Refferences

Matthew Otwinowski

Scaling Analysis Of Acid Rock Drainage

REFFERENCES 

[Al] R. Albright, in Proceedings of Acid Mine Drainage Workshop; Environment Canada and Transport Canada, Halifax, NS, p. 146, 1987.

[ApK] R.J. Applegate and M.Kraatz, Rehabilitation of Rum Jungle Mine, Proc. MEND Conf, Montreal 1991.

[Ar] R. Aris, A Mathematical Theory of Reacting and Diffusing Systems, Wiley 1981.

[BaZ] G.I. Barenblatt and Y.B. Zeldovich, "A. Rev. Fluid Mech.", 4, 285 (1972).

[BaO] J.E. Bailey and D.F. Ollis, Biochemical engineering fundamentals, McGraw-Hill, New York 1986.

[Be] T. Beer, Applied Environmetrics Hydrological Tables, Applied Environmetrics, Victoria, Australia, 1991.

[BeK] B.B. Benson and D. Krouse, Jr., The concentration and isotopic fractionation of gases dissolved in fresh water in equilibrium with the atmosphere, I Oxygen, "Limnol. Oceanog.", 25, 662 (1980).

[BeP] J.W. Bennet and G. Pantelis, Construction of a waste rock dump to minimise acid mine drainage a case study, Proc. MEND Conf., Montreal 1991.

[BeR] J.W. Bennett and A.I.M. Ritchie, Measurements of the transport of oxygen into two rehabilitated waste rock dumps, Proc. MEND Conf., Montreal 1991.

[BoB] F.C. Boogerd, C. van denBeemd, T. Stoelwinder, P.Bos and J.G. Kuenen, Relative contributions of biological and chemical reactions to the overall rate of pyrite oxidation at temperatures between 30°C and 70°C, "Biotechnology and Bioengineering", 38, 109 (1991).

[BrC] British Columbia Acid Mine Drainage Task Force, Acid Rock Drainage Technical Guide, BC AMD Task Force Report, 1989.

[CaG] F.T. Carruccio, G. Geidel and M. Pelletier, "J. Energy (Amer. Soc. Civil Engineers)", 107, (1981).

[CaW] M.C. Campbell, D. Wadden, A. Marchbank, R.G.L. McCready and G. Ferroni, In-place leaching of uranium at Denison Mines Ltd., Proc. IAEA Meeting, Vienna 1985.

[CdO] J. Chadam and P. Ortoleva, "Earth Science Review", 29, 175 (1990).

[Ch] F. Chaiken, private communication (1992).

[ChB] K. Ye Cheng and P.L. Bishop, "J. Air&Waste Manage. Assoc.", 42, 164 (1992).

[Co] W.R. Cowan, "Hazardous Mat.Mgmt.", 3, 12 (1991).

[CoR1] Coastech Research, Investigation of Prediction Techniques for Acid Mine Drainage, MEND Project, West Vancouver, BC, Canada, 1989.

[CoR2] Coastech Research, Acid Rock Drainage Prediction Manual: A manual of Chemical Evaluation Procedures for the prediction of Acid Generation from Mine Wastes, Mend Project, North Vancouver, BC, Canada 1990.

[Da] S. Day, Long term kinetic acid generation studies. Cinola project, B.C. AMD Task Force, 1993.

[DaR1] G.B. Davis and A.I.M. Ritchie, A model of oxidation in pyritic mine wastes: part 1: equations and approximate solutions, "Appl. Math. Modelling", 10, 314 (1986).

[DaR2] G.B. Davis and A.I.M. Ritchie, A model of oxidation in pyritic mine wastes: part 2: comparison of numerical and approximate solutions, "Appl. Math. Modelling", 10, 323 (1986).

[DaR3] G.B. Davis and A.I.M. Ritchie, A model of oxidation in pyritic mine wastes: part 3: importance of particle size distribution, "Appl. Math. Modelling", 11, 417 (1987).

[DrA] D.W. Drott and R. Aris, Communications on the theory of diffusion and reaction-I A complete parametric study of the first-order, irreversible exothermic reaction in a flat slab of catalyst, "Chem. Engng. Sci.", 24, 541, (1969).

[DuB] D.W. Duncan and A. Bruynsteyn, Determination of the Acid Production Potential of Waste Materials, Metallurgical Society of AIME Annual Meeting, New Orleans (1979).

[DuD] J.G. Dunn, G.C. De and P.G. Fernandez, The effect of experimental variables on the multiple peaking phenomenon observed during the oxidation of pyrite, "Thermochimica Acta", 135, 267 (1988).

[ErH] P.M. Erickson, R.W. Hammack and R.L.P. Kleinman, Prediction of acid drainage potential in advance of mining, in Control of Acid Mine Drainage, USBM IC 9027.

[Es] H. Ese, Acidic mine drainage from the Killingdal Mine Norway, Proc. MEND Conf., Montreal 1991.

