# Numerical Analysis on Flow and Solute Transmission during Heap Leaching Processes

Authors: J. Z. Liu; A. X. Wu; L. W. Zhang
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

Based on fluid flow and rock skeleton elastic deformation during heap leaching process, a deformation-flow coupling model is developed. Regarding a leaching column with 1 m 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 and concentration of solvent decrease with bed’s depth increasing; while the concentration of dissolved mineral increases firstly and decreases from a certain position, the peak values of concentration curves move leftward with time. The comparison between experimental results and numerical solutions is given, which shows these two are in agreement on the whole trend.


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## Body

## 1. Introduction

Solution mining is conceptualized as the removal of dissolved metals from original solid matrix [1–3]. In general, in situ leaching and heap leaching are adopted, and the latter is more often used. During heap leaching processes, factors, such as fluid flow, pore pressure, chemical or biochemical reaction between target metals and leaching solution, target metals dissolution, and reaction byproduct deposition, all result in deformation of the heap, affecting the leaching rate [4]. Of all these factors, elastic deformation caused by pore pressure is the main skeleton deformation. In recent years, some mathematical models have been developed to describe the processes of heap leaching. Bouffard and Dixon studied the hydrodynamics of heap leaching processes deeply. They derived three mathematical models in dimensionless form to simulate the transport of solutes through the flowing channels and the stagnant pores of an unsaturated heap [5]. Lasaga investigated the chemical kinetics of water-rock interactions and gave the description of rock deformation regularity [6]. Solute transport and flow through porous media with applications to heap leaching of copper were studied deeply [7–9]. Sheikhzadeh et al. developed an unsteady and two-dimensional model based on the mass conservation equations of liquid phase in the ore bed and in the ore particle, respectively. The model equations were solved using a fully implicit finite difference method, and the results gave the distributions of the degree of saturation and the vertical flowing velocity in the bed [10]. Wu et al. built the basic equations describing the mass transmission in heap leaching. They gave the analytic solution omitting convection with small application rate and determine the hydrodiffusion coefficient [11]. The models discussed above concentrated on the steady flowing conditions without considering the effect of elastic deformation.The purpose of this work is to apply an elastic deformation model for simulating the column leaching processes and develop the governing equations of coupled flow and deformation behavior with mass transfer. These equations are solved numerically by Comsol Multiphysics. The changeable regularity of volumetric strain and concentration distributions of the solvent and the solute is given. The validation of the mathematical model and numerical analysis is concerned through experiment.


## 2. Model Development

### 2.1. Flow and Solid Elastic Deformation Model

Supposing the solution flows in a deformational and homogeneous porous medium, the basic seepage equation for column leaching is [12](1)χp1-n+χfn∂p∂t+∇·-κη(∇p+ρfg∇e)+∂εv∂t=Qs,where χp, χf are pore deformation coefficient and fluid deformation coefficient, p is liquid pressure, e is elevation, κ is permeability, η is viscosity, g is acceleration of gravity, ρf is the liquid density, εv is volumetric strain, Qs is the source term, and n is the porosity.Solid elastic deformation equations describing the plain strain deformation state are [13](2)∇·σ+∇p=0,σ=Dε,where σ is stress matrix; ε is strain matrix; D, the elasticity matrix, is a function of Young’s modulus E and Poisson’s ratio ν. Consider(3)σ=σxxσyyσxyT,ε=εxxεyyεxyT,D=E(1-υ)(1+υ)(1-2υ)1υ1-υ0υ1-υ10001-2υ2(1-υ).With S being the displacement vector, strain matrix ε and volumetric strain εv can be expressed as follows: (4)S=SxSyT,εxy=12(∂Sx∂y+∂Sy∂x),εxx=∂Sx∂x,εyy=∂Sy∂y,εv=εxx+εyy.


