4.0 Scaling Analysis Of Coupling Between Chemical Kinetics And Transport Phenomena

Matthew Otwinowski

Scaling Analysis Of Acid Rock Drainage

4.0 SCALING ANALYSIS OF COUPLING BETWEEN CHEMICAL KINETICS AND TRANSPORT PHENOMENA

Mass and energy transport can be modelled by nonlinear partial differential equations which describe the spatially nonhomogeneous distribution of oxygen, water, reaction products and temperature as a function of physical and mineralogical properties such as pile porosity, oxygen transport rate, thermal conductivity, etc. [ScD], [Sch], [Ar], [Ot1], [Ot2].

In this Chapter, multiple steady state solutions are calculated exactly and the bifurcation diagram is analyzed. The scaling analysis is performed in order to establish the relative importance of individual factors responsible for acid rock drainage. The relative importance of various physical and chemical factors responsible for acid rock drainage is described by means of a dimensionless scaling parameter δ which contains quantitative information about the different processes involved. The scaling model covers a broad range of chemical and physical parameters.

Long term predictions can be usually based on quantitative results obtained for steady state solutions of reaction-transport models. One has to remember, however, that rock texture undergoes changes during weathering processes, and parameters such as the reactive surface area change over time. To our knowledge, the only detailed study of temporal changes in the reactive surface area has been performed for tailings [Ni], [NiG], A similar long term study for waste rock would provide useful information for the present scaling analysis. Because of the lack of quantitative experimental data we do not attempt to model temporal changes in the reactive surface area, S.

4.1 Formulation of the Reaction-Transport Model

From a physico-chemical point of view a pile of waste rock reacting with oxygen and water is an open non-equilibrium system evolving towards an equilibrium state in which pyrite is transformed into ferrous/ferric iron and acid.

The overall process of pyrite oxidation in waste rock piles can be described by a set of two coupled nonlinear parabolic equations in two fields, oxygen concentration, Y and temperature, T:

Equations (4.1) and (4.2) are the energy and mass conservation equations, respectively. (T - temperature, Y - oxygen concentration in the gas phase, ε - porosity of the pile, ρ_a - air density, C_a - specific heat of air, ρ_r - rock density, C_r - specific heat of rock, λ - thermal conductivity of the pile (air+rock), h - heat of reaction of oxygen with pyrite, D - diffusion coefficient of oxygen in the gas phase, q - effective thermal exponent, p - effective order of chemical reaction of oxygen with pyrite.) 

The values of a, p and q are different for different pH values. For pH less than 4 p=0.6, q=2.1. For pH greater than 4, p=0.6, q=2.8. In order to simplify the scaling analysis we will use the value of p=1. We plan to properly calibrate our results when the results of thermokinetic tests at different values of oxygen partial pressure become available for rock samples.

We consider a pile characterized by the linear dimension L, with the oxygen concentration Y_b at the boundaries, and temperature T_b at the boundaries. For symmetric boundary conditions, for a steady state we have:

(For non symmetric boundary conditions

where C is a constant depending on the boundary conditions).

From equations (4.1) and (4.2) one obtains for steady state solutions:

where

The integration constant C is not a free parameter, but (for given values of parameters) is determined by T_b and is implicitly related to L. For different values of C we obtain different values of the maximum temperature, T_m in the pile. The linear size, L is given by the formula

Formula (4.6) defines implicitly the distribution of temperature T(x).

The maximum temperature in the pile cannot exceed the value:

At the point where the temperature T_m* is reached, the concentration of oxygen is equal to zero. Note that T_m* does not depend on the rate constant aS. Thus the state with zero oxygen concentration at the point of maximum temperature is controlled by oxygen transport.

In the next step we derive a dimensionless parameter δ which will define the scaling law involving all the parameters in equations (4.4) and (4.5).

We introduce the variables:

In terms of the new variables, the linear size can be expressed as an integral which depends only on the parameter r: 

The universal function I[U_m; U_b] is presented in Figs. 4.2 and 4.3 for q=2.1 corresponding to low pH values and two different values of U_b (see p. 27). 

4.2 Scaling Properties and Sensitivity Analysis of Acid Rock Drainage 

It is useful to define the dimensionless parameter:

which after substituting expressions (4.5) defining the parameters A and B takes the form:

The scaling parameter δ is a combination of parameters describing physico-chemical properties of pyritic rock and storage conditions. The scaling parameter δ can be used as a quantitative indicator of acid rock drainage. Values of δ define different regimes of acid rock drainage. The greater the value of δ the greater the rate of acid generation. δ shows a strong nonlinear dependence on parameters characterizing acid rock drainage. Some of these parameters, like pile porosity and pile size, can be controlled. Maintaining the smallest possible value of δ is important in order to minimize the environmental impact of acid rock drainage. The expected environmental impact increases with the value of δ. (On the other hand, one may consider a controlled preemptive leaching over a short time in order to reduce the long term environmental impact and to increase production [Go]. In this scenario the large values of δ may be required in order to intensify leaching.)

