Matthew Otwinowski

In this Section, four different sets of entry data are used for our simple reaction-diffusion model. The two examples illustrate the possible range of variability of the dimensionless parameter δ and possible quantitative change in the type of nonlinear behaviour for different values of Pile porosity. For different sets of entry data we find the conditions for the chemical kinetic coefficient α and pile size L necessary for the waste rock pile to remain in the low temperature regime. Scenarios 1 and 2 use small and intermediate porosity values. Scenarios 4 and 5 assume presence of impermeable covers resulting in a smaller than atmospheric concentration of oxygen at the cover - waste rock interface.

At first we calculate the Maximum Temperature T*_m. T*_m is the upper bound on the maximum temperature in a pile. T*_m can be reached only in infinitely large piles. For the value of T*_m we obtain: 

Note that when ε goes to zero, there is no access of oxygen and T*_m equals the temperature at the surface, T_b. For large values of ε, in the absence of impermeable covers, the formula (4.7) cannot be used because the convective effects can no longer be neglected. For the values of ε between 0 and 0.25 the formula (4.7) gives a reasonable estimate. 

For the adopted values of entry data the lower limit of the integral in formula (4.10) is given by:

By using the formula T=UT_m*(q+2)/(q+1) (see eq. (4.4); q=2.1) we obtain the dependence of the maximum temperature in the pile as the function of the dimensionless physicochemical universal scaling parameter δ. The plot presented in Fig. 5.1 is obtained from the universal plot in Fig. 4.3 by rescaling the vertical axis by the factor:

Because γ defines the temperature scale, we call γ the thermophysical coefficient. Note that the thermophysical coefficient does not depend on the effective active surface area. The effective active surface area, S, appears in the formula for the scaling coefficient δ which defines the horizontal scale: 

where α=aS. The effective active surface area S seems to be the most difficult coefficient for estimation from the first principles.

In Fig. 5.1 two scales on the horizontal axis are used. For a given value of κ=hεD/λ and T_b, the plot 5.1 is valid for any combination of parameter values entering the formula (5.2) and for which U_b remains unchanged. We have chosen the pile size, L, as a variable parameter. For fixed values of other parameters Fig. 5.1 illustrates the dependence of the maximum temperature in the pile as the function of the pile height, 2L. The length scale in Fig. 5.1 is defined by the formula (α=aS; see also eqs. (4.9), (4.10) and (4.12)): 

In the analogous way, plots for any other values of the active surface area S can be obtained from a single plot 5.1. This property illustrates the universal character of the parameter δ given by formula (5.4). One can calculate the value of the scaling parameter δ based on the experimental data. The waste rock pile should be designed so that:

In our example (see Table 5.1), for ε=0.13 and S=0.5 m² the pile height should be less than 2L^*=15.2 m. When the condition (5.6) is satisfied, the maximum temperature in the pile is less than 30°C. Analogous estimates can be made also in the presence of bacteria. In the presence of bacteria pyrite oxidation rates are much higher and the critical value of pile size, L^* will be smaller. (One must remember, however, that the bacterial oxidation rates decrease with temperature at temperatures greater than 40°C. For this reason qualitative behaviour of waste rock piles in the presence of bacteria is expected to be different than for abiotic oxidation. A separate scaling analysis is required when appropriate field studies will provide information about distribution of bacteria concentrations inside waste rock piles.)

The temperature profiles, T(x), for two different points P₁ and P₂ in Fig. 5.1, corresponding to two different values of δ, are presented in Fig.5.2. The value δ₁=2.0 is smaller than the cross-over value δ^*=2.35 at which a thermodynamic catastrophe occurs. The value δ₂=3.0 is greater than δ^*. The value of δ^* is defined by the maximum slope of the curve in Fig. 5.1. (In our next example the value of δ^* is defined by the value of δ at which the discontinuity occurs (see discussion in Section 4.3). 