[Fe] K.D. Ferguson, Static and Kinetic Methods to Predict Acid Mine Drainage, Env. Canada, 1985.

[FeE] K.D. Ferguson and P.M. Erickson, Approaching the AMD Problem-from Prediction and early Detection, In. Proc. Int. Conf, on Control of Environmental Problems from Metal Mines, 1988, Roros, Norway.

[FeE1] K.D. Ferguson and P.M. Erickson, Will It Generate Acid? An Overview of Methods to Predict Acid Mine Drainage.

[FeMe] K.D. Ferguson and P.E. Mehling, Acid Mine Drainage in B.C. The Problem and Search for Solutions, Environment Canada, 1986.

[FeM] K.D. Ferguson and K.A. Morin, The prediction of Acid Rock Drainage - Lesson from the Database, Environment Canada Report, 1992.

[Ga] D.M. Galbraith, The mount Washington acid mine drainage reclamation project, Proc. MEND Conf., Montreal 1991.

[Ge] P. Gelinas and R. Guay, Acid Mine Drainage Generation from a Waste Rock Dump and Evaluation of Dry Covers Using Natural Minerals: La Mine Doyon Case Study, MEND Report (1990).

[Ge] P. Gelinas, private communication (1993).

[Go] B. Godin, private communication.

[Gol] M B. Goldhaber, Experimental study of metastable sulpfur oxyanion formation during pyrite oxidation, "Am. J. Sci.", 283, 193 (1983).

[Ha] Sophie Bierens de Haan, A review of the rate of pyrite oxidation in aqueous systems at low temperature, "Earth-Science Reviews", 31, 1, (1991).

[HoS] George R. Holdren, Jr. and Patricia M. Speyer, Reaction rate - surface area relationships during the early stages of weathering - I. Initial observations, "Geochimica et Cosmochimica Acta", 49, 675 (1985).

[HwS] C.C. Hwang, R.C. Streeter, R.K. Young and Y.T. Shah, Kinetics of the ozonation of pyrite in aqueous suspension, "Fuel", 66, 1574 (1987).

[KeW1] G.H. Kelsall and R.A. Williams, Electrochemical behavior of ferrosilicides Fe_xSi) in Neutral Alkaline Aqueous Electrolytes. I. Thermodynamics of Fe-Si-H₂O systems at 298 K, "J. Electrochem. Soc.", 138, 941 (1991).

[KoB] A. Kok, N. Bolt and J.H.N. Jergersma, "KEMA Sci.&Tech. Reports", 7, 63 (1989).

[Kw] Y.T.J. Kwong, Acid generation in waste rock as exemplified by the Mount Washingtonminesite, B.C., Canada, Proc. MEND Conf., Montreal 1991.

[Las] Antonio C. Lasaga, Chemical Kinetics of Water-Rock Interactions, "Journal of Geophysical Research", 89, 4009, (1984).

[LawR] R.W. Lawrence, G.M. Ritcey, G.W. Poling, P.B. Marchant, Strategies for the Prediction of Acid Mine Drainage, Proceedings of the 13 th Mine Reclamation Symposium, Ministry of Energy, Mines and Petroleum Resources; Victoria, 1989.

[LeW] J.H. Levy and T.J. White, The reaction ofpyrite with water vapour, "Fuel", 67, 1336 (1988).

[LGI] R. Lefebvre, P. Gelinas and D. Isabel, Heat transfer during acid mine drainage production in a waste rock dump, La Mine Doyon (Quebec), MEND Report 1.14.2.

[Low] Richard T. Lowson, Aqueous Oxidation of Pyrite by Molecular oxygen, "Chemical Reviews", 82, 461 (1982).

[Lu] George W. Luther III, Pyrite oxidation and reduction: Molecular orbital theory considerations, "Geochimica et Cosmochimica Acta", 51, 3193 (1987).

[LuF] G.W. Luther III, T.G. Ferdelman, J.E. Kostka, E.J. Tsamakis and T.M. Church, Temporal and spatial variability of reducedulpfur species andporewater parameters in salt marsh sediments, "Biogeochemistry", 14, 57 (1991).

[MaR1] C.T. Mathews and R.G. Robins, "Australais. Ins. Mining Metall", 242, 47 (1972).

[MaR2] C.T. Mathews and R.G. Robins, "Aust. Chem. Eng.", 15, 19 (1975).

[MaR3] C.T. Mathews and R.G. Robins, "Aust. Chem. Eng.", 13, 129 (1972).

[McC] R. McCandless, private communication, 1993.

[McK] Michael A. McKibben and Hubert L. Barnes, Oxidation of pyrite in low temperature acidic solutions: Rate laws and surface textures, "Geochimica et Cosmochimica Acta", 50, 1509-20 (1986).

