AN AIR SPARGING CFD MODEL FOR HEAP BIOLEACHING OF COPPER-SULPHIDE

Transcription

1 Third Interntionl Conference on CFD in the Minerls nd Process Industries CSIRO, Melbourne, Austrli December 2003 AN AIR SPARGING CFD MODEL FOR HEAP BIOLEACHING OF COPPER-SULPHIDE Mrtin J. LEAHY 1, 2, M. Philip SCHWARZ 1 nd Mlcolm R. DAVIDSON 2 1 CSIRO Division of Minerls, Clyton, Victori 3169, AUSTRALIA 2 Deprtment of Chemicl nd Biomoleculr Engineering, The University of Melbourne, Victori, 3010, AUSTRALIA ABSTRACT A computtionl fluid dynmics (CFD) solver CFX is used to implement stedy stte model of hep bioleching of copper-sulphide, which includes ir sprging (forced ertion) bsed on previous model (Css et l., 1998) with nturl convection. A prmeter nlysis is performed which shows tht the fctors importnt to copper leching re liquid nd ir flow rtes, nd permebility. The bility to control which prts of the bed received the highest extrction s function of the liquid nd ir flow rtes ws estblished. Sprging is found to increse the oxygen concentrtion throughout the hep compred to the circumstnce with no sprging (nturl convection), nd consequently improves the copper extrction significntly. The results show tht sprging does not provide ny better copper extrction for very high hep permebilities. NOMENCLATURE hep top hlf-width, (m) b hep bottom hlf-width, (m) B body force vector, (N/m 3 ) c upper x vlue of computtionl domin, (m) C ir concentrtion, (kg/m 3 ) C L oxygen concentrtion in liquid, (kg/m 3 ) C P,L specific het of liquid, (J/kg/K) d upper y vlue of computtionl domin, (m) D ir diffusion coefficient, (m 2 /s) FPY kg pyrite to kg chlcocite leched g grvittionl constnt vector, (m/s 2 ) G o copper grde, (kg Cu/kg bed) H bed height, (m) H ch chlcocite enthlpy, (J/kg) He Henry s lw functionl h L liquid flow enthlpy, (J/m/s) H Py pyrite enthlpy, (J/kg) H R rection enthlpy, (J/kg) K bed permebility, (m 2 ) m lower x vlue of sprging inlet, (m) M ch moleculr weight chlcocite, (kg/mol) M ox moleculr weight oxygen, (kg/mol) M py moleculr weight pyrite, (kg/mol) n upper x vlue of sprging inlet, (m) p ir pressure, (N/m 2 ) q L liquid flow rte, (m/s) q l,0 fixed liquid flow rte, (m/s) R resistivity due to porous medi, (kg/m 3 /s) t time, (s) T ir temperture, (K) t * ( 1/(dα/dt)) chrcteristic rection time, (dys) buoyncy reference temperture, (K) T 0 u ir velocity, (m/s) V in verticl sprged ir speed, (m/s) V M mximum bcteril respirtion rte, (kg O 2 / bcteri/s) W o oxygen mss frction, (kg O 2 / kg ir) X bcteri concentrtion, (bcteri/m 3 ) (x, y) two-dimensionl sptil coordinte system, (m) Greek α copper extrcted, (Cu/Cu(initil)) β stoichiometric coefficient (kg Cu/ kg O 2 ) γ ir therml expnsion coefficient (1/K) ε bed porosity, (void m 3 / totl m 3 ) λ ir therml conductivity, (W/m/K) µ dynmic ir viscosity, (kg/m/s) µ L dynmic liquid viscosity, (kg/m/s) ρ ir density, (kg/ m 3 ) ρ,0 ir density (=ρ ), (kg/ m 3 ) ρ b bed density, (kg/ m 3 ) ρ L liquid density, (kg/ m 3 ) INTRODUCTION Hep bioleching is n importnt minerl extrction process used in industry for the removl of metls such s copper nd gold from low grde ores (Brtlett, 1998). The process is fvoured becuse of the low mintennce nd lrge volume in processing, only requiring the control of cid ppliction, inocultion of bcteri nd provisions for supplying sufficient oxygen throughout the hep. The leching process occurs s result of infiltrtion of sulphuric cid into the hep, which forms film over the ore prticles. Oxygen in the ir phse rects chemiclly with the ferric ions in the cid nd copper sulphide in the ore, to