### 2.2. Mass Transfer

Both H+ of solvent and Cu2+ of solute are transported by the leaching solution. The couple mass relationship is deduced based on the continuous reaction rates between them.The equations describing mass transfer in pore during leaching processes are(5)∂C1∂t+2b∂s∂t+u∂C1∂y-D∂2C1∂y2+2Jdb=-βRi,(6)∂C2∂t+2b∂s∂t+u∂C2∂y-D∂2C2∂y2+2Jdb=Ri,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 sy,t, is the absorbed solute mass on unit pore area; u is the flowing velocity; D is the dispersion coefficient; b is opening width of pore; Jd is diffusion flux; Ri is chemical reaction rate; β is the stoichiometric coefficient.The chemical reaction rateRi can be expressed as follows [6]:(7)Ri=∂C-2∂t=ρsρln0∂n∂t=Cmax⁡G∂n∂t,where Cmax⁡ is the maximum concentration of dissolved metal in 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 dissolved term and absorption term is(8)s=dsdC1C1=kfC1.That is,(9)∂s∂t=kf∂C1∂t,where kf is distributed coefficient [7].Considering the diffusion fluxJd, according to the first Fick theorem,(10)Jd=-nD∂C1∂x.Substituting (6) and (7) into (3) and (4) introduces the retardation coefficient R. Consider(11)R=1+2bkf.The solute transmission equations can be written as follows: (12)∂C1∂t+uR∂C1∂y-DR∂2C1∂y2-2nDbR∂C1∂x=-βCmax⁡G∂n∂t,∂C2∂t+uR∂C2∂y-DR∂2C2∂y2-2nDbR∂C2∂x=Cmax⁡G∂n∂t.


## 2.1. Flow and Solid Elastic Deformation Model

Supposing the solution flows in a deformational and homogeneous porous medium, the basic seepage equation for column leaching is [12](1)χp1-n+χfn∂p∂t+∇·-κη(∇p+ρfg∇e)+∂εv∂t=Qs,where χp, χf are pore deformation coefficient and fluid deformation coefficient, p is liquid pressure, e is elevation, κ is permeability, η is viscosity, g is acceleration of gravity, ρf is the liquid density, εv is volumetric strain, Qs is the source term, and n is the porosity.Solid elastic deformation equations describing the plain strain deformation state are [13](2)∇·σ+∇p=0,σ=Dε,where σ is stress matrix; ε is strain matrix; D, the elasticity matrix, is a function of Young’s modulus E and Poisson’s ratio ν. Consider(3)σ=σxxσyyσxyT,ε=εxxεyyεxyT,D=E(1-υ)(1+υ)(1-2υ)1υ1-υ0υ1-υ10001-2υ2(1-υ).With S being the displacement vector, strain matrix ε and volumetric strain εv can be expressed as follows: (4)S=SxSyT,εxy=12(∂Sx∂y+∂Sy∂x),εxx=∂Sx∂x,εyy=∂Sy∂y,εv=εxx+εyy.


## 2.2. Mass Transfer

Both H+ of solvent and Cu2+ of solute are transported by the leaching solution. The couple mass relationship is deduced based on the continuous reaction rates between them.The equations describing mass transfer in pore during leaching processes are(5)∂C1∂t+2b∂s∂t+u∂C1∂y-D∂2C1∂y2+2Jdb=-βRi,(6)∂C2∂t+2b∂s∂t+u∂C2∂y-D∂2C2∂y2+2Jdb=Ri,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 sy,t, is the absorbed solute mass on unit pore area; u is the flowing velocity; D is the dispersion coefficient; b is opening width of pore; Jd is diffusion flux; Ri is chemical reaction rate; β is the stoichiometric coefficient.The chemical reaction rateRi can be expressed as follows [6]:(7)Ri=∂C-2∂t=ρsρln0∂n∂t=Cmax⁡G∂n∂t,where Cmax⁡ is the maximum concentration of dissolved metal in 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 dissolved term and absorption term is(8)s=dsdC1C1=kfC1.That is,(9)∂s∂t=kf∂C1∂t,where kf is distributed coefficient [7].Considering the diffusion fluxJd, according to the first Fick theorem,(10)Jd=-nD∂C1∂x.Substituting (6) and (7) into (3) and (4) introduces the retardation coefficient R. Consider(11)R=1+2bkf.The solute transmission equations can be written as follows: (12)∂C1∂t+uR∂C1∂y-DR∂2C1∂y2-2nDbR∂C1∂x=-βCmax⁡G∂n∂t,∂C2∂t+uR∂C2∂y-DR∂2C2∂y2-2nDbR∂C2∂x=Cmax⁡G∂n∂t.


