3.0 Effective Kinetic Equations For The Oxidation Of Pyrite

Matthew Otwinowski

Scaling Analysis Of Acid Rock Drainage

3.0 EFFECTIVE KINETIC EQUATIONS FOR THE OXIDATION OF PYRITE

This Chapter presents the results of our mathematical analysis of numerical results obtained for a set of kinetic equations discussed in Chapter 2. The purpose of our analysis is to derive closed form formulae for overall oxygen consumption rates and thermal energy generation rates. The effective formulae have a simple form suitable for the scaling analysis of coupled physical and chemical processes.

The scaling parameters discussed in this Chapter allow the estimation of the temperature dependent rates for processes contributing to acid rock drainage, without solving the differential equations.

In Sections 3.1 and 3.2 we analyze two aspects of pyrite oxidation:

i) Scaling relations determined by the quantitative analysis of the chemical reactions described in the previous section; and

ii) Scaling relations for the concentration of oxygen dissolved in water as a function of temperature and concentration of oxygen in the gas phase.

In Section 3.3 effective equations for oxygen consumption rates and energy generation rates are derived for different pH values.

3.1 Scaling Relations for Chemical Reactions

The scaling parameters should be useful for analyzing certain site-specific situations where values for material parameters and initial conditions are different than the particular values used in this report. The scaling analysis for chemical rates will be used subsequently in our bifurcation analysis of the reaction-transport scaling model which will include processes of mass and energy transport.

There is one obvious scaling parameter, S/V, which relates time scales of the surface reactions (R1) and (R3) to the active surface area and water volume. We can also define other scaling factors which relate quantities characterizing acidic drainage for different conditions labelled by i and j.

For the reaction (R1) we define the scaling factors: 

and 

The thermodynamic scaling factor f₁(T₀,ΔT) tells us that when we increase temperature by ΔT, the rate constant will change f₁(T₀,ΔT) times. The thermodynamic scaling factors satisfy a recursion relation:

It turns out that for activation energy values on the order of 10⁵J and temperatures between T_min=283 K and T_max=333 K the scaling factor defined by eq.(3.2), can be approximated by the factor f₁(ΔT) which does not depend on T₀. At first we find the ratio between the greatest and the smallest values of the rate constant k₁(T):

Then the thermodynamic scaling factor f₁(ΔT) is defined as:

which satisfies a recursion relation:

The accuracy of the approximation f₁(ΔT)≈f₁(T₀,ΔT) is better than 10%. Subscripts l and h label results obtained for low (at pH<4) and high pH values respectively. For the reaction (R1) at pH<4, when E_1l=57 kJ/mol, we obtain:

which means that if we increase the temperature by 10K, then the rate constant for the oxidation by dissolved oxygen increases 2.1 times. For high pH values, when E_1h=88 kJ/mol we obtain: 

We may combine factors r₁ and f₁ and define the scaling factor:

which takes into account the effective active surface area, the amount of water, oxygen concentration and temperature. We can use the scaling factor F₁ to transform the results obtained for one set of material parameters and initial conditions into results for different values of parameters and initial conditions. The conclusions are based on our analysis of experimental data and numerical results presented in [SyT] (See Appendix A and Appendix B). For example, by comparing Figs. A3(21) and A3(31) we see that Fig. A3(21) can be obtained from Fig. A3(31) after rescaling the time axis by the factor F₁(ΔT=20K) equal to 8.2 (calculated by using the eqs. (3.1), (3.8) and (3.9) and taking into account the decreasing concentration of dissolved oxygen - see Table 3.1).

In a similar way we may derive scaling factors f₂ and f₃ for the reactions (R2) and (R3). At low pH values we can use the average activation energy (E_1l+E_2l)/2 to define:

For the reaction (R3) (for which the activation energy E₃=90 kJ/mol [MaR2], [WiR]) we obtain:

For ΔT=20 K the thermodynamic scaling factor f₃(20K)=10.7. Fig. 3.1 presents numerical solutions to eqs. (D2) for large initial values of [Fe³⁺] at different temperatures and for different values of S/V.

Fig. 3.1 Numerical solutions to eqs. (D2) for the initial values [Fe²⁺(t₀)]=Fe³⁺(t₀)]=0.01M, [H⁺(t₀)]=10^-4M. Plots (a), (b) and (c) obtained for S/V=1m²/l and temperatures 283K, 303K and 323K respectively; (d) obtained for S/V=10m² and T=303 K. Vertical axis: molar concentrations; horizontal axis: time in seconds (30 000 s = 8 hours).

As we see, Figures 3.1 (a) and 3.1 (d) are almost identical. They are related by the scaling factor F₃=f₃(20K)S(d)/S(a)=1.07.  Figs. 3.1 (a), (b) and (c) are related by the scaling factor f₃(20K)=10.7.

More complex symmetries of differential equations and more complicated scaling relations will be used in Chapter 4 to analyze reaction-transport models defined by coupled partial differential equations [Ar], [BaZ]. In our further analysis we follow principles used for other known examples of reaction-transport problems for which the scaling relations allow the determination of a bifurcation structure and reduce computation time [Ar], [Ot], [SyT], The results of this report can be used as a starting point for analyzing the scaling properties of a waste rock model or a model for underwater disposal.

