research article numerical analysis on flow and solute...

Research ArticleNumerical Analysis on Flow and Solute Transmission duringHeap Leaching Processes

J. Z. Liu,1 A. X. Wu,2 and L. W. Zhang1

1College of Information Technology, Shanghai Ocean University, Shanghai 201306, China2School of Civil and Environment Engineering, University of Science and Technology Beijing, Beijing 100083, China

Correspondence should be addressed to J. Z. Liu; [email protected] and A. X. Wu; [email protected]

Received 26 August 2014; Accepted 30 October 2014

Academic Editor: Kim M. Liew

Copyright © 2015 J. Z. Liu et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Based on fluid flow and rock skeleton elastic deformation during heap leaching process, a deformation-flow coupling model isdeveloped. Regarding a leaching column with 1m height, solution concentration 1 unit, and the leaching time being 10 days,numerical simulations and indoors experiment are conducted, respectively. Numerical results indicate that volumetric strain andconcentration of solvent decrease with bed’s depth increasing; while the concentration of dissolved mineral increases firstly anddecreases from a certain position, the peak values of concentration curves move leftward with time. The comparison betweenexperimental results and numerical solutions is given, which shows these two are in agreement on the whole trend.

1. Introduction

Solutionmining is conceptualized as the removal of dissolvedmetals from original solid matrix [1–3]. In general, in situleaching and heap leaching are adopted, and the latter ismore often used. During heap leaching processes, factors,such as fluid flow, pore pressure, chemical or biochemicalreaction between target metals and leaching solution, targetmetals dissolution, and reaction byproduct deposition, allresult in deformation of the heap, affecting the leaching rate[4]. Of all these factors, elastic deformation caused by porepressure is the main skeleton deformation. In recent years,some mathematical models have been developed to describethe processes of heap leaching. Bouffard and Dixon studiedthe hydrodynamics of heap leaching processes deeply. Theyderived three mathematical models in dimensionless formto simulate the transport of solutes through the flowingchannels and the stagnant pores of an unsaturated heap [5].Lasaga investigated the chemical kinetics of water-rock inter-actions and gave the description of rock deformation regu-larity [6]. Solute transport and flow through porous mediawith applications to heap leaching of copper were studied

deeply [7–9]. Sheikhzadeh et al. developed an unsteady andtwo-dimensional model based on the mass conservationequations of liquid phase in the ore bed and in the oreparticle, respectively. The model equations were solved usinga fully implicit finite difference method, and the resultsgave the distributions of the degree of saturation and thevertical flowing velocity in the bed [10]. Wu et al. built thebasic equations describing the mass transmission in heapleaching.They gave the analytic solution omitting convectionwith small application rate and determine the hydrodiffusioncoefficient [11]. The models discussed above concentrated onthe steady flowing conditions without considering the effectof elastic deformation.

The purpose of this work is to apply an elastic defor-mation model for simulating the column leaching processesand develop the governing equations of coupled flow anddeformation behavior withmass transfer.These equations aresolved numerically by Comsol Multiphysics. The changeableregularity of volumetric strain and concentration distribu-tions of the solvent and the solute is given. The validation ofthe mathematical model and numerical analysis is concernedthrough experiment.

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 982436, 5 pageshttp://dx.doi.org/10.1155/2015/982436

2 Mathematical Problems in Engineering

2. Model Development

2.1. Flow and Solid Elastic DeformationModel. Supposing thesolution flows in a deformational and homogeneous porousmedium, the basic seepage equation for column leaching is[12]

[𝜒𝑝 (1 − 𝑛) + 𝜒𝑓𝑛]𝜕𝑝𝜕𝑡

+ ∇ ⋅ [−𝜅𝜂(∇𝑝 + 𝜌𝑓𝑔∇𝑒)]+

𝜕𝜀V𝜕𝑡

= 𝑄𝑠,

(1)

where 𝜒𝑝, 𝜒𝑓 are pore deformation coefficient and fluiddeformation coefficient, 𝑝 is liquid pressure, 𝑒 is elevation, 𝜅is permeability, 𝜂 is viscosity, 𝑔 is acceleration of gravity, 𝜌𝑓is the liquid density, 𝜀V is volumetric strain, 𝑄𝑠 is the sourceterm, and 𝑛 is the porosity.