[MiS] F.J. Millero, S. Sotolongo and M. Izaguirre, The oxidation kinetics of Fe(II) in Seawater, "Geochimica et Cosmochimica Acta", 51, 793-801 (1987).

[MoJ] K.A. Morin, E. Gerencher, C.E. Jones, D. Konasewich and J.R. Harries, Critical literature review of acid drainage from waste-rock, Nothwest Geochem, 1990.

[MoM] John W. Morse, Frank J. Millero, Jeffrey C. Cornwell and D. Rickard, The chemistry of the Hydrogen Sulfide and Iron Sulfide Systems inatural Waters, "Earth-Science Review", 24, 1 (1987).

[MsH] Carl O. Moses, Alan T. Herlihy, Janet. S. Herman and Aaron L. Molls, Ion-chromatographic analysis of mixtures of ferrous and ferric iron, "Talanta", 35, 15 (1988).

[MsH] Carl O. Moses and Janet S. Herman, Homogeneous Oxidation Kinetics of Aqueous Ferrous Iron at circumneutral pH, "Journal of Solution Chemistry", 18, 705 (1989).

[MsH] Carl O. Moses and Janet S. Herman, Pyrite oxidation at circumneutral pH, "Geochimica et Cosmochimica Acta", 55, 471 (1991).

[MsN] Carl O. Moses, D. Kirk Nordstrom, Janet S. Herman and Aaron L. Mills, Aqueous pyrite oxidation by dissolved oxygen and by ferric iron, "Geochimica et Cosmochimica Acta", 51, 1561 (1987).

[Ni] R.V. Nicholson, private communication (1993).

[NiG] Ronald V. Nicholson, Robert W. Gillham and Eric J. Reardon, Pyrite oxidation in carbonate-buffered solution: 1. Experimental kinetics, "Geochimica et Cosmochimica Acta", 52, 1077 (1988).

[NiG] Ronald V. Nicholson, Robert W. Gillham and Eric J. Reardon, Pyrite oxidation in carbonate-buffered solution: 2. Rate control by oxide coatings, "Geochimica et Cosmochimica Acta", 54, 395 (1990).

[NiP] G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems, Wiley, 1977.

[NiS] R.V. Nicholson and J.M. Scharer, Laboratory Studies of Pyrrhotite Oxidation Kinetics, preprint.

[No] D.K. Nordstrom, Aqueous pyrite oxidation and the consequent formation of secondary iron minerals in Acid sulfate wearhering: pedogeochemistry and relationship to manipulation of soil materials, Soil Science Soc. Amer. Press, Madison, 1982.

[NoD] Nolan, Davis and Associates, Heath Steele Waste Rock Study, MEND Report 2.23.1a, 1992.

[NoJ] D.K. Nordstrom, E.A. Jenne and J.W. Ball, Redox equilibria of iron in acid mine waters, "ACS Symposium Series", 93, 51-59 (1979).

[Ot1] M. Otwinowski, A thermodynamic model for the spontaneous heating and combustion of coal. I: Elementary exothermic processes, "Fuel", submitted; also Alberta Energy Report, Edmonton 1991.

[Ot2] M. Otwinowski, A thermodynamic model for the spontaneous heating and combustion of coal. II. Reaction-diffusion model, "Fuel", submitted; also Alberta Energy Report, Edmonton 1991.

[Ot3] M. Otwinowski, Static and Dynamic Patterns in Equilibrium and Nonequilibrium Systems in Nonlinear Structures in Physical Systems, Chaos Pattern formation and Waves, Springer, 1990.

[OtL1] M. Otwinowski, W.G. Laidlaw and R. Paul, Structural instability of the Brusselator model with reversible chemical reactions, "Can. J. Phys.", 68, 743 (1990).

[OtL2] M. Otwinowski, W.G. Laidlaw and R. Paul, Explicit solutions to a class of nonlinear diffusion equations, "Phys. Lett.", 124, 149 (1988).

[PaR] G. Pantelis and A.I.M. Ritchie, Macroscopic transport mechanisms as a rate-limiting factor in dump leaching of pyrite ores, "Appl. Math. Modellings", 15, 136 (1991).

[PrL] I. Prigogine and R. Lefevre, "J. Chem. Phys.", 48, 1695 (1968).

[PrT] William A. Pryor and Umberto Tonnellato, Nucleophilic Displacements at Sulfur. III. The Exchange of Oxygen-18 between Sodium Thiosulfate O^-18 and Water, "Journal of the American Chemical Society", 89, 3379 (1967).

[Res] Rescan Env. Services Ltd., Kutcho Creek project. Acid generation testwork, Report, 1990, Vancouver, Canada.

[ScG] J.M. Scharer, V. Garga, R. Smith and B.E. Halbert, Use of Steady-State Models for Assessing Acid Generation in Pyritic Mine Tailings in Proceedings of the 2nd International Conference on Abatement, Montreal, p. 211, Tome 2, 1991.

