Leaching Theory

HYDROMETALLURGY: 4) LEACHING THEORY A distinction is made between homogeneous and heterogeneous reaction systems.

A homogeneous reaction takes place in one phase i.e. the reactant and the product are either in the gas or the liquid phase. A heterogeneous reaction takes place at the interface between several phases: solid/liquid, solid/gas, gas/liquid. All hydrometallurgical reactions are heterogeneous since solids are brought into a liquid solution and in many cases there is also a gas phase is involved.

Figure 4.1. Basic sketch of the leaching process As is shown in Figure 4.1 above, a leaching process takes place in several steps: 1. 2. 3. 4. 5.

Diffusion of the reactant through the diffusion layer ((x) Adsorption of the reactant on the solid Chemical reaction between the reactant and the solid Desorption of the product from the solid Diffusion of the leaching product through the diffusion layer ((x)

The rate of diffusion in a solution is governed by Fick's Law:

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Where J = the amount of substance that diffuses through a surface per time unit A = the area of the reacting particle D = the diffusion constant (unit of surface/unit of time)

= the concentration gradient The leaching rate is thus dependent on the area of the leaching body, the diffusion coefficient and the concentration gradient . In addition, the concentration gradient is dependent on the thickness of the diffusion layer ((x), which is shown in Figure 4.2.

Figure 4.2 Concentration of reactant at the surface of solid subjected to leaching. The slowest step in the leaching reaction is the rate-controlling step. Depending on which process is rate-controlling, three different type reactions may be obtained: 1. Reaction controlled leaching The chemical reaction at the surface is much slower than the diffusion

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When the chemical reaction is the rate controlling step the activation energy is usually in the range 40- 100 kJ/mole. 2. Diffusion controlled leaching The chemical reaction at the interface is much faster than the diffusion

For leaching where diffusion is rate controlling the activation energy is usually 21 kJ/mole or less. 3. Intermediate controlled leaching The chemical reaction on the surface is approximately the same rate as the diffusion

The activation energy for intermediate controlled leaching fall in the range of 21-40 kJ/mole.

Reaction controlled leaching For chemically controlled leaching the reaction between the material to be leached and the reactant is much slower than the diffusion, and the concentration of the reactant at the surface of the solid is approximately the same as the concentration in the bulk solution. The reaction rate of a heterogeneous reaction, for example that of a mineral with an acid, may, provided the acid concentration is constant, may be expressed through the formula:

Where W = the weight of the leached particle at time t k = the rate constant A = the surface area of the leached particle C = the concentration of reactant (constant) Depending on the geometry of the solid to be leached different kinetic expressions can be derived.

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For example: Flat surface. The surface is constant during the leaching

Out of this expression the rate constant (k) may be determined by plotting (Wo - W) against time (t) in a diagram. Sphere. The surface area is reduced with time

Equation (6) is inserted into (4)

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Out of this expression the rate constant (k´) may be determined by plotting time (t) in a diagram, where the slope of the line is equal to the rate constant.

against

The above equation is valid for a reaction controlled leaching process where the initial weight and the weight of the sample at time (t) are used for evaluating leaching progress. However, it is more common to express the leaching progress either as percentage or fraction leached. Kinetic equations expressed as fraction leached, (. Fraction leached (() is defined as:

Fraction leached for a spherical particle will then be:

Chemically controlled leaching of spherical particles is often described as the "shrinking particle" model, as is visualised in Figure 4.3.

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Figure 4.3 Schematic of leaching according to shrinking particle model. The general expression for the leaching rate of a heterogeneous reaction, as of above, is:

The expressions for the area (A) and the weight (W) for a sphere were respectively:

and

Differentiation of (5)

By inserting equations (4) and (12) into (2) we get the following expression:

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Equation (14) is the expression for chemical reaction controlled leaching according to the shrinking particle model. As is evident from the equation, the leaching rate is inversely proportional to the radius of the particle. Provided that the concentration of the leach solution can be assumed to be constant and that the density is known, the rate constant may be determined by plotting the left hand side against time in a diagram. Naturally, similar expressions may be calculated also for surfaces other than spheres provided their geometry can be described mathematically.