Solid elastic deformation equations describing the plainstrain deformation state are [13]

∇ ⋅ 𝜎 + ∇𝑝 = 0,

𝜎 = D𝜀,(2)

where 𝜎 is stress matrix; 𝜀 is strain matrix; D, the elasticitymatrix, is a function of Young’s modulus 𝐸 and Poisson’s ratio]. Consider

𝜎 = [𝜎𝑥𝑥 𝜎𝑦𝑦 𝜎𝑥𝑦]𝑇

, 𝜀 = [𝜀𝑥𝑥 𝜀𝑦𝑦 𝜀𝑥𝑦]𝑇

,

D = 𝐸 (1 − 𝜐)(1 + 𝜐) (1 − 2𝜐)

[[[[[

[

1 𝜐1 − 𝜐

0𝜐

1 − 𝜐1 0

0 0 1 − 2𝜐2 (1 − 𝜐)

]]]]]

]

.(3)

With S being the displacement vector, strain matrix 𝜀 andvolumetric strain 𝜀V can be expressed as follows:

S = [𝑆𝑥 𝑆𝑦]𝑇 ,

𝜀𝑥𝑦 =12(𝜕𝑆𝑥𝜕𝑦

+𝜕𝑆𝑦𝜕𝑥

) , 𝜀𝑥𝑥 =𝜕𝑆𝑥𝜕𝑥

, 𝜀𝑦𝑦 =𝜕𝑆𝑦𝜕𝑦

,

𝜀V = 𝜀𝑥𝑥 + 𝜀𝑦𝑦.(4)

2.2. Mass Transfer. Both H+ of solvent and Cu2+ of soluteare transported by the leaching solution. The couple massrelationship is deduced based on the continuous reactionrates between them.

The equations describing mass transfer in pore duringleaching processes are

𝜕𝐶1𝜕𝑡

+ 2𝑏𝜕𝑠𝜕𝑡

+ 𝑢𝜕𝐶1𝜕𝑦

− 𝐷𝜕2𝐶1𝜕𝑦2

+2𝐽𝑑𝑏

= −𝛽𝑅𝑖, (5)

𝜕𝐶2𝜕𝑡

+ 2𝑏𝜕𝑠𝜕𝑡

+ 𝑢𝜕𝐶2𝜕𝑦

− 𝐷𝜕2𝐶2𝜕𝑦2

+2𝐽𝑑𝑏

= 𝑅𝑖, (6)

where 𝑦 is the axis along ore column; 𝑡 is leaching time; 𝐶1,𝐶2 are the concentrations of reagent and dissolved metal; 𝑠,which can be written as 𝑠(𝑦, 𝑡), is the absorbed solute masson unit pore area; 𝑢 is the flowing velocity;𝐷 is the dispersioncoefficient; 𝑏 is opening width of pore; 𝐽𝑑 is diffusion flux; 𝑅𝑖is chemical reaction rate; 𝛽 is the stoichiometric coefficient.

The chemical reaction rate 𝑅𝑖 can be expressed as follows[6]:

𝑅𝑖 =𝜕𝐶2𝜕𝑡

=𝜌𝑠𝜌𝑙𝑛0

𝜕𝑛𝜕𝑡

=𝐶max𝐺

𝜕𝑛𝜕𝑡

, (7)

where𝐶max is themaximum concentration of dissolvedmetalin solution and 𝑛 and 𝑛0 are instant and initial porosity.

Assuming that the absorption on pore surface is linear,balanced, and thermal, the relationship between dissolvedterm and absorption term is

𝑠 = 𝑑𝑠𝑑𝐶1

𝐶1 = 𝑘𝑓𝐶1. (8)

That is,

𝜕𝑠𝜕𝑡

= 𝑘𝑓𝜕𝐶1𝜕𝑡

, (9)

where 𝑘𝑓 is distributed coefficient [7].Considering the diffusion flux 𝐽𝑑, according to the first

Fick theorem,

𝐽𝑑 = −𝑛𝐷𝜕𝐶1𝜕𝑥

. (10)

Substituting (6) and (7) into (3) and (4) introduces theretardation coefficient 𝑅. Consider

𝑅 = 1 + 2𝑏𝑘𝑓. (11)

The solute transmission equations can be written as follows:

𝜕𝐶1𝜕𝑡

+ 𝑢𝑅𝜕𝐶1𝜕𝑦

− 𝐷𝑅𝜕2𝐶1𝜕𝑦2

− 2𝑛𝐷𝑏𝑅

𝜕𝐶1𝜕𝑥

= −𝛽𝐶max𝐺

𝜕𝑛𝜕𝑡

,

𝜕𝐶2𝜕𝑡

+ 𝑢𝑅𝜕𝐶2𝜕𝑦

− 𝐷𝑅𝜕2𝐶2𝜕𝑦2

− 2𝑛𝐷𝑏𝑅

𝜕𝐶2𝜕𝑥

=𝐶max𝐺

𝜕𝑛𝜕𝑡

.