[Sch] A. Schmal; Ph.D. Theses, Utrecht, 1983.

[ScD] A. Schmal, J.H. Duyzer and J.W. van Heuven, "Fuel", 64, 1969 (1985).

[Se] Senes Consultants, Description of RATAP. BMT3 Component Modules, Report, 1991.

[SiS] P.C. Singer and W. Stumm, Acidic Mine Drainage: The Rate Determining Step, "Science", 167, 1121 (1970).

[SnL] G.A. Singleton and L.M. Lavkulich, "Soil Science", 42, 984 (1978). 

[SoS] A.A. Sobek, W.A. Schuller, J.R. Freeman and R.M. Smith; US EPA Report; EPA 600/2-78-054, 1978.

[St] W. Stumm ed., Aquatic Chemical Kinetics. Reaction Rates of Processes in Natural Waters, Wiley, 1991.

[StL1] W. Stumm and G.F. Lee, "Schwei. Z. Hydrol.", 22, 295 (1960).

[StL2] W. Stumm and G.F. Lee, "Ind. Eng. Chem.", 53, 143 (1961).

[StM] W. Stumm and J.J. Morgan, Aquatic Chemistry. An introduction Emphasizing Equilibria in Natural Waters, Wiley, 1986.

[Sul] P.J. Sullivan and A.A. Sobek, Minerals and the Environment, 4, 9 (1982).

[SyT] Synergetic Technology, Quantitative Analysis of Chemical and Biological Kinetics for the Acid Mine Drainage Problem, MEND Report (1993).

[TaW2] B.E. Taylor and M.C. Wheeler and D.K. Nordstrom, Stable isotope geochemistry of acid mine drainage: Experimental oxidation of pyrite, "Geochimica et Cosmochimica Acta", 48, 2669 (1984).

[WaK] E.I. Wallick, H.R. Krouse and A. Shakur, Environmental isotopes: principles and applications in ground water geochemical studies in Alberta, Canada, in First Can. Conf. on Hydrology, 1984.

[We] Bernhard Wehrli, Redox reactions of metal ions at mineral surfaces, in Aquatic Chemical Kinetics W. Stumm ed., Willey, 1990.

[WhJ] W.W. White III and T.H. Jeffers, Chemical predictive modelling of acid mine drainage from metallic sulfide-bearing waste rock, U.S. Department of Interior, Bureau of Mines, 1992.

[WiR] C.L. Wiersma and J.D. Rimstidt, Rates of reaction of pyrite and marcasite with ferric iron at pH 2, "Geochimica et Cosmochimica Acta", 48, 85 (1984).

[YaW] E.K. Yanful, K.G. Wheeland, C. Luc and N. Kuyucak, Overview of Noranda Research on Prevention and Control of Acid Mine Drainage, Environmental Workshop 1991, Australian Mining Industry Council.

[ZiM] Z.D. Zivkovic, N. Molosavljevic and J. Sestak, Kinetics and Mechanism of Pyrite Oxidation, "Thermochimica Acta", 157, 215 (1990).

Appendix B: Numerical solutions by the finite elements method

Matthew Otwinowski

Scaling Analysis Of Acid Rock Drainage

APPENDIX B
Numerical solutions by the finite elements method

Figs. B1 and B2 present numerical results for the two piles, 12 m and 18 m high, analyzed in Scenario 1 in Section 5.1. The numerical values are obtained for trapezoidal piles with the base 100 meters long. The maximum temperatures in both cases are by about 1°C lower than the values obtained analytically. This difference is due to the cooling effect resulting from the heat escape through the side boundaries. The numerical results confirm the results of the scaling analysis.

Fig. B1(a). Temperature profile for the pile 12 m high (y-axis) and 100 m long (x-axis); L=6 m; all entry data listed in Table 5.1. Analytical results are presented in Fig. 5.2.

Fig. B1(b). Oxygen concentration profile for the pile 12 m high (y-axis) and 100 m long (x-axis); L=6 m; all entry data listed in Table 5.1. 

Fig. B2(a). Temperature profile for the pile 18 m high (y-axis) and 100 m long (x-axis); L=9m; all entry data listed in Table 5.1. Analytical results are presented in Fig. 5.3.

Fig. B2(b). Oxygen concentration profile for the pile 18 m high (y-axis) and 100 m long (x-axis); L=9m; all entry data listed in Table 5.1.

Appendix A: Numerical results for the chemical kinetics at different pH values

Matthew Otwinowski

Scaling Analysis Of Acid Rock Drainage

APPENDIX A
Numerical results for the chemical kinetics at different pH values 

Fig. A1. The oxidation kinetics due to reactions (R1)-(R4) for different temperatures; S/V=10 m²/l and the initial conditions: [Fe²⁺(t=0)]=0, [SO₄^2-(t=0)]=0, [Fe(OH)₃(t=0)]=0 and pH(t=0)=7. Constant in time concentration of dissolved oxygen, [O₂] for [O₂]_gas=21%. Horizontal axis: time in seconds, vertical axis: concentration in moles per litre. (180 000 sec = 5 hours).

