Simulation equation for leaching kinetics

It represents metal leaching rate versus time function, related to the rate of mass transfer of reactants or chemical reaction rate of the solid surface. But for the purpose of determining process control under conditions, the most convenient way is to use a mathematical equation to fit the curve without mentioning these factors.

This advancement and retreat of the surface affects the leaching kinetics as the surface of the solid reactant in a leaching system advances or retreats as the reaction progresses. This is also the case when solid reaction products are continuously formed around the reactive ions. Based on this behavior, mathematical equations for several models were obtained and used to describe the shape of the rate curve obtained by leaching minerals under different conditions. The chemical behavior of the leaching system is extremely important when determining the chemical conditions used, especially when developing new processes.

First, the shrinkage particle model

Exporting a contracted particle model assumes that the particle is a sphere, but the resulting final equation can be applied to any other shape of equal volume particle.

Let the number of moles n of unreacted spheres be

n=4Ï€r ³ ∕3V (1)

Where V-molar volume is equal to m∕ρ, where m is the mass and ρ is the solid density.

The reaction rate of the surface of the sphere with radius r can be written as

-dn∕dt=4Ï€r ² Ch' (2)

Where k' is the first order reaction rate constant of the reactant at a concentration of C in the solution. The equation (1) is differentiated from time and substituted into equation (2) to obtain a linear rate

-dr∕dt=VCk' (3)

Where VCk' is the linear rate constant k ₁ . If C is the concentration of the reaction solute expressed in mol ∕ cm ³ , the linear rate at which the radius r decreases is expressed in cm ∕ s. If C is left unchanged, equation (3) represents a constant rate of movement of the reaction interface, which is the definition of linear dynamics.

If the initial radius of the reactive particles is r ₀ and α is the reaction score, then

(4)

The above formula is different for time

(5)

Combine equations (3), (4), and (5)

(6)

For the initial condition t=0, α=0, it can be assumed that C is constant and the integral of equation (6) is obtained.

(7)

Where k = Ck ₁ ∕r ₀ (time ^{- 1} ). To plot the left side of equation (7) against t, we should get a straight line with a slope of k, and the dimension of k is 1∕t.

The above model is derived from a single particle. If all the ore particles in the slurry have the same initial diameter, the reaction combination rate of many ore particles will also obey this equation. For the slurry with a certain particle size distribution, the mass fraction w _{i of} each particle size distribution must be known. Assuming that the initial average ore radius of the mass fraction w _i is r _i0 , then equation (8) becomes

(8)

(9)

Where α _i is the reacted fraction with a mass fraction of w _i . The total amount of reaction solids is α= .

If the concentration changes as the reaction progresses, the change in concentration must be considered in the integral basic rate expression. For surface control reactions, or for diffusion through a boundary film of the limit, equation (9) must account for the concentration of the reactants, thus obtaining

(10)

Wherein C ₀ - initial concentration;

Σ-stoichiometric coefficient;

b = n ₀ ∕V ₀ C ₀ , C ₀ , n ₀ are the total moles of minerals in the system, respectively.

In the specific case of σb=1, the integral is obtained.

(11)

Second, the shrinking core model

In many cases mineral particles contain a variety of metals, while only one metal is eluted when leaching. This situation can result in the formation of a porous solid reaction product around each of the ore particles that are reacting. Only one special case is considered here, that is, the particle radius always remains equal to the original radius r _{0 of the} particle before the reaction starts. The shrinking core of radius r continues to react at a rate determined by the rate at which the reactants diffuse through the product layer toward the reaction interface. If the reaction particles are spherical, the reaction rate can be written as:

(12)

The number of moles of unreacted mineral in the n-core;

Σ-metering factor;

C- the number of moles of diffusing material required to leach 1 mole of metal from the core;

The effective diffusion coefficient in the D-product layer is the concentration of the reactants at the interface of the shrinking core.

In the steady state condition, from r to r ₀ integral, when the concentration C of the reactant on the interface is much smaller than the bulk concentration C ₀ ,

(13)

The equations for combining the equations (12) and (13) to obtain the core and reaction product boundary movement rate expressed by the radius r of the unreacted core are as follows

(14)

Combining equations (14), (4) and (5), obtains a reaction rate equation expressed by the fractional α of the reaction.

(15)

When the boundary condition t=0, α=0, the integral is obtained.

(16)

The left side of the above equation is plotted against time to get a straight money, and the slope is proportional to r ₀ ^-2 .

Similarly, when it is necessary to consider the change in the concentration of the reactants, the equation (15) becomes a diffusion through the product or the residual layer.

(17)

In the specific case of σb=1, the integral is obtained.

(18)

In the general case of σb ≠ 1, equations (13) and (17) need to be solved by numerical integration.

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