VI. Reaction - Diffusion - Convection Model

     

Matthew Otwinowski

Non-linear waste rock modelling

VI. REACTION - DIFFUSION - CONVECTION MODEL

Ra is the thermal Rayleigh number for air. The velocity components v_x and v_z are defined by the derivatives of the stream field. The change of energy density associated with the convective vapour transport is expressed as

X(Y,T) denotes the local concentration of oxygen dissolved in water which is given by a closed form formula as a function of temperature and oxygen concentration in the gas phase (see Table I), r is the relative humidity (greater than zero and less than one), v is flow velocity, Q_w - latent heat of evaporation; Q_w=2.45·10⁶ J kg^-1. At velocity values on the order of 10^-4 m/s the energy transport rate due to convective vapour transport becomes comparable to the rate of heat transfer by thermal conduction. 

The individual terms represent¹:

1. the total (local) rate at which oxygen concentration changes in the gas phase inside the pile
2. the diffusive transport of oxygen through the pile pores
3. the convective transport of oxygen through the pile pores
4. the rate at which oxygen is depleted due to the oxidation reactions
5. the total (local) rate at which the thermal energy density changes inside the pile
6. the thermal conductivity of waste rock + air
7. the convective cooling due to convective flow of air with water vapour
8. the energy generation due to exothermic sulphide oxidation

One of our objectives was to obtain numerical results which would indicate how sulphide oxidation rates per unit rock mass are affected by the pile height, volume and shape. Two different types of piles analysed in our simulations are depicted in Fig. 6.1. Pile A has a square base of length sl and a square top of a length s2; sl=100m, s2=90m. Pile B has dimensions sl=100m and s2=60m. Pile C is much smaller and has the values of sl=40m and s2=30m. We have obtained numerical solutions for piles with different height L.

Fig. 6.1. Two types of piles with different slopes considered in numerical simulations.

Table II contains the input parameters used during the numerical simulation. The boundary conditions for oxygen concentration Y_s1 assume that the bedrock porosity is five times smaller than the pile porosity.

Table III summarizes the important information about: 

- total acid generation rates measured as the total number of moles of sulphate generated inside the pile (denoted by SULtot)

- average acid generation rate per one cubic metre of waste rock per hour (SULav)

- spatial variability of acid generation rates measured as the minimum and the maximum local sulphate generation rates in moles of sulphate generated per one cubic metre of waste rock per hour (SULmin and SULmax)

- average concentration of oxygen (Yav)

- maximum temperature inside the pile (Tmax)

- total energy generation rate (Etot)

- maximum local energy generation rates per one cubic metre per hour (Emax)

- average energy generation rate (Eav)

- total thermal energy stored (Est) calculated as the volume integral

(Only the difference between the stored energy for different piles has a physical meaning because the energy is determined up to an additive constant. An additive constant is different for piles of different size and only piles of the same type A, B or C can be compared).

Piles with different shapes and size but the same height L have the same values of maximum temperature.

The maximum temperature and the oxidation rates increase with the pile height L. For example, piles A6.0, B6.0 and C6.0 have exactly the same maximum temperature and the same maximum (local) sulphate generation rates.

Piles A6.0 and C6.0 have the same average sulphate generation rates per unit waste rock mass, despite a significant difference in the lateral dimensions (piles A have the base 100m by 100m, piles C have the base 40m by 40m and contain much less waste rock than piles A). Piles B do not have steep slopes and produce less acid per unit waste rock mass than the piles A and B of the same maximum height.

The rate of energy generation increases with pile height. Energy generation rates are proportional to the acid generation rates. Piles A, B and C have different volumes and for this reason they generate different amounts of energy per unit time (Etot). The total stored energy Est also increases with the pile size.

Numerical results for piles of different size, shape and different sulphide content

TABLE III. The symbols in the first column refer to the pile lateral dimensions and height in metres. Piles A, B and C have different shapes and are defined in Fig. 6.1. The value of the reactive surface area σ per unit volume is equal to 0.5 m^-1 for piles A, B and C. Piles SA11.3/8 and SA17/12 have σ=0.25 m^-1 and height greater by a factor of 2^1/2 than the height of piles A8.0 and A12.0 respectively. The ratio between the total mass of waste rock in piles A, B and C of the same height is about 7/5/1. (See text for further explanation and discussion).

Fig. 6.1. Temperature distribution in piles (from left to right) A10.0, A11.5 and A13.0 (here and in other figures pile height shown along Y-axis). Tire three-dimensional plots show a strongly nonlinear increase of temperature when the pile height increases.

Fig. 6.2. Sulphate generation rates in piles (from left to right) A10.0, A11.5, and A13.0. Note that the maximum oxidation rates coincide with the temperature maxima in Fig. 6.1.

SCALING PROPERTIES

Piles SA11.3/8 and SA17/12 have the values of L greater by a factor of 2^1/2 and contain a less reactive waste rock, with σ=0.25, smaller by a factor of two than σ=0.5 in their counterparts, piles A8.0 and A12.0.

Piles A8.0 and SA11.3/8 have exactly the same maximum temperature, and average thermal energies different by a factor of two. Piles A12.0 and SA17/12 also have exactly the same maximum temperature, and average thermal energies different by a factor of two. The average acid generation rate in pile SA17/12 are also slower by a factor of two than in pile A12.0. The ratios between the height and rock reactivity of piles SA17/12 and A12 are again 2^1/2 and ½, respectively. In pile SA17/12 the to tal acid generation rate is by a factor of 2^1/2 smaller than in pile A12.

The numerical results confirm our earlier results of the scaling analysis². The large-scale properties of waste rock piles are governed by a scaling parameter³

Because the oxidation rate and energy generation rate are proportional to σ it is obvious that after a transformation L → 2^1/2L and σ → σ/2 we should obtain a pile with the same maximum temperature but the average oxidation rate per unit mass reduced by a factor of 2^1/2.

Piles with the same values of δ are expected to have the same properties.

The numerical and analytical (scaling) results show that the physical features of the pile design are as important as the geochemical properties of the waste rock.

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¹ The finite element method with an adaptive grid generator for a fully coupled set of the nonlinear reaction-diffusion-convection equations has been used to analyse the nonlinear processes responsible for ARD. Iterative schemes which decouple the set of equations do not produce reliable results.

² M. Otwinowski, Scaling Analysis of Acid Rock Drainage, MEND Report (1995).

³ A complete scaling formula which includes the remaining model parameters will be presented in our forthcoming publications.