produce ferrous ions nd copper in solution which flows out of the hep with the liquid. The process occurs in the presence of bcteri, such s Thiobcillus ferrooxidns, which convert ferrous ions bck into ferric ions. In ddition to the infiltrtion of cid, the ore prticles during hep construction re coted with liquid (cid), which is inoculted with bcteri. There re number of fctors to be ccounted for in hep bioleching modelling, including the liquid nd ir flow, bcteril ction (possibly multiple species), nd copper extrction. The chemicl rections of these components (oxygen, liquid, bcteri, copper) re resonbly well known, however rection rte depends on mss trnsfer processes, specificlly oxygen supply in mny cses. In this work we ssume the liquid flow behviour in the hep is uniform, nd look t the effect on copper extrction of ir flow in the hep under ir sprging (forced ertion), for vrious ssumed bcteril concentrtions. Without the Copyright 2003 CSIRO Austrli 581

2 use of sprging, oxygen supply to the hep depends on nturl convection which occurs when tempertures inside the hep re greter thn outside. The onset of such effects hs been studied by Lu nd Zhng (1997). For lrge heps like those used in industry, this does not provide enough oxygen deep within the hep for copper extrction (Brtlett nd Prisbrey, 1998). In ddition, Css et l. (1998) found tht heps with height nd width lrger thn 10 by 20 metres, respectively, were ineffective in terms of copper recovery due to lck of ir flow. Previous models of ir flow in heps hve included nturl convection (Css et l., 1998; Pntelis nd Ritchie, 1991, 1992; Pntelis et l., 2002) with the ir flow through the bed described by Drcy s Lw (Ber, 1972). Of the models mentioned, none hs included sprging of ir; moreover none hs included the full solution of the Nvier-Stokes equtions, which cn become importnt t high ir flow rtes, or very high permebilities. The nlysis here uses the N-S equtions with resistnce force bsed on Drcy s Lw. MODEL DESCRIPTION Assumptions Consider two dimensionl uniform porous medium of trpezoidl shpe (see Figure 1) with given permebility nd porosity. In prctice the prticle size distribution cn be widely vrying, with regions of different permebility nd porosity through the bed. Figure 1 shows the hep in bold, nd the computtionl domin used. include dynmic nd sptilly vrible kinetics (Monod, temperture, ph), nd mobility/ttchment vi liquid flow. Rections The ore being considered in this work is composed of pyrite (iron sulphide) nd chlcocite (copper sulphide). The rection of metl sulphide (MS) in the hep is twostge rection. The solution contins ferric ions (Fe 3+ ) tht rects with copper sulphide to produce ferrous ions (Fe 2+ ) s MS+ 2Fe M + 2Fe + S. Ferrous ions re re-oxidized to ferric ions in the presence of bcteri (only T. ferrooxidns is considered) s 2+ + bcteri 3+ 2Fe + 0.5O2+ 2H 2Fe + H2O where M is the metl copper or iron, present in chlcocite nd pyrite respectively. We follow the pproch of Css et l. (1998) who ssume the bcteri nd oxygen in the liquid re limiting the rection rte, whilst the ferrous ions re in excess so tht α (copper extrcted) cn be described by the Michelis-Menton form dα β C = XV L M, (1) dt ρbgo KM + CL where X, V M,K M nd C L, re the bcteri concentrtion, mximum specific respirtion rte of bcteri, Michelis- Menton hlf growth rte constnt nd liquid concentrtion respectively. The overll rections for chlcocite Cu 2 S nd pyrite FeS 2 hve the stoichiometries FeS O2+ H2O FeSO4+ H2SO4 Cu2 S+ 2.5O2+ H2SO4 2CuSO4+ H2O. Figure 1: schemtic hep Acid solution is ssumed