to diffusion controlled. At

high temperatures, both chemical reaction and mass transfer control the dissolution rate of the

zinc ore particles containing silica.

Acknowledgement

The authors would like to thank Professor Fathi Habashi from University of Laval, Canada,

for his valuable comments during the work.

Nomenclature

CA acid concentration at the surface of the core (kg.m-3)

CA0 acid concentration in the bulk (kg.m-3)

dp particle diameter (m)

Dg diffusivity of gel (m2.s-1)

DZnSO4 diffusion coefficient of zinc sulfate in solution (m2.s-1)

G zinc oxide grade

hD mass transfer coefficient (m.s-1)

ACC

EPTE

D M

ANU

SCR

IPT

ACCEPTED MANUSCRIPT

12

k reaction rate constant (m.s-1)

MZnO zinc oxide molecular weight (kg.kmol-1)

nZnO moles of zinc oxide (kmol)

rc core radius (m)

r0 initial radius of the particle (m)

R gas constant (8.314 J.mol-1.K-1)

Rr rate of reaction for dissolution of zinc oxide (kg.m-2.s-1)

Re Reynolds number; Re=ρupdp/µ

Sc surface of the core (m2)

Sc Schmidt number; Sc=µ/ρDZnSO4

Sh Sherwood number; Sh=hDdp/µ

t time (sec.)

T temperature (K)

up terminal velocity (m.s-1)

X volumetric conversion

Greek letters

δ gelatinous layer thickness (m)

µ viscosity of solution (kg.m-1.s-1)

ρOre density of zinc ore (kg.m-3)

ρ density of solution (kg.m-3)

References

Abdel-Aal, E. A., 2000. Kinetics of sulfuric acid leaching of low-grade zinc silicate ore.

Hydrometallurgy 55, 247-254.

ACC

EPTE

D M

ANU

SCR

IPT

ACCEPTED MANUSCRIPT

13

Aydoğan, S., Erdemoğlu, M., Aras, A., Uçar, G., Özkan, A., 2006. Dissolution kinetics of

celestite (SrSO4) in HCl solution with BaCl2, Hydrometallurgy 84, 239–246

Espiari, S., Rashchi, F., Sadrnezhaad, S. K., 2006. Hydrometallurgical treatment of tailings

with high zinc content. Hydrometallurgy 82, 54-62.

Guerra, E., Bestetti, M., 2006. Physicochemical properties of ZnSO4-H2SO4-H2O electrolytes

of relevance to zinc electrowinning, J. Chem. Eng. Data 51, 1491-1497.

Kristovnikov, A. H., Davydovskaya, E. A., 1936. Zh. Fiz. Khim (In Russian). 8, 77-84.

Liddell, K. C., 2005. Shrinking core models in hydrometallurgy: What students are not being

told about the pseudo-steady approximation, Hydrometallurgy 79, 62–68.

Liu, Y., Qi, T., Chu, J., Tong, Q., Zhang, Y., 2006. Decomposition of ilmenite by

concentrated KOH solution under atmospheric pressure, Int. J. Miner. Process. 81, 79–84.

Matthew, G., Elsner, D., 1977. Hydrometallurgical treatment of zinc silicate ores,

Metallurgical Transactions 8B, 73-83.

Monhemius, A. J., Terry., 1983. Acid dissolution of willemite and hemimorphite,

Metallurgical Transactions 14B, 335-346.

Mulak, W., Miazga, B., Szymczycha, A., 2005. Kinetics of nickel leaching from spent

catalyst in sulphuric acid solution, Int. J. Miner Process 77, 231-235.

Pecina, T., Franco, T., Castillo, P., Orrantia, E., 2007. Leaching of zinc concentrate in H2SO4

solutions containing H2O2 and complexing agents, Minerals Eng. 21(1), 23-30.

Ranz, W. E., Marshall, W. R. 1952, Evaporation from drops: part 1, Chem. Eng. Prog. 48,

141-150.

Souza, A. D., Peina, P. S., Lima, E. V. O., daSilva, C. A., Leão, V. A., 2007. Kinetics of

sulphuric acid leaching of a zinc silicate calcine, Hydrometallurgy 89, 337-345.

ACC

EPTE

D M

ANU

SCR

IPT

ACCEPTED MANUSCRIPT

14

Szubert, A., Łupiński, M., Sadowski, Z., 2006. Application of shrinking core model to

bioleaching of black shale particles, Physicochemical Problems of Mineral Processing,

40, 211-225.

Velardo, A., Giona, M., Adrover, A., Pagnanelli, F., Toro, L., 2002. Two-layer shrinking-

core model: parameter estimation for the reaction order in leaching processes, Chem.

Eng. J. 90, 231–240.

Weisener, C. G., Smart, R. St. C., Gerson, A. R., 2004. A comparison of the kinetics and

mechanism of acid leaching of sphalerite containing low and high concentrations of iron,

Int. J. Miner. Process 74, 239-249.

ACC

EPTE

D M

ANU

SCR

IPT

ACCEPTED MANUSCRIPT

15

Figure Captions

Figure 1. Schematics of different mechanisms of leaching

Figure 2. Model verification with and without gelatinous layer

Figure 3. Gel diffusivity versus temperature

Figure 4. Comparing reaction and gel diffusion resistances.

ACC

EPTE

D M

ANU

SCR

IPT

ACCEPTED MANUSCRIPT

16

Leaching Time

(c) Shrinking Core – Shrinking Particle

(b) Shrinking Core – Constant Particle Size

(a) Shrinking Particle

Figure 1. Schematics of different mechanisms of leaching

ACC

EPTE

D M

ANU

SCR

IPT

ACCEPTED MANUSCRIPT

17

0

0.25

0.5

0.75

1

0 50 100 150

time (min)

X

Espiari et al. (2006)

Present model

Neglecting gel

Figure 2. Model verification with and without gelatinous layer

ACC

EPTE

D M

ANU

SCR

IPT

ACCEPTED MANUSCRIPT

18

-25

-24

-23

0.0029 0.003 0.0031 0.0032 0.0033 0.0034 0.0035

1/T (K-1)

ln D

g

Figure 3. Gel diffusivity versus temperature

ACC

EPTE

D M

ANU

SCR

IPT

ACCEPTED MANUSCRIPT

19

0

20000

40000

60000

80000

270 290 310 330 350 370

Temperature (K)

Res

ista

nce

(min

./m)

Diffusion

Reaction

Figure 4. Comparing reaction and gel diffusion resistances.