Fig. A2. The oxidation kinetics due to reactions (R1)-(R4) for different temperatures, S/V=1 m²/l; the initial conditions: [Fe²⁺(t=0)]=0, [SO₄^2-(t=0)]=0, [Fe(OH)₃(t=0)]=0. Constant in time pH(t=0)=6.9 concentration of dissolved oxygen, [O₂] for [O₂]_gas=21%. Horizontal axis: time in seconds, vertical axis:concentration in moles per litre. (1 800 000 sec = 3 weeks).

Fig. A3. The oxidation kinetics due to reactions (R1)-(R4) for different temperatures, S/V=1 m²/l; the initial conditions: [Fe²⁺(t=0)]=0, [SO₄^2-(t=0)]=0, [Fe(OH)₃(t=0)]=0. Constant in time pH(t=0)=6.3 concentration of dissolved oxygen, [O₂] for [O₂]_gas=21%. Horizontal axis: time in seconds, vertical axis: concentration in moles per litre. (1 800 000 sec = 3 weeks).

Fig. A4. The consumption of dissolved oxygen at pH=6.9 for different values of S/V and different temperatures. (a) S/V=1m²/l, T=283 K; (b) S/V=1m²/l, T=303 K; (c) S/V=1m²/l, T=323 K; (d) S/V=10m²/l, T=303 K. Horizontal axis: time in seconds; vertical axis: concentration in moles per litre (3 000 000 s = 35 days).

Fig. A5. The consumption of dissolved oxygen at pH=5.5 for different values of S/V and different temperatures. (a) S/V=1m²/l, T=283 K; (b) S/V=1m²/l, T=303 K; (c) S/V=1m²/l, T=323 K; (d) S/V=10m²/l, T=303 K. Horizontal axis: time in seconds; vertical axis: concentration in moles per litre (3 000 000 s = 35 days).

6.0 Conclusions And Recommendations

Matthew Otwinowski

Scaling Analysis Of Acid Rock Drainage

6.0 CONCLUSIONS AND RECOMMENDATIONS

A physico-chemical model can produce a realistic description of a large-scale thermo-chemical behaviour of waste rock piles. Having demonstrated that the simple model derived from fundamental physical and chemical principles gives reasonable results, we feel confident that after a further refinement it is possible to construct a realistic and useful waste rock model.

Scaling analysis of a nonlinear reaction-transport model has been performed. The acid generation rates can be characterized by a physico-chemical scaling parameter δ which depends on pile porosity, pile size, ambient temperature, effective reactive pyrite surface area, heat of oxidation and thermal conductivity. The scaling parameter δ can be used as a practical indicator of ARD. The scaling parameter provides quantitative information about the relative importance of the factors responsible for acidic drainage. In particular, the effectiveness of impermeable covers increases with the pyrite concentration. Sample calculations of temperature profiles and acid generation rates have been performed for realistic parameter values in the absence of bacteria and without neutralization. Alternative scenarios with and without impermeable covers are discussed.

The scaling analysis indicates that geochemical and transport processes operate at the meso-scale in a way fundamentally different from the full-scale. When oxygen supply is controlled by the diffusion process, the waste rock piles can exhibit thermodynamic catastrophes. A thermodynamic catastrophe occurs when relatively small changes of parameters, such as pile size, pile porosity, ambient temperature or effective active surface area result in a dramatic increase of the acid generation rate. The scaling parameter δ provides quantitative information about waste rock storage conditions, such as pile porosity and pile size, which have to be satisfied in order to avoid a thermodynamic catastrophe leading to accelerated acid generation rates.

The scaling analysis indicates that in order to avoid a thermodynamic catastrophe leading to high acid generation rates, the waste rock piles should be designed so that the value of the physico-chemical scaling parameter δ is less than a critical value δ^*. Illustrative examples of the practical applications of the critical scaling criterion are presented.

In the next step, air convection and water transport should be included in the model and quantitative results for the expected water quality should be obtained.

In an additional study one should analyze the problem of optimum pile shape as the function of the cover permeability and the total amount of waste rock to be stored.

All the quantities involved in the scaling analysis can be measured in independent simple laboratory experiments. The accuracy of simple experiments sufficient for a predictive waste rock model has been analyzed. Because realistic results are obtained with no additional adjustable parameters, we believe that after experimental validation and additional refinement (involving nonsymmetric boundary conditions, water transport and convection of air) the results of the scaling analysis can be used as simple practical guidelines for design of waste rock sites.