to feed uniformly downwrd under the influence of grvity, though in prctice the solution flow cn tke tortuous flow pths, with both chnnelling nd liquid stgnnt regions, due to heterogeneities nd properties of the unsturted liquid flow (Brtlett, 1998). The liquid is ssumed to cot the grins of the hep such tht its ffect on the pore spce vilble to the gs cn be ignored. Due to the lrge grin size, cpillry pressure is neglected in this work nd the gs flow is determined from single-phse, rther thn two-phse clcultion. It cn be shown tht it is good pproximtion to ssume tht the temperture of the ir is in locl thermodynmic equilibrium with the liquid nd ore bed. Consequently, it is sufficient to solve for the het trnsport of ir, tking into ccount the het dvected by the liquid. The bcteril concentrtion would vry in time nd spce in prctice, but is kept constnt in this work for simplicity, however this spect will be expnded in future work to From these rections, we cn derive the stoichiometric fctor β, the mss rtio of copper produced to oxygen consumed s MChM Py β =. (2) 2.5MOxM Py + 3.5(FPY)MOxMCh In (1), ρ b nd G o re the bed density nd copper grde respectively, nd in (2) FPY is the rtio of mss of pyrite to the mss of chlcocite leched, which is kept constnt in the work, lthough in prctice it will vry in time nd cuse the copper extrction described in (1) to slow in time. The mximum respirtion rte for T. ferrooxidns bcteri is given by Css et l. (1998), dependent on temperture s x10 Texp( -7000/T) VM =. (3) 1 + exp( /t) This reltion exhibits mximum vlue for temperture round 37 o C, nd conforms to the temperture rnge 4-46 o C in the literture in which bcteri cn survive (Ahonen nd Tuovinen, 1989). The oxygen concentrtion C is written in terms of oxygen mss frction W o s C = ρw o (4) Henry s lw gives the oxygen concentrtion in the liquid C L C L = He C (5) where He=-7x10 8 (T ) x10 5 (T ) 2-1.2x10 3 (T )+5.23x10-2 (6) 582

3 by fitting oxygen solubilities in wter s function of temperture (Perry nd Green, 1984). Air Flow The stedy stte flow occurring in the porous hep is described by the eqution of continuity nd the stedy stte N-S equtions given by u =0 (7) 2 ( uu) = + µ u B ρ p + (8) where p, u, B, µ nd ρ re the pressure, velocity, body forces, ir viscosity nd density. The body force B ccounts for the Drcy resistnce to flow in the porous medium nd grvity by tking the form B = R u + ρ g (9) where g is the grvittionl vector nd R is the porous resistivity (Al-Khlift nd Arstoopour, 1997) given by εµ R =. (10) K Here ε nd K re the porosity nd permebility of the hep respectively. For low flow rtes (s in this work), it cn be shown equtions (8), (9) nd (10) represents Drcy flow. Although the ir is not considered compressible, the ir density ρ vries due to differences in the ir temperture, nd induces buoyncy effect. The Boussinesq pproximtion is used to describe the ir density for the body force terms for ρ in (9) s ρ = ρ 1 γ ( Τ Τ )), (11),0 ( 0 where γ is the therml expnsion coefficient of ir, ρ,0 is constnt nd equl to the ir density in tmospheric conditions, nd T 0 is the buoyncy reference temperture. The density on the left hnd side of (8) is set equl to the tmospheric vlue, so tht the momentum eqution is given by 2 ( uu) = p + u ρ µ,0 Liquid Flow εµ u ρ,0( 1 γ ( Τ Τ0)) g. (12) K The liquid is ssumed to flow verticlly nd to hve negligible effect on the flow of ir, except to cool it. A more sophisticted liquid flow model could be included, involving liquid hold-up (Bouffrd nd Dixon, 2001), or unsturted liquid flow (Pntelis et l., 2002). However, for