3.2 Scaling Formula for Concentration of Oxygen Dissolved in Water 

Table 3.1 presents data on the concentrations of oxygen dissolved in water, [O₂] (in units of mol per litre) as a function of the temperature and oxygen concentration in the gas phase, [O₂]_gas (in volume %). The temperature values correspond to those observed in waste rock piles. [O₂] increases slightly faster than linearly with [O₂]_gas at constant temperature. At 50°C and [O₂]_gas=21%, [O₂] is less than 50% of its value at 10°C. In this way we observe a competitive effect of temperature and dissolved oxygen concentration on the reaction rates.

The values in Table 3.1 are obtained for water without any dissolved iron or sulphate ions. Unfortunately we do not have any experimental data on the concentrations of dissolved oxygen in the presence of iron and sulphate ions at various pH values. We will simply assume that the maximum concentration of oxygen dissolved in water does not depend on pH. This assumption is supported by the fact that the order of reaction (R3) with respect to oxygen does not depend on pH, as discussed byMcKibben and Barnes [McK].

The solubility of oxygen in water can be described by the relation:

The last formula gives the values of concentration of dissolved oxygen in water in micromoles of oxygen per litre when concentration of oxygen in the gas phase is measured as a percentage of the total volume of air and temperature is measured in °C. This formula gives better than 10% average agreement with the experimental data presented in Table 3.1.

Eq. (3.12) introduces the temperature dependent factor which inhibits oxidation rates at high temperatures. To our knowledge, this effect has not been analyzed in previous models for acid rock drainage.

In Chapters 4 and 5 we will use the variable Y which will denote [O₂]_gas expressed in units of mol per cubic meter and appropriately transformed formula (3.12) will be used.

Table 3.1 Saturation concentrations of dissolved oxygen in water, [O₂], at atmospheric pressure (1013.25 hPa) for different oxygen concentrations in the gaseous phase [O₂]_gas. The solubilities of oxygen are defined in terms of the mass of oxygen dissolved in one litre of the water in equilibrium with an atmosphere saturated with water vapour at various temperatures. The data are calculated following Benson and Krause [BeK] and using a computer program published by Beer [Be].

3.3 Effective Kinetic Equations

Rates of oxygen consumption and energy release can be derived as effective rates taking into consideration the summary effect of all reactions involved. Our quantitative analysis of experimental data and numerical analysis of eqs. (D1) and (D2) issummarized below.

3.3.1 pH Between 4 and 7

1) The acid production rates are directly proportional to the coefficient S/V which measures the ratio between the active surface area of pyritic rock per volume of water.

2) Oxygen consumption rates increase at the rate of 3.1 times per 10K. This corresponds to the power law increase T^3.30 with temperature T measured in °C. This formula gives better than 5% agreement with both laboratory data and our previous numerical results over the temperature range from 0°C to 60°C typical for waste rock piles.

3) The total rate of iron production (as Fe²⁺ and Fe(OH)₃) is proportional to the rate of oxygen consumption with the molar proportionality coefficient (0.28±0.01) in the temperature range between 0°C and 60°C (see Figs. A1 and A2).

4) The rate of sulphate ion production follows the stoichiometry of the reactions (R1)-(R4) and is twice as high as that of the ferrous iron production. The molar coefficient with respect to oxygen is 0.56±02 (i.e. 0.56mol of sulphate produced per 1mol of oxygen consumed).

5) The solubility of oxygen in water can be described by the relation (μmol/1 of oxygen in water)=100 μmol/1 {[O₂]_gas/(10%)}^1.2(30°C/T)^0.5 which describes the dependence of dissolved oxygen as a function of temperature and oxygen concentration in the gas phase.

Based on the results summarized above the effective kinetic equations for oxygen consumption and energy release have the form:

where h is the heat of pyrite oxidation per mol of oxygen consumed.

3.3.2 pH Less Than 4

1) The acid production rates are directly proportional to the coefficient S/V which measures the ratio between the active surface area of pyritic rock per volume of water.

2) Oxygen consumption rates increase at the rate of 2.5 times per 10K. This corresponds to the power law increase T^2.6 with temperature T.

3) The rate of ferrous iron production is proportional to the rate of oxygen consumption with the molar proportionality coefficient (0.29±0.01) in the temperature range between 0°C and 60°C.

4) The rate of sulphate ion production follows the stoichiometry of the reactions (R1)-(R4) and is twice as high as that of the ferrous iron production. The molar coefficient with respect to oxygen is 0.58±02 (i.e. 0.58mol of sulphate produced per 1mol of oxygen consumed).

5) The solubility of oxygen in water can be described by the relation (μmol/1 of oxygen in water) = 100 μmol {[O₂]_gas/(10%)}^1.2(30°C/T)^0.5 which describes the dependence of dissolved oxygen as a function of temperature and oxygen concentration in the gas phase.

The molar coefficients are significantly affected by the bacterial oxidation. (See ref. [SyT] for details).

The kinetic equations have the form:

In Chapter 4 the rate equations (3.13)-(3.16) will be used in the reaction-transport scaling model. The effective kinetic equations derived in this Chapter are based on laboratory data obtained for small samples of pyrite in water. In our further analysis we replace the coefficient kS/V by the coefficient aS in which S is the reactive surface area of pyrite in one cubic meter of waste rock and a is the rate constant in the appropriate units (see Tables 5.1 and 5.2).