(12)

3. Numerical Analysis

Regarding a leaching column with 1m height and solutionconcentration 1 unit being continually supplied from thetop of the column for 10 days, application rate is 𝑤 =1.25 × 10−6m3/(m2⋅s). The calculated model is illustrated inFigure 1.

Mathematical Problems in Engineering 3

Table 1: The chemical analysis of main element contained in ore sample.

Component Cu Fe S Mo SiO2 Al2O3 CaO MgO AsPercentage (%) 0.56 4.40 0.91 0.012 67.73 13.21 0.20 1.64 0.013

𝜕C1

𝜕n=

𝜕C2

𝜕n= 0𝜕C1

𝜕n=

𝜕C2

𝜕n= 0

y Barren solution

Pregnant solution

Figure 1: Schematic of numerical calculation.

The initial conditions, top boundary conditions, andbottom boundary conditions for the flow, deformation, andmass transfer coupled equations are

𝑝 (𝑦, 0) = 0,

n ⋅ [−𝜅𝜂(∇𝑝 + 𝜌𝑓𝑔∇𝑒)] (1, 𝑡) = 1.25 × 10−6,

𝑝 (0, 𝑡) = 0,

S (𝑦, 0) = 0,

S (1, 𝑡) free,

S (0, 𝑡) = 0,

𝐶1 (𝑦, 0) = 0, 𝐶2 (𝑦, 0) = 0,

𝐶1 (1, 𝑡) = 1, 𝐶2 (1, 𝑡) = 0,

n ⋅ [𝜃𝐷∇𝐶1 (0, 𝑡)] = 0, n ⋅ [𝜃𝐷∇𝐶2 (0, 𝑡)] = 0.

(13)

During calculation process, initial porosity 𝑛0 and finalporosity 𝑛𝑓 are assumed to be 0.30 and 0.35, respectively;the stoichiometric coefficient 𝛽 is 1. Equations (1), (2), and(12) are solved by ComsolMultiphysics Software for the givenproblem.

Figure 2 shows the variations of the volumetric strain inleaching column with respect to the bed’s depth at differenttime intervals. It indicates the volumetric strain decreaseswith the bed’s depth increasing.This is because reagent reactswith valuable metal and consumes gradually.

Figure 3 shows the spatial and temporal distributions ofdissolved mineral and reagent at different time. (a) indicates

×10−3

1.2

1

0.8

0.6

0.4

0.2

0

−0.20 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time (d)

Heap height y (m)

1

2

34

5

6

7

8

9

10

Volu

met

ric st

rain

Figure 2: Distribution of volumetric strain in leaching column.

the solute concentration increases rapidly at the first stage andreaches the peak value and decreases gradually towards theheap bottom.The peak values move rightwards with leachingduration. The reason is that, at the beginning of leaching,solvent concentration is higher and chemical reaction speedis quicker. Moreover, the content of target metals is alsohigher. (b) indicates the concentration of solvent decreaseswith the depth increasing which is because chemical reactionconsumes reagent.

4. Experiment and Discussion

To verify the numerical simulations, indoors physical exper-iment is done according to dump leaching in Dexing coppermine, Jiangxi province. The chemical content analysis of oresample is 0.20% sulphide copper, 0.17% sulphide copper,0.12% free oxide copper, and 0.072% combined oxide copper.Ore component analysis is conducted by X-Ray Diffractome-ter M21X and is shown in Table 1.

The maximum diameter of ore particle in dump leachingfield in Dexing mine is 800mm. It is very difficult to carryon experiment according to field situation. What is more,the general apparatus is not large enough to hold such largeore sample, so most theoretical research works are conducted

4 Mathematical Problems in Engineering

Table 2: The distribution of ore particle diameter after crashing.

Particle diameter (mm) <0.1 0.1∼0.2 0.2∼0.4 0.4∼0.7 0.7∼2 2∼5 5∼8 8∼10Content (%) 1.94 6.3 4.12 3.63 4.6 12.59 35.84 30.98

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Heap height y (m)

Solu

te co

ncen

trat

ion(k

g/m

3)

0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0

Time (d)1

2

34

5

6

7

8

9

10

(a) Solute concentration distribution

Reag

ent c

once

ntra

tion(k

g/m

3)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time (d)

Heap height y (m)

1

2

34

5

6

7

8

9

10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b) Reagent concentration distribution

Figure 3: Spatial and temporal distributions of reagent (a) and dissolved mineral (b).

indoors. The inner diameter of the column leaching cylinderused in experiment is 50mm; it is necessary to crash oresample to let the diameter be less than 10mmaccording to theresearch conclusions obtained by Bear [13]. The distributionof ore particle diameter after crashing is shown in Table 2.