the purposes of the ir phse modelling in this work, we use constnt liquid velocity within the hep. Energy Blnce The temperture is described by the stedy-stte het eqution with source term given by h 2 L ρbg o d α HR = λ T - u.t (13) y β dt where λ is the therml conductivity of ir. The first term on the left hnd side of (13) represents the het tken up by the liquid, whilst the second term is the het produced by the rections. In (13) h L =-q L C P,L ρ L (T-T ref ) where C P,L is the specific het of liquid, nd H R is the het of rection given by Css et l. (1998) s H R = H py + FPY H ch, where H py nd H ch re the het of rections for pyrite nd chlcocite respectively. Oxygen Blnce Trnsport of oxygen in the hep is described by dvection nd moleculr diffusion (we hve not included n incresed effective diffusion due to porous medium), modified by rections, nd is described by ρbg β dα dt 2 = ρ D Wo - ρ.(wou) (14) where D is the diffusion coefficient of ir, nd the term on the left hnd side of (14) is the sink term due to rections. Boundry Conditions The conditions imposed t the boundries of the computtionl domin (see Figure 1) re s follows; u (x,0) = (0,Vin ), T(x,0) = 298K m x n (15) T(x,0) = 0 y 0 x m nd n x c (16) T(0,y) = 0 (17) x p(x,d)=1tm for 0 x c, p(c,y)=1tm for 0 y d. (18) Severl inlet velocities V in t y=0, x (m, n) re used in the results section. We lso present the results without sprging, however unlike Css et l. (1998), we do not force the flow to be perpendiculr to the hep surfce s it enters or leves. Symmetry of the solution pplies round the line x=0 llows the computtion domin in Figure 1 to be restricted to the region 0 x c, 0 y d. The computtionl domin used ws lrger thn the ctul hep (see Figure 1), with pressure boundries t the outer edges of the computtionl domin. The resultnt flow ws such tht ir leves the hep nd then flows through the rest of the computtionl domin, so tht boundry conditions for the ctul hep edges re not required. Numericl Method The stedy stte equtions (1)-(7), (12)-(18) were solved using CFX4, nd the simultions required n orthogonl grid (250 by 250 cells) to void instbility. The grid used ws rectngulr nd stircse of computtionl cells represents the slnted top of the hep. The results for grid with 125 by 125 cells ws found to be insignificntly different to tht of 250 by 250 cells. The liquid enthlpy flow in (13) ws pproximted using upwind differencing to ensure stbility, but ll sptil derivtives were pproximted using centrl differencing. The results were ssumed converged when the normlized residuls were ll less thn 1x10-3. Liquid ws pplied evenly cross entire exposed boundry of the hep with given flow velocity q L. 583

4 RESULTS Prmeter Anlysis The concentrtion of bcteri in the hep, in prticulr T. ferrooxidns, is lrgely unknown, nd it is pproprite to look t the effect of such levels on the temperture, oxygen nd hence copper extrction. Hence, the results re split into two prts for low nd high bcteril concentrtions, with vlues of X=5x10 13 bcteri/m 3 nd X=5x10 12 bcteri/m 3 respectively. We hve not investigted the effect of vrition of the FPY fctor, unlike Css et l (1998), due to spce constrints. In the next two sections the following prmeters were used (Tble 1). Further prmeters nd vribles re specified where pproprite. Bed porosity ε =0.3(m 3 /m 3 ) Bed height H=10 (m) Top nd bottom length =8 (m), b=20 (m) Upper computtionl domin x, y c=10 (m), d=25 (m) Air density ρ =1.208 (kg/m 3 ) Air viscosity µ =1.812x10-5 ( kg/m/s) Air diffusion coefficient D =1.44x10-5 (m 2 /s) Air therml conductivity