The scaling indicator δ provides important quantitative information about relative importance of individual factors contributing to ARD. Because δ is proportional to S^1/2 the pile design parameters become increasingly important for large values of pyrite concentration. In particular, the effectiveness of impermeable covers increases with the pyrite concentration. In Chapter 5 we demonstrate how to use the scaling indicator δ for the predictive analysis of ARD.

The scaling formula (4.12) indicates the required precision of individual laboratory tests in which parameter values are determined. This aspect is analyzed in Section 5.4.

Laboratory tests usually measure acid generation over time and provide information about the chemical activity of pyritic rock (factor aS). Acid/base tests should also determine a detailed temperature dependence of acid generation rates at temperatures between 0°C and 60°C. Experimental validation of the formula describing the dependence of reaction rates on oxygen concentration in the gas phase for different pH values also seems necessary. Such experimental tests would determine the values of parameters a, S and q which are crucial for reliable estimates based on the scaling parameter δ.

For given values of other parameters we obtain the condition:

which defines the optimum size of the pile as the function pile porosity, ε. The form of condition (4.13) indicates a very strong nonlinear dependence of acid rock drainage on ε. The porosity factor:

which enters the relation (4.14), affects very significantly the total rates of acid generation.

Fig. 4.1. Dependence of the porosity factor defined by eq. (4.14 on porosity values for high and low pH values (q=2.8 and 2.1 respectively).

Fig. 4.1 shows the plots of P(ε) for values of q characteristic for low and high pH values. Additional strongly nonlinear effects appear as the result of discontinuities and cross-over effects discussed in the next section.

4.3 Bifurcations and Thermodynamic Catastrophes in Acid Generation Rates

Numerical values of the integral I[U_m; U_b] show that the value of δ should be less than a certain critical value δ* for which a transition to accelerated rates of acid production is expected. The values of δ* depend on the scaling exponent, q, and are different for low and high pH values. The values of U_b and U_m in the integral define the temperature at the pile surface and the maximum temperature in the pile respectively (see eq. (4.8)). Figs. 4.2 and 4.3 present plots of the maximum value of U as the function of the scaling parameter δ for q=2.1 and two different values of U_b: U_b=0.0792 and U_b=0.13636 used later in Chapter 5 when various alternative scenarios are discussed. For δ=δ*=2.75 in Fig. 4.2 we observe a discontinuity in U_m. Because the maximum temperature in the pile is proportional to U_m, we call this discontinuity a thermodynamic catastrophe. The thermodynamic catastrophe is responsible for accelerated rates of acid generation.

For given values of a and S the values of pile porosity and pile size should be controlled in such a way that the value of δ* is not exceeded. The values of generation rates at δ>δ* can exceed by several times the acid generation rates at δ<δ*. The critical values, δ*(pH) can be determined from numerical results obtained for the integral I(U_m; U_b] which is a multivalued function of U_m and U_b.

The curve U_m(δ*) in Fig. 4.3 is continuous but also shows a dramatic increase in the vicinity of δ*=2.35. The critical value of the scaling parameter is defined in this case as the value of δ at which the maximum slope of the curve U_m(δ) occurs. The values of δ* depend strongly on the temperature at the pile boundary. The greater the ambient temperature, the smaller the value of δ that should be maintained in order to avoid the thermodynamic catastrophe.

Chapter 5 demonstrates the practical use of the scaling parameter δ as the practical indicator of acid rock drainage. The qualitative features of the integral I[U_m; U_b] for other values of the thermokinetic exponent q are the same as for the particular value of q=2.1 used in our examples. Thus the occurrence of thermodynamic catastrophes is a general feature of waste rock piles.

Fig. 4.2 Dependence of the maximum value of U (defined by eqs. (4.8)-(4.10)) on the values of the parameter δ; U_b=0.0792. At δ=δ*=2.75 a discontinuity in the maximum temperature and acid generation rates occurs. (Note: The top branch extends beyond the point P₂ - this feature is not visualised here and is responsible for the thermodynamic hysteresis not discussed in this report).

Fig. 4.3 Dependence of the maximum value of U (defined by eqs. (4.8)-(4.10)) on the values of the parameter δ; U_b=0.13636. In the vicinity of δ=δ*=2.35 a dramatic increase in the maximum temperature and acid generation rates occurs.