Quantitative information about the oxygen consumption by bacteria and the spatial distribution of bacteria concentration in waste rock piles is necessary for extending the present scaling analysis to the case of bacterial oxidation by Thiobacillus ferrooxidans. When such data become available, the quantitative results for bacterial oxidation reviewed in our previous study [SyT] can be used to modify the present scaling analysis. 

One should rely on the results of simple laboratory tests. In addition to the usual acid/base accounting tests, thermokinetic tests should be performed for different values of temperature, pH and different oxygen partial pressures in order to define the scaling curves Y^p and αT^q which are expected to be different for rocks with different neutralization potential. Representative samples of waste rock should be used for such tests. For the needs of a predictive model it would be sufficient to perform oxidation tests under isothermic conditions at temperature intervals of 15°C in the temperature range between 5°C and 80°C and oxygen partial pressures varying between 0.21 atm and 0.01 atm at 0.05 atm intervals. The same experiments should also measure the cumulative heat of oxidation/neutralization reactions. Such isothermic tests should be performed separately for pH greater than four and pH less than four in the presence of bacteria, at water saturation values characteristic for waste rock piles. For small pH values bacteria concentration and oxygen consumption should be monitored for different values of oxygen partial pressure. In this way one can produce in a relatively inexpensive way the input data necessary for a predictive model which should help to optimize the storage of waste rock and to provide a reliable tool useful for environmental assessment. Usual acid/base accounting tests do not provide data necessary for a predictive model which should generate quantitative information about the effluent.

The value of heat, h, generated during the pyrite oxidation process can be different for different values of the acid neutralization potential. We could not find reliable data on the heat generated by calcite dissolution or the neutralization reactions, which would modify the value of h used in this report. We do not recommend, however, to measure these quantities separately. The cumulative heat generated during the weathering process is sufficient for the predictive reaction- transport model. The heat generated can be measured in the same isothermal experiment (proposed above) as the value of cooling necessary to preserve constant temperature of the oxidation process. In this way we can keep the same number of measurable coefficients as in our scaling model in which only the effective heat of pyrite oxidation is analyzed.

5.4 Discussion Of Results

Matthew Otwinowski

Scaling Analysis Of Acid Rock Drainage

5.4 Discussion of Results

The examples (Scenarios 1-4 in Section 5.1) illustrate the practical power of the scaling relations. Results for various combinations of physical and chemical parameters characterizing waste rock and pile design, can be obtained without repeating the numerical calculations for every new set of entry data.

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 lower. The activation energy of pyrite oxidation by ferric iron is greater than the activation energy of oxidation by oxygen. For this reason the model involving the bacterial oxidation will use a different value of the nonlinear thermokinetic exponent q and additional nonlinearity will be introduced by the temperature dependence of bacterial oxidation [SyT], [Se].

While the sample calculations are performed for pyrite, the same model with only slight modifications can be used for other acid generating minerals. Useful scaling relations can also be derived in the presence of water transport and air convection.

Unlike the existing waste rock models, the kinetic equations used in this study do not contain adjustable parameters. All the parameters entering eqs. (4.1) and (4.2) can be measured in a series of independent experiments. At the same time we do not pretend that the equations (4.1), (4.2) and their steady state solutions provide a complete description of waste rock piles. The kinetic model used for our scaling analysis assumes that:

• the convective transport of oxygen is slow;

• the water flow is slow;

• the pile is large and seasonal temperature changes affect a small external portion of the pile;

• the waste material has a uniform pyrite concentration throughout the pile; and

• water vapour concentration is uniform inside the pile so that the diffusion coefficient D remains constant.

The first two assumptions should be adopted as a practical rule for waste rock management and therefore should be considered as realistic features of waste rock piles. (Different principles have to be adopted when the preemptive leaching is used - see comment on p.24). The first four assumptions are not satisfied in the case of most field-study reports available through MEND. In particular, piles 17, 18a and 18b in the Heath Steele study were small (only 4-5 m high), had high porosity and were not placed on impermeable lining (thus allowing high flux of oxygen from the base). In such cases the oxidation rates are limited not by diffusive oxygen transport but by the convective transport of thermal energy and convective oxygen transport. This is an undesirable situation leading to temperatures above 40°C despite a small pile size. In the case of Mine Doyon only the third assumption is satisfied and the estimated thermal energy stored in the pile is equivalent to about 2.5 years of the thermal activity of the waste material. This results in relatively small seasonal temperature variations at distances greater than 5 meters from the pile surface (see Fig. 4 on p. 33 of [LGI]). Because of the large thermal inertia the seasonal changes in Mine Doyon's dump remain small despite high convection rates. 