## 3. Numerical Analysis

Regarding a leaching column with 1 m height and solution concentration 1 unit being continually supplied from the top of the column for 10 days, application rate isw=1.25×10-6m3/(m2·s). The calculated model is illustrated in Figure 1.Figure 1
Schematic of numerical calculation.The initial conditions, top boundary conditions, and bottom boundary conditions for the flow, deformation, and mass transfer coupled equations are(13)py,0=0,n·-κη(∇p+ρfg∇e)1,t=1.25×10-6,p0,t=0,Sy,0=0,S1,tfree,S0,t=0,C1y,0=0,C2y,0=0,C11,t=1,C21,t=0,n·θD∇C10,t=0,n·θD∇C20,t=0.During calculation process, initial porosityn0 and final porosity 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 Comsol Multiphysics Software for the given problem.Figure2 shows the variations of the volumetric strain in leaching column with respect to the bed’s depth at different time intervals. It indicates the volumetric strain decreases with the bed’s depth increasing. This is because reagent reacts with valuable metal and consumes gradually.Figure 2
Distribution of volumetric strain in leaching column.Figure3 shows the spatial and temporal distributions of dissolved mineral and reagent at different time. (a) indicates the solute concentration increases rapidly at the first stage and reaches the peak value and decreases gradually towards the heap bottom. The peak values move rightwards with leaching duration. The reason is that, at the beginning of leaching, solvent concentration is higher and chemical reaction speed is quicker. Moreover, the content of target metals is also higher. (b) indicates the concentration of solvent decreases with the depth increasing which is because chemical reaction consumes reagent.Figure 3
Spatial and temporal distributions of reagent (a) and dissolved mineral (b).
(a)
Solute concentration distribution
(b)
Reagent concentration distribution


## 4. Experiment and Discussion

To verify the numerical simulations, indoors physical experiment is done according to dump leaching in Dexing copper mine, Jiangxi province. The chemical content analysis of ore sample 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 Diffractometer M21X and is shown in Table1.Table 1
The chemical analysis of main element contained in ore sample.
Component
Cu
Fe
S
Mo
SiO2
Al2O3
CaO
MgO
As
Percentage (%)
0.56
4.40
0.91
0.012
67.73
13.21
0.20
1.64
0.013The maximum diameter of ore particle in dump leaching field in Dexing mine is 800 mm. It is very difficult to carry on experiment according to field situation. What is more, the general apparatus is not large enough to hold such large ore sample, so most theoretical research works are conducted indoors. The inner diameter of the column leaching cylinder used in experiment is 50 mm; it is necessary to crash ore sample to let the diameter be less than 10 mm according to the research conclusions obtained by Bear [13]. The distribution of ore particle diameter after crashing is shown in Table 2.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
8~10
Content (%)
1.94
6.3
4.12
3.63
4.6
12.59
35.84
30.98The samples were bioleached in PVC (5 cm in diameter and 100 cm in height) for 10 days. Solution with a concentration of 1 unit is continually supplied from the top of the column; the application rate isw=1.25×10-6m3/(m2·s).As shown in Figure4, numerical results and experimental values of copper ion concentration at a certain point (nearly the middle part of the trunk) are consistent on the whole trend, which indicates that the mathematical model, the numerical method, and parameters can describe the transmission process in leaching ore column.Figure 4
Comparison between calculated result and experimental result of the concentration of copper ions in ore heap.


## 5. Conclusions

(i)
With respect to the mineral skeleton deformation, a flow and solid elastic model is developed to describe the flow reaction and mass transfer processes in heap leaching.
(ii)
The model equations are solved by Comsol Multiphysics Software. The distributions of volumetric strain and concentrations of reagent and dissolved mineral are given based on numerical results.
(iii)
The numerical simulation results show that the straight strain decreases with the bed’s depth increasing; the concentration of the solvent decreases with the bed’s depth increasing; the concentration of dissolved mineral increases firstly and decreases from a certain position: the peak values of the curves move leftward with time.
(iv)
The numerical results are compared with the experimental results; these two are in agreement on the whole trend, which indicates that the mathematical model, the numerical method, and parameters can describe the multifactor coupled processes in heap leaching.



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*Source: 982436-2015-03-30.xml*