λ =2.88x10-2 (W/m/K) Air therml expnsion coefficient γ =3.43x10-3 (1/K) Liquid specific het C P,L = 4000 (J/kg/K) Reltive liquid flow rte q L,0 =1.4x10-6 m/s Monod hlf growth rte constnt K m =1x10-3 (kg/m 3 ) Het of rection for pyrite H py =-1.26 x10 7 (J/kg) Het of rection for chlcocite H ch =-6.0x10 6 (J/kg) Stoichiometric fctor β = Copper grde G o =0.5 % wt FPY FPY = 5 Atmospheric oxygen mss frction W o(tmospheric)=22%wt Tble 1: prmeters for ll results. Low Bcteril Levels The results found for reltively low bcteril levels re described in this section, with the constnt vlue X=5x10 12 bcteri/m 3. In Figures 2-5 we see (in order) the 2-D plots (with stremlines) of vectors coloured by speed, nd flooded contours of % oxygen mss frction normlized by tmospheric oxygen mss frction (100xW o /0.22), temperture nd % copper extrcted (α x100, fter 1 yer) for sprging velocity V in =1x10-4 m/s. In Figure 5, the copper extrcted fter 1 yer is clculted by extrpolting linerly from the initil rection rte. This corresponds to typicl ir flow rte used in prctice (Brtlett, 1998). The figures show tht some nturl convection occurs, whilst sprging domintes the ir flow. The oxygen mss frction stys reltively high through out the hep, showing tht the process of sprging voids low oxygen levels tht inhibit copper extrction. Figure 3: Stremlines with contours of normlised mss frction oxygen % (100xW o /0.22), V in =1.0x10-4, q l =0.5q L,0, K=5x10-10 Figure 4: Stremlines with contours of T, V in =1.0x10-4, q l =0.5q L,0, K=5x10-10 Figure 5: Stremlines with contours of 100xα (1 yer), V in =1.0x10-4 q L =0.5q L,0, K=5x10-10 In Figure 3, the minimum oxygen level occurs ner the bottom right hnd corner of the hep, which is cused by the low flow rtes (Figure 2) in tht re. This low oxygented re occurs in spite of the fct there is some nturl convection in tht re. In Figure 4 nd 5 the temperture nd copper extrction (fter 1 yer) re shown reveling the strong dependence of the copper extrction upon the temperture. In Figures 6 nd 7 comprison is mde between cses with nd without sprging, for fixed liquid flow rte, bcteril concentrtion nd permebility. Figure 7 shows tht oxygen becomes depleted without the use of sprging, limiting copper extrction with chrcteristic extrction time of 13,258 dys, compred to 8,013 dys when sprging t V in =1x10-4 (Figure 6). Figure 6: Stremlines with contours of normlised oxygen mss frction oxygen % (100xW o /0.22),V in =1x10-4, q l =1.5q L,0, K=5x10-10 Figure 2: Stremlines with vector plot nd coloured by speed, V in =1.0x10-4, q l =0.5q L,0 K=5x10-10 Figure 7: Stremlines with contours of normlised oxygen mss frction oxygen % (100xW o /0.22), no sprging, q l =1.5q L,0, K=5x

5 In Tble 2 comprison of the chrcteristic copper extrction time t * 1/(dα/dt) is mde for different flow rtes, showing tht t * is highly dependent on the liquid flow rte, with minimum occurring round 0.5q L,0 where q L,0 =1.4x10-6. This dependence occurs due to the cooling effect the liquid hs on the ir. Thus liquid flow rte my be used to control the temperture distribution to ensure tht it remins s close s possible to the optiml vlue of round 311K. q L (x q L,0) t * (dys) Tble 2: t * versus liquid flow rte q L (x q L,0 ), K=5x10-10, V in =1x10-4 In Tble 3 the effect of the permebility is shown on the chrcteristic copper extrction time t *. It is only dependent on the permebility to smll extent for n ir sprging velocity V in =1.0x10-4 nd V in =1.0x10-5, s explined by the t * rtio in Tble 3. The reltive difference is negligible in terms of long term extrction. t * (dys) K V in=1.0x10-4 V in=1.0x10-5 t * rtio 5x x x Tble 3: t * versus K for q L =1.5q L,0, nd ssocited t * rtio of V in =1.0x10-5 to V in =1.0x10-4 High Bcteril Levels In this section we look t the results in the cse of higher constnt bcteril concentrtion, X=5x10 13 bcteri/m 3, compred to the lst section where X=5x One of the most significnt chnges ws the decrese in chrcteristic copper extrction time t * (compre Tble 2 nd Tble 4). Agin, the liquid flow rte hs substntil effect on the copper extrction (see Tble 4), due to the bility of the liquid to cool the hep, nd push the temperture closer to the optiml temperture (~311K). Tble 4 shows the best liquid flow rte occurs round 5q L,0, nd such estimtes could be useful in prctice. The computed results for this optiml liquid flow rte re shown in Figures Figure 8 shows some entrinment of ir occurs into the hep, whilst in Figure 9 we see the oxygen mss frction becomes depleted well before ir leves the hep, suggesting tht higher inlet velocity should be used (see Figures 12-14). Figure 10 shows tht the temperture is distributed round the optiml temperture of 311K for lrge re, leding to extrction in some res s high s 92% fter one yer. q L (x q L,0) t * (dys) Tble 4: t * versus liquid flow rte q L (x q L,0 ), V in =1x10-4 K=5x10-10 Figure 8: Stremlines with vector plot nd coloured by speed, V in =1.0x10-4, q l =5q L,0 K=5x10-10 Figure 9: Stremlines with contours of normlised oxygen mss frction % (100xW o /0.22), V in =1.0x10-4, q l =5q L,0, K=5x10-10 Figure 10: Stremlines with contours of T, V in =1.0x10-4, q l =5q L,0,K=5x10-10 This estimte does not represent n extrction vlue obtinble in prctice, since we hve not ccounted for vrition in FPY s discussed erlier. Further work will be required to ccount for this. Overll the predicted chrcteristic extrction time t * is 1119 dys (see Tble 4). Figure 11: Stremlines with contours of 100xα (1 yer), V in =1.0x10-4, q l =5q L,0,K=5x10-10 Figure 14 shows tht for higher inlet velocity of V in =1.0x10-3, completely different copper extrction profile is obtined, with most of the extrction occurring in the middle of the bed. This suggests the possibility of controlling copper extrction through different regions of the bed, by using certin combintions of liquid nd ir flow rtes. This higher ir velocity pushes the chrcteristic copper extrction time down to 676 dys (see Tble 5), since more oxygen is present in the hep (Figure 12) nd the temperture distribution is closer to the optiml vlue of 311K (Figure 13). An even higher sprging velocity of V in =1x10-2 does not provide ny better copper extrction (see Tble 5), becuse the temperture becomes higher thn optiml over lrge region. For the permebility K=5x10-10, the chrcteristic copper extrction time rtio (see Tble 6) depends in the inlet velocity, but the dependence is less for K=5x10-9.This 585

6 comprison shows tht if the hep permebility cn be incresed to K=5x10-9, lower sprging rte cn be used to obtin similr copper extrction. In fct, chrcteristic copper extrction time without sprging nd K=5x10-9 hs been found to be 2,212 dys, indicting tht for the size of hep studied here, there is essentilly no benefit to be gined from sprging with such high permebilities. This nlysis shows the benefits of creting high permebility, to reduce costs of sprging. Figure 12: Stremlines with contours of normlised mss frction oxygen %, (100xW o/0.22), V in=1.0x10-3, q l=5q L,0, K=5x10-10 Figure 13: Stremlines with contours of T, V in =1.0x10-3, q l =5q L,0, K=5x10-10 Figure 14: Stremlines with contours of copper