We concentrate our attention on diffusive transport because it seems that the convective transport should be eliminated in properly managed waste rock piles. (A model which will properly describe convective effects should be analyzed, however, in order to understand ARD for a broad range of conditions). Even when convective transport is eliminated, the nonlinear effects which were disregarded in existing numerical models, are responsible for thermal catastrophes leading to an abrupt increase in the acid generation rates when the critical value δ^* of the physico-chemical scaling parameter is reached. For large values of the scaling indicator δ, the pyrite oxidation rates with diffusive oxygen transport become as high as in the presence of convection. In order to understand this feature one has to realize that rates of oxygen supply while increasing due to convection, are accompanied by increased rates of energy transport to the surroundings. The scaling analysis can be easily performed also when convective mass and energy transport becomes dominant. One of the main differences is the proportionality of acid generation rates to α instead of α^1/2 - this results in a more dramatic increase of the acid generation rates with respect to the pyrite content of waste rock.

A sample of numerical solutions obtained by the finite elements method is presented in Appendix B. The numerical results confirm the analytical results of the scaling analysis.

Our main purpose was to demonstrate that the chemical parameters measured in small- scale and meso-scale laboratory experiments can be combined with large-scale physical parameters (like pile size, for example) and that a physico-chemical model can produce a realistic description of the large-scale thermo-chemical behaviour of waste rock piles. Having demonstrated that the simple model derived from fundamental physical and chemical principles gives reasonable results, we feel confident that after a further refinement it is possible to construct a realistic and useful waste rock model.

5.3 Required Accuracy Of Laboratory And Field Tests

Matthew Otwinowski

Scaling Analysis Of Acid Rock Drainage

5.3 Required Accuracy of Laboratory and Field Tests

Acid generation rates depend critically on the value of the physico-chemical scaling parameter δ given by eq. (5.4). The scaling parameter δ can be calculated for a wide range of physical conditions characterized by the temperature T_b at the pile surface and pile porosity, ε. The chemical properties of waste rock can be characterized by the kinetic coefficient, a, and the effective active surface area, S. The value of S depends on several factors including the total content of pyrite, pyrrhotite and other sulphur compounds, the effective value of rock porosity, rock morphology, etc. In particular, one may measure rock porosity by means of the fractal dimension of pores. We feel, however, that the complexity of elementary factors responsible for the ARD potential is so high, that one should rely on the results of simple laboratory tests. Such tests should be performed for different values of temperature and different oxygen partial pressures in order to define the scaling curve αT^q and the exponent, p, for the oxidation process. Representative samples of waste rock should be used for such tests. For the needs of a predictive model it would be sufficient to perform oxidation tests under isothermic conditions at temperature intervals of 15°C in the temperature range between 5°C and 80°C and oxygen partial pressures varying between 0.21 atm and 0.01 atm at 0.05 atm intervals. Such isothermic tests should be performed at water saturation values characteristic for waste rock piles. In this way one can produce in a relatively inexpensive way the input data sufficient for a predictive model. This approach seems to be more practical than the approach based on a more direct measurement of the effective reactive surface area, S, and fractal dimensions of pores. One has to realize that microscopic factors such as the content of different morphological forms of pyrite can vary the value of S by a factor of two. The surface areas exhibited by pyrite of different morphologies vary from 6.5x10^-3 m²/g for euhedral morphology to 1.2 x 10^-2 m²/g for subhedral/framboidal morphologies [WhJ], Variation of surface area as the function of pyrite particle size is another important microscopic factor. 

The value of heat, h, generated during the pyrite oxidation process can be different for different values of acid neutralization potential. We could not find reliable data on the heat generated by calcite dissolution or the neutralization reactions, which would modify the value of h used in this report. We do not recommend, however, to measure these quantities separately. The cumulative heat generated during the weathering process is sufficient for the predictive reaction- transport model. The heat generated can be measured in the same isothermal experiment (proposed above) as the value of cooling necessary to preserve constant temperature of the oxidation process. In this way we can keep the same number of measurable coefficients as in our scaling model in which only the effective heat of pyrite oxidation is analyzed.

We also perform a simple error analysis for the scaling coefficient δ. The relative contributions due to the variation of different measurable parameters indicate the required accuracy of experimental tests. The estimated relative variation of δ, is given by:

The above formula assumes that the exact value of D is known. Our rather lengthy analysis which we do not present here, shows that the error in q has a negligibly small effect on δ/δ^* and for this reason there are no terms proportional to Δq in eq.(5.13). 

If we assume, somewhat arbitrarily, the relative accuracy Δδ/δ should be better than 25%, then we obtain the following estimate for the accuracy of the parameters involved when q=2.1:
 

We have allowed a 10% error for the directly measured simple quantities, and a 15% error for the effective coefficient α which has to be determined from a series of results and usually varies even in waste rock from the same source. A sample of waste rock with the particle size representative for the pile should be used to determine the parameter α=aS. Usual acid/base accounting tests do not provide data necessary for a predictive model which should generate quantitative information about the effluent. 

If the indicated accuracy of the experimental data is achieved, the waste rock pile should be designed so that the condition:

is satisfied.