The samples were bioleached in PVC (5 cm in diameterand 100 cm in height) for 10 days. Solution with a concen-tration of 1 unit is continually supplied from the top of thecolumn; the application rate is 𝑤 = 1.25 × 10−6m3/(m2 ⋅ s).

As shown in Figure 4, numerical results and experimentalvalues of copper ion concentration at a certain point (nearlythe middle part of the trunk) are consistent on the wholetrend, which indicates that the mathematical model, thenumerical method, and parameters can describe the trans-mission process in leaching ore column.

5. Conclusions

(i) With respect to the mineral skeleton deformation, aflow and solid elastic model is developed to describe

Calculated valuesExperimental values

Con

cent

ratio

n (m

ol/L

)

Time (min)

0.04

0.035

0.03

0.025

0.02

0.015

0.01

0.005

00 50 100 150

Figure 4: Comparison between calculated result and experimentalresult of the concentration of copper ions in ore heap.

Mathematical Problems in Engineering 5

the flow reaction and mass transfer processes in heapleaching.

(ii) The model equations are solved by Comsol Mul-tiphysics Software. The distributions of volumetricstrain and concentrations of reagent and dissolvedmineral are given based on numerical results.

(iii) The numerical simulation results show that thestraight strain decreases with the bed’s depth increas-ing; the concentration of the solvent decreases withthe bed’s depth increasing; the concentration of dis-solved mineral increases firstly and decreases from acertain position: the peak values of the curves moveleftward with time.

(iv) The numerical results are compared with the exper-imental results; these two are in agreement on thewhole trend, which indicates that the mathematicalmodel, the numerical method, and parameters candescribe the multifactor coupled processes in heapleaching.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

This project is supported by the Natural Science Fund ofChina (51104100, 51304076, and 51074013).

References

[1] Z.-K. Li, B.-W. Gui, and X.-X. Duan, “The production practiceand technical study in heap leaching mill of Dexing CopperMine,” Mining and Metallurgical Engineering, vol. 22, no. 1, p.46, 2002 (Chinese).

[2] H. Z. Liu and Y. K. Zhang, “The status and prospect ofintractable low-grade gold mineral bacterium heap leaching,”Journal of Sichuan Geology, vol. 18, no. 3, pp. 234–240, 1998(Chinese).

[3] T. Yuan, G. Zibin, and G. Renxi, “The advance of the uraniumand gold ore heap leaching,” Uranium Metallurgy, vol. 17, no. 2,pp. 121–126, 1998 (Chinese).

[4] R. W. Bartlett, Solution Mining: Leaching and Fluid Recoveryof Minerals, Gordon and Breach Science Publisher, Singapore,1992.

[5] S. C. Bouffard and D. G. Dixon, “Investigative study into thehydrodynamics of heap leaching processes,” Metallurgical andMaterials Transactions B, vol. 3, pp. 393–408, 2001.

[6] A. C. Lasaga, “Chemical kinetics of water-rock interations,”Journal of Geophysical Research, vol. 89, no. 6, pp. 4009–4025,1982.

[7] G. E. Grisak and J. A. Cherry, “Solute transport throughfractured media: 2. Column study of fractured till,” WaterResources Research, vol. 16, no. 4, pp. 40–45, 1998.

[8] M. Sidborn, J. Casas, J. Martınez, and L. Moreno, “Two-dimensional dynamic model of a copper sulphide ore bed,”Hydrometallurgy, vol. 71, no. 1-2, pp. 67–74, 2003.

[9] E. Cariaga, F. Concha, andM. Sepulveda, “Flow through porousmedia with applications to heap leaching of copper ores,”Chemical Engineering Journal, vol. 111, no. 2-3, pp. 151–165, 2005.

[10] G. A. Sheikhzadeh, M. A. Mehrabian, S. H. Mansouri, and A.Sarrafi, “Computational modelling of unsaturated flow of liquidin heap leaching—using the results of column tests to calibratethemodel,” International Journal of Heat andMass Transfer, vol.48, no. 2, pp. 279–292, 2005.

[11] A. X. Wu, J. Z. Liu, and S. H. Yin, “Mathematical model andanalytic solution of mass transfer in heap leaching process,”Mining and Metallurgical Engineering, vol. 5, pp. 7–10, 2005.

[12] F. A. L. Dullien, Porous Media: Fluid Transport and PoreStructure, Academic Press, New York, NY, USA, 1979.

[13] J. Bear,Dynamics of Fluids in Porous Media, translated by J. S. Li& Z. X. Chen, China Architecture & Building Press, 1983.

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