extrcted 100xα (fter 1 yer), V in =1.0x10-3, q l =5q L,0, K=5x10-10 V in t * (dys) 1x x x Tble 5: t * versus V in for K for q L =5q L,0 t * (dys) K V in =1x10-4 V in =1x10-5 t * rtio 5x x Tble 6: t * for q L =1q L,0, nd ssocited rtio of t * for V in =1.0x10-5 to V in =1.0x10-4 CONCLUSION The model simultion presented in this work hs shown tht ir sprging hs criticl effect on the hep leching process, in ddition to the effect of the liquid flow rte. This is the first time this effect hs been simulted in numericl model. Copper extrction hs been shown to increse gretly with the use of sprging, in comprison to relying on nturl convection. However, where the oxygen ws kept t high levels, one must be cutious bout the copper extrction vlues, since other re fctors re likely to become rte limiting, such s liquid flow. The bility to determine optiml ir nd liquid flow rtes ws illustrted, nd this bility could be of use in prctice. The bility to control copper extrction in prticulr prts of the bed ws shown, which lso could be of interest to opertors. It hs been shown tht for permebilities s lrge s K=10-9, one does not chieve ny better copper extrction by sprging. Other interesting spects for further work include investigting cooler mbient tempertures nd its ffect on bed temperture nd ir entrinment. It my lso be possible to turn off sprging (t lest temporrily) once hep tempertures become high enough to llow nturl convection to tke over, thus sving the cost of sprging. The model presented cn be improved by including these spects nd by relxing the simplifictions in the model of liquid flow, bcteril growth nd rtio of pyrite to chlcocite leched (FPY). The effect of hep dimensions could lso be investigted with regrd to mny of the spects mentioned in this work. ACKNOWLEDGMENTS The help of Jkub Bujlski nd Peter Witt on severl issues were gretly pprecited. Funding from APA nd CSIRO top-up scholrship ws gretly pprecited. REFERENCES AHONEN, L. nd TUOVINEN, O.H., (1989), Microbiologicl oxidtion of ferrous iron t low tempertures, Appl. Environ. Microbiol., 55(2), AL-KHLAIFAT, A. nd ARASTOOPOUR, H. (1997), Simultion of two-phse flow through low-permebility porous medi, AEA technology interntionl user conference 97. BARTLETT, R., (1998), Solution Mining: Leching nd fluid recovery of mterils, 2 nd edition, The Netherlnds: Gordon nd Brech Science Publishers. BARTLETT, R. nd PRISBREY, K.A. (1996), Convection nd diffusion limittion ertion during biooxidtion of shllow ore heps, Int. J. Miner. Process., 47, BEAR, J., (1972), Dynmics of fluids in porous medi, Americn Elsevier Science Publishing Co. Inc. BOUFFARD, S.C. nd DIXON, D. (2001), Investigtive study into the hydrodynmics hep leching processes, Met Mt. Trns., 32B, CASAS, J.M., MARTINEZ, J., MORENO, L., nd VARGOS, T., (1998), Bioleching model of coppersulphide ore bed in hep nd dump configurtions, Met. Mt. Trns., 29B, LU, N. nd ZHANG, Y., (1997), Onset of thermlly induced ir convection in mine wstes, Int. J. Het Mss Trnsfer, 40 (11), PANTELIS, G. nd RITCHIE, A.I.M., (1991), Rtelimiting fctors in dump leching of pyritic ores, Appl. Mth. Modelling, 16, PANTELIS, G. nd RITCHIE, A.I.M., (1992), Mcroscopic trnsport mechnisms s rte-limiting fctor in dump leching of pyritic ores, Appl. Mth. Modelling, 15, PANTELIS, G., RITCHIE, A.I.M. nd STEPANYANTS, Y.A. (2002), A conceptul model for the description of oxidtion nd trnsport process in sulphidic wste rock dumps, Appl. Mth. Modelling, 26 (2), PERRY, J.H. nd GREEN, D.W., (1984), Perry s Chemicl Engineers Hndbook, 6 th edition, McGrw- Hill Book Compny, Jpn. 586