We hope that the last condition can be used as a practical and simple guideline for designing waste rock piles. Better estimates can be produced by a numerical model which should use nonsymmetric boundary conditions and include convective effects. When the waste rock pile is placed on the impermeable lining, the maximum allowed value of δ will increase provided the impermeable lining does not lower the heat transport rate. The present model has to be validated by using results of available laboratory and field tests before adopting the results of this study as practical guidelines.

5.2 Acid Generation Rates

Matthew Otwinowski

Scaling Analysis Of Acid Rock Drainage

5.2 Acid Generation Rates 

In the absence of bacteria, the acid generation rate is practically proportional to the oxygen consumption rate. We limit our attention to the qualitative features of the temperature dependence of acid generation rates.

Several existing linear models produce incorrect profiles of acid generation rate as the function of distance from pile surface. For the symmetric boundary conditions used in this study, the iron and sulphate production rates are given by the relations:

and 

The molar coefficients M_Fe=0.29 and M_SO42-=0.58 define the rates of iron and sulphate production in absence of bacteria, when quasi-equilibrium concentrations of ferric iron are small [SyT]. In general, the values of p and q have to be determined by laboratory tests discussed in Section 5.3.

Fig.5.8 presents the dependence of pyrite oxidation rate as the function of temperature for the same entry data as in Table 5.1 (Temperature profiles, T(x), are presented in Figs. 5.2 and 5.4; q=2.1 and p=1).

According to our mathematical analysis, the maximum acid generation rates occur at some distance from the pile surface, at the temperature T_Rm(x_Rm)=37.5°C. For L=7.6 m in Scenario 1, the maximum pyrite oxidation rates occur close to the center of the pile at the distance x_Rm=5 m from the pile surface. The maximum value of pyrite oxidation rate, 1.35 mol/(m³s) corresponds to about 510 grams of pyrite per year per cubic meter.

Fig. 5.8 Temperature dependence of pyrite oxidation rates expressed in units [mol/(m³s)]. The maximum pyrite oxidation rate occurs for an intermediate temperature value T_Rm= 37.5°C. The value of T_Rm is determined by the competitive effects of increasing temperature and decreasing oxygen concentration. The oxidation rate goes to zero when the temperature approaches the value T_m^*=54.5°C; T_b=10°C. The entry data listed in Table 5.1 are used.

The total acid generation rate increases 4 times as the result of the thermodynamic catastrophe at δ^*=2.75 in Scenario 2. This is a dramatic effect indicating the importance of waste rock pile design. (The total acid generation rates can be obtained after integrating the profile R(x) over the pile volume).

For nonsymmetric boundary conditions, the oxygen concentration and temperature are related by eq. (4.3a). For the stronglynonsymmetric boundary conditions, the pyrite oxidation rates may show spatial oscillations. Such spatially oscillating patterns have been observed in Heath Steele [NoD], (We have obtained approximate solutions expressed in terms of Bessel function for thenonsymmetric boundary problem - the discussion of this problem goes beyond the scope of the scaling analysis presented in this report).

5.1 Alternative Scenarios Of Acid Rock Drainage

Matthew Otwinowski

Scaling Analysis Of Acid Rock Drainage

5.1 Alternative Scenarios Of Acid Rock Drainage 

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.

Scenario 1: Small value of pile porosity

We use the following set of entry data:

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)): 

Fig. 5.1 Dependence of the maximum temperature (in °C) in the waste rock pile as the function of δ (upper horizontal scale) and pile size L (lower horizontal scale; L in meters). (The pile height is 2L)

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).

Fig. 5.2 Temperature profile T(x) as the function of the distance x from the pile surface for the point P₁ in Fig. 5.1. Pile height 2L=12 m. Parameter values are given in Table 5.1.

Fig. 5.3 Temperature profile T(x) as the function of the distance x from the pile surface for the point P₁ in Fig. 5.1. Pile height 2L=12 m. Parameter values are given in Table 5.1.

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: 

For the pile of height 2L=13m, the thermal catastrophe occurs when the effective active surface area reaches the value S^*=0.5 m². (From the scaling relation (5.10), S^* is proportional to 1/L²; for the pile height of 15.2 m the critical value of S^* becomes equal to 0.37)

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.

Fig. 5.5 Temperature profiles as the function of the distance x from the pile surface for the two points P₁ and P₂ on the lower and upper thermodynamic branches in Fig. 5.4. Parameter values are given in Table 5.2.

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. 

In general, for any set of entry data it is sufficient to:

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. 

Scenario 3: Large value of pile porosity and moderate-permeability cover

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.

Fig. 5.6 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 (lower scale).

As in the previous examples, the temperature scale in the plot in Fig. 5.6 is determined 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=0.28, 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.

Scenario 4: Large value ofpile porosity and poor cover

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.

Fig. 5.7 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 (lower scale). 

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.