Figs. 5.2 and 5.3 present plots of temperature as the function of the distance, x from the pile surface. Two particular values L₁=6.0 m and L₂=9.0 meters, which correspond to δ₁=2.0 and δ₂=3.0 are used. This example illustrates the cross-over effect. While L₂ is about 50% greater than L₁, one observes a dramatic effect of size increase on the maximum temperature which changes from 15°C to 45°C. The pyrite oxidation rate at 45°C is about 10 times faster than at 15°C. Our analytical results are also supported by the numerical results obtained for the two trapezoidal piles with L₁=6.0 m and L₂=9.0 m and the base length equal to 100 meters. The temperature and oxygen profiles are presented in Appendix B. (The scaling analysis is also a good test for the numerical algorithm which we plan to use for more complex scenarios of ARD).

Scenario 2: Moderate value of pile porosity

In the second example, there is a greater value for pile porosity with a resulting smaller value of thermal conductivity. The following set of entry data is used: 

The same steps are followed as in the previous example. At first the temperature T*_m is calculated, which is the upper bound on the maximum temperature in a pile. For the value of T*_m we obtain: 

Once again we stress that T*_m cannot be reached in finite-size piles. To our knowledge, the maximum reported temperature measured during field tests is 75°C.  

As a result of greater pile porosity and a smaller value of thermal conductivity, the present value of T*_m is greater than in Scenario 1. (The value of T_b=10°C is the same as in Scenario 1.) The dependence of thermal conductivity on pile porosity is discussed Syhmal [Sch]. 

For the adopted values of entry data the lower limit of the integral in formula (4.10) is given by:  

In the next step we calculate the temperature profiles for two different points P₁ and P₂ in Fig. 5.4. P₁ and P₂ belong to two different thermodynamic branches at the critical value of δ=δ^*=2.75 at which the thermodynamic catastrophe occurs. (In Fig. 5.1 the value of δ^* is defined by the maximum slope of the curve T(δ)). In the present example the value of the pile porosity is responsible for a smaller value of U_band this leads to the discontinuity at δ=2.75.

Fig. 5.4 Dependence of the maximum temperature in the waste rock pile as the function of: (a) scaling parameter δ; (b) pile size L for S=0.5 m2 (middle scale); (c) active surface area when L=6.5 m (lower scale).

In order to minimize the overall rate of acid generation the waste rock pile should be designed so that:





 As in the previous example, the plot presented in Fig. 5.4 is obtained from the universal plot in Fig. 4.2 by rescaling the vertical axis by the thermophysical coefficient:  

which does not depend on the effective active surface area, S. (S appears in the formula for δ).

In the present example, for ε=0.20 and the same value of S as in Scenario 1, the pile height should be less than 2L^*=13 m. This illustrates the general tendency that the cross-over value of L decreases with the increase of pile porosity. (2L^* is equal to 15.2 m for ε=0.13 in Scenario 1). 

In Fig. 5.4 three scales on the horizontal axis are used. For a given value of κ and T_b the plot 5.4 is valid for any combination of parameter values entering the formula (5.8) and preserving the value of U_b=0.0792. When the horizontal scale in the middle is used, Fig. 5.4 represents the plot of the maximum temperature T_m as a function of the pile size, L, when other parameters have fixed values given in Table 5.2. In addition to our previous example, we have also chosen the active surface area, S as a variable parameter. For the lowest horizontal scale, the pile size parameter has the fixed value 6.5 m. For fixed values of other parameters Fig. 5.4 with the lowest horizontal scale illustrates the dependence of the maximum temperature in the pile as the function of the effective active surface area, S. The lowest horizontal scale in Fig. 5.4 is defined by the formula: 

Fig. 5.5 presents plots of temperature as the function of distance, x, from the pile surface for the critical value of L*=6.5 m when the thermodynamic catastrophe occurs (S=0.5 m²). The maximum values of temperature reached for the two curves of T(x) are the same as the values of T_m(P₁)=26°C and T_m(P₂)=58°C in Fig. 5.4. 

The oxygen concentration plots can be easily obtained by using eq. (4.3) (See Appendix B). The temperature and oxygen concentration profiles define the rate of acid generation discussed in the next section.

