Mathematical Problems in Engineering

Volume 2015, Article ID 982436, 5 pages

http://dx.doi.org/10.1155/2015/982436

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

^{1}College of Information Technology, Shanghai Ocean University, Shanghai 201306, China^{2}School of Civil and Environment Engineering, University of Science and Technology Beijing, Beijing 100083, China

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, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 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]where , are pore deformation coefficient and fluid deformation coefficient, is liquid pressure, is elevation, is permeability, is viscosity, is acceleration of gravity, is the liquid density, is volumetric strain, is the source term, and is the porosity.

Solid elastic deformation equations describing the plain strain deformation state are [13]where is stress matrix; is strain matrix; , the elasticity matrix, is a function of Young’s modulus and Poisson’s ratio . ConsiderWith being the displacement vector, strain matrix and volumetric strain can be expressed as follows:

##### 2.2. Mass Transfer

Both H^{+} of solvent and Cu^{2+} 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 arewhere is the axis along ore column; is leaching time; , are the concentrations of reagent and dissolved metal; , which can be written as , is the absorbed solute mass on unit pore area; is the flowing velocity; is the dispersion coefficient; is opening width of pore; is diffusion flux; is chemical reaction rate; is the stoichiometric coefficient.

The chemical reaction rate can be expressed as follows [6]:where is the maximum concentration of dissolved metal in solution and and 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

That is,where is distributed coefficient [7].

Considering the diffusion flux , according to the first Fick theorem,

Substituting (6) and (7) into (3) and (4) introduces the retardation coefficient . ConsiderThe solute transmission equations can be written as follows:

#### 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 is . The calculated model is illustrated in Figure 1.