Diffusion-controlled leaching In the cases in which the chemical reaction is much faster than the diffusion the leaching is said to be diffusion-controlled. The leaching mechanism often becomes diffusion-controlled when, during the leaching, a porous product layer forms on the surface of the particle to be leached. This can for example happen in the case of leaching of sulphides where a layer of elemental sulphur can be deposited on the sulphide surface. The mechanism of diffusion-controlled leaching of a spherical particle is often called the "shrinking core" model. It is shown schematically in Figure 4.4. Concentration of reactant at surface = Ci

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Figure 4.4 Leaching through the "shrinking core" model For a sphere, Fick's Law may be formulated as follows:

where J is the number of molecules per unit time that pass through the product layer

According to the definition of diffusion controlled leaching the concentration of reagent at the surface is zero, Ci = 0.

Fraction leached for a sphere, as of above, is:

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The number of moles that has not reacted in the shrinking core = N

Where ( is the density and M the molecular weight. Differentiating equation (18)

The number of moles diffusing into the surface is proportional to the number of moles reacting.

Where ( is a stoichiometric factor.

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Insertion of (17) into (21)

After a series of transfers, we arrive at the following expression:

Equation (22) is the expression for diffusion-controlled leaching according to the shrinking core model. As is evident from the equation, the leach rate is inversely proportional to the square of the radius of the particle. Assuming that the concentration (C) is constant, everything except the diffusion constant (D) are constants in the left hand of the equation. The diffusion constant can be determined by plotting the left hand side against time in a diagram. Given the assumptions made that C is constant and that volume changes has not been taken into account, the model is accurate until 80-90% has been leached.

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Factors influencing leaching kinetics The equations arrived at (14) and (22) for reaction controlled and diffusion controlled leaching, respectively, have implications on the operating cost for leaching operations. Such costs are for example size reduction by grinding, leaching temperature and agitation rate. Depending on the leaching mechanism, i.e. if the leaching process is reaction or diffusion controlled the leaching kinetics are influenced differently by variations of these parameters.

Particle size As mentioned above, and as is evident from equations (14) and (22), the kinetics is affected in different ways by the particle size during chemically reaction controlled and diffusion-controlled leaching respectively. Generally, smaller particle size yields faster leaching kinetics.

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For chemically controlled leaching the dependence is proportional to

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For diffusion-controlled leaching the dependence is proportional to

Agitation rate Increased stirring may reduce the thickness of the diffusion layer and it has the following effect on the two leaching mechanisms: 

Chemically controlled leaching is not affected, or is affected to a limited extent by the agitation rate, as the chemical reaction is much slower than the diffusion through the diffusion layer.

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With diffusion-controlled leaching the leach rate increases with increased agitation as the diffusion layer diminishes. The diffusion layer cannot be completely eliminated, which results in variations of the dependence of the leaching rate on the stirring speed, as in Figure 4.5:

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Figure 4.5. Leaching rate as a function of agitation rate for diffusion-controlled leaching The dependence of the diffusion layer on the agitation during leaching in a tank is described by the following formula:

Where (x is the thickness of the diffusion layer and L is the characteristic length. N re and NSc are Reynold's number and Schmidt's number as defined below.

Here ( is the dynamic viscosity, ( is the pulp density, N is the rotation speed of the stirrer and R is the radius of the stirrer.

In Schmidt's number D is the diffusion constant.

Reagent concentration In general, the leaching rate increases with increased concentration of reagent, but only up to a certain maximum level. The leaching mechanism may be changed as a result of changes in the concentration of reagent. The leaching mechanism may for instance be changed from chemically controlled to diffusion-controlled when the concentration of reagent is changed from high to low.

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