The same plots can be also used for other combinations of fixed and variable parameters. The same critical dependence of T_m on δ occurs for arbitrary combinations of ε and T_b which produce the same value of U_b. For example, when in Scenario 2 the value of T_b=10°C is replaced by a new value of temperature at the pile surface, T_b=18.5°C, we obtain a plot of U_m(δ) with the same value of δ^* as in Fig. 4.3. This happens because ε=0.20 and T_b=18.5°C produce the same value of U_b=0.13636 as ε=0.13 and T_b=10°C (see eq.(5.8)). In order to obtain the plots for T_m(δ), T_m(L), T_m(S) and T(x) one simply has to rescale the vertical axis of Fig. 4.3 by the thermophysical coefficient γ and use eqs.(5.5), and (5.10) which define the horizontal scale. 

i) find the value of U_b (eq. (5.2)); 

ii) calculate the universal function I(U_m;U_b) to obtain a plot of U_m(δ) analogous to those in Figs. 4.2 and 4.3; 

iii) use the definition of thethermophysical coefficient γ (eq.(5.3)) to define the temperature scale (vertical axis); and 

iv) use the definition of the scaling indicator δ to define the horizontal scale in terms of any variable (L, S, etc) entering the formula (5.4) for δ. 

This discussion illustrates the general value of the scaling relations. In the decision making process one can analyze different ways of managing the waste rock piles by using equivalent scenarios offered by the results of scaling analysis. 

In the third example, there is a large value for pile porosity with a resulting smaller value of thermal conductivity. In order to eliminate the detrimental effect of large pile porosity the entry data assume that a low permeability cover is applied in order to reduce oxygen access. This results in a smaller than atmospheric value of oxygen concentration at the interface between the cover and waste rock. We use a smaller value of Y_b than in previous examples. The following set of entry data is used: 

The present value of Y_b corresponds to 5% oxygen concentration (21% is the normal atmospheric concentration.) The same steps are followed as in the previous examples. At first the temperature T*_m is calculated, which is the upper bound on the maximum temperature in a pile. For the value of T*_m we obtain: 

Because of the impermeable cover, the present value of T*_m is similar to that in Scenario 1 despite the greater pile porosity and a smaller value of thermal conductivity (compare Table 5.1; the present value of T_b=15°C is slightly higher as in Scenario 1.)

For the adopted values of entry data the lower limit of the integral in formula (4.10) is given by: 

Fig. 5.6 presents the dependence of the maximum temperature in the pile as the function of the scaling parameter δ and the pile size L. The critical values are δ^*=2.0 and L^*=13 meters.

As in the previous examples, the temperature scale in the plot in Fig. 5.6 is determined by the thermophysical coefficient:

In the present example, for ε=0.28, Y_b=1.88 mol/m³ and the same value of S as in Scenarios 1 and 2, the cross-over value of L, L^*=13 m is obtained. L^* is equal to 7.6 m for ε=0.13 in Scenario 1 (well compacted pile) and L^* is equal to 6.5 m for ε=0.20 (moderately compacted pile) in Scenario 2 when no covers are used. The present example suggests that low- permeability covers should have a better effect than the pile compaction. 

In general, the boundary value of oxygen concentration, Y_b, is a function of cover permeability and the total amount of oxygen consumed inside the pile per unit time. This aspect should be addressed in a more detailed numerical study.

In the fourth example, we again use a large value for the pile porosity with a resulting smaller value of thermal conductivity. A poor cover reduces the value of oxygen concentration at the interface between the cover and waste rock to a much lesser degree than in Scenario 3. We use the value Y_b two times greater than in Scenario 3. The following set of entry data is used:

The present value of Y_b corresponds to 10.5% oxygen concentration (21% is the normal atmospheric concentration). For the value of T*_m we obtain:

As a result of the higher cover permeability the present value of T*_m is much greater than in Scenario 3.

For the adopted values of entry data the lower limit of the integral in formula (4.10) is given by: 

Fig. 5.7 presents the dependence of the maximum temperature in the pile as the function of the scaling parameter δ and the pile size L.

The critical values are δ^*=2.4 and L^*=8.0 meters at which the maximum temperature T_m=40°C. In Scenario 3 the same maximum temperature was reached in a much larger pile when L=19 m (see Fig. 5.6). This illustrates the general tendency that the allowed value of L increases with the decrease of cover permeability.

At large values of the maximum temperature, when T_m is greater than 40°C, the results for Scenario 4 are not expected to be very accurate because of the neglected convective air flow. In future work the results of scaling analysis should be extended to the situations when convective air flow and water infiltration rates are significant.