3 (2006) 21–28www.elsevier.com/locate/hydromet

Hydrometallurgy 8

A review of rate equations proposed for microbial ferrous-ironoxidation with a view to application to heap bioleaching

T.V. Ojumu, J. Petersen ⁎, G.E. Searby, G.S. Hansford

Department of Chemical Engineering, University of Cape Town, Rondebosch 7701, South Africa

Available online 15 May 2006

Abstract

In view of the fact that the microbial oxidation of ferrous iron to the ferric form is an essential sub-process in the bioleaching ofsulphide minerals, the development of a comprehensive rate equation for this sub-process is critical. Such a rate equation isnecessary for the design and modelling of both tank and heap bioleach systems.

Most of the rate equations presented in the literature define the specific microbial growth rate using a Monod-type form forferrous substrate limitation, with further terms added to account for ferric product inhibition, ferrous substrate limitation andinhibition. A few of the published rate equations describe the specific substrate utilization rate in terms of a modified Michaelis–Menten equation and include the maximum yield constant and cell maintenance via the Pirt equation. Other rate equations arebased on chemiosmotic theory or an analogy with an electrochemical cell.

In the present paper a selection of rate equations are compared against each other by calibrating them against the same set ofdata and comparing the fits. It was found that none fits the data particularly well and that some of the underlying assumptions needto be questioned. In particular, it appears that ferric inhibition is perhaps not as significant a factor than previously assumed and thatrate control by the availability of ferrous is more significant.

Some rate equations include terms to account for the effects of temperature, pH, biomass concentration, ionic strength as well asinhibition due to arsenic. In general these effects have been studied in isolation and in ranges not too far off the optimum. Few rate equationscombine more than 2 effects and there is no clarity on how a comprehensive model to account for all effects should be constructed.

Rate equations have been applied to tank bioleach systems, which usually operate under controlled conditions near theoptimum. Heap bioleach systems, on the other hand, often operate far from optimum conditions with respect to temperature, pH,solution conditions, etc., at the same time. The kinetics of such sub-optimal systems are still poorly understood. Future studiesshould be directed towards the development of a comprehensive rate equation useful for describing the kinetics of heap bioleachingover a wide range of conditions.© 2006 Elsevier B.V. All rights reserved.

Keywords: Bioleaching; Rate equations; Modelling

⁎ Corresponding author. Tel.: +27 21 650 5766; fax: +27 21 6505501.

E-mail address: jp@chemeng.uct.ac.za (J. Petersen).

0304-386X/$ - see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.hydromet.2006.03.033

1. Introduction

It is generally accepted that the bioleaching ofsulphide minerals involves three major sub-processes,viz., the acid ferric leaching of the sulphide mineral,microbial oxidation of the sulphur moiety and themicrobial oxidation of ferrous-iron to the ferric form.

22 T.V. Ojumu et al. / Hydrometallurgy 83 (2006) 21–28

Although the stoichiometry of the overall reaction willvary according to the particular metal sulphide beingbioleached, a typical reaction for a metal sulphide (MS)as follows:

2MS þ 4Fe3þ→2M2þ þ 4Fe2þ þ 2S0 ð1Þ

The ferrous-iron is then re-oxidized to the ferric formby microbial action:

4Fe2þ þ 4Hþ þ O2→4Fe3þ þ 2H2O ð2Þ

It is the kinetics of this reaction and the subsequentbiomass synthesis and maintenance that is the mainfocus of this paper.

There are many publications on the kinetics ofmicrobial ferrous-iron oxidation [1–15]. These include:

Table 1Selected published kinetic models for ferrous iron oxidation with At. ferroox

Author Model

Lacey and Lawson [4]l ¼ lmax½Fe2þ�

YSXKm þ ½Fe2þ�

MacDonald and Clark [5] l ¼ lmax½Fe2þ�Km þ ½Fe2þ�

Jones and Kelly [6] l ¼ lmax½Fe2þ�Fe2þ� �þ Km 1þ ½Fe3þ�

Ki

� �

Braddock et al. [7]l ¼ lmaxð½Fe2þ�−½Fe2þ�tÞ

Km þ ð½Fe2þ�−½Fe2þ�tÞ

Liu et al. [8] l ¼ lmax½Fe2þ�½Fe2þ� þ Ksð1þ Ki½Fe3þ�Þ

Lizama and Suzuki [9] −rO2 ¼k3V½X�½Fe2þ�

Fe2þ� �þ Km 1þ ½X�

KiVþ ½Fe3þ�

Kifþ ½X

a

h

Nikolov and Karamanev [10] l ¼ lmax½Fe2þ�Fe2þ� �þ Ks þ Fe3þ

� �KsKiþ ð½Fe2þ�Þ2

Ksi

Huberts [11] −rFe2þ ¼ a1pO2

kBþpO2

� �½Fe2þ�

Fe2þ½ �þKFe2þ 1þ½Fe3þ�K V

� ��

Crundwell [12] −rFe2þ ¼ k ½Fe2þ�=½Hþ�KFe2þþ½Fe2þ�=½Hþ�þKi ½Fe3þ�

� �0:5

kB

�

Hansford [3] qFe2þ ¼ qmaxFe2þ

1þ KFe2þ½Fe3þ�½Fe2þ�

Nemati and Webb [13]d½Fe2þ�

dt¼ K0e−

EaRT X½ � Fe2þ� �

1þ ½Fe3þ�Ki

� �Km þ Fe2þ

� �� �Boon et al. [14] qO2 ¼

qmaxO2

1þ Ks

½Fe2þ�−½Fe2þ�t þKsKid ½Fe3þ�½Fe2þ�−½Fe2þ�t

Meruane et al. [15] qFe2þ ¼ K⁎1 exp

nF2RT Em−E0

h

� �� �1−exp nF

RT

�1þ K⁎

2

½Fe2þ� þ K⁎3 exp

nFRT Eh−

��

initial rate studies investigations in batch and continuousculture and investigations using iso-potential devices.The data has been fitted with modified Monod orMichaelis equations or to models derived from chemi-osmotic theory or electrochemical analogies. Moststudies have used Acidithiobacillus ferrooxidans butsome have studied Leptospirillum ferrooxidans orLeptospirillum ferriphilum, and recently the results of athermophilic culture has been presented [16]. There islittle information on the effects of dissolved oxygen,carbon dioxide, metal cations and anions. The effects oftemperature and pH have largely been limited toconditions near the optimum or to those used in tankbioleaching operations.

It is the objective of this paper to review publishedstudies and to attempt to identify a rate equation that is

idans

Conditions

Batch STR, T=25–30 °C, pH=2–2.3, FeT=6 g L−1

Continuous, T=28 °C, pH=2. 2

Continuous, T=30 °C, pH=1.6, FeT=5–400 mM

Continuous, T=22 °C, FeT=9–22 mM, Isolate AK1

Continuous, T=35 °C, pH=1.8, FeT=0.52–3.29 g L−1

�½Fe3þ�KiVKif

i Initial Rate, T=29 °C, pH=1.8–2.0, FeT=0.25–26 mM

Continuous, T=29 °C, pH=1.8–2.0, FeT=2–70.8g L−1

Continuous, T=30 °C, pH=2.0, Leptospirillumferrooxidans

pO2þpO2

�0:5Theoretical, fitted to data from Huberts [11]

Only fitted to Leptospirillum data

Initial Rate, T=30 °C, pH=2.0, FeT=0.45–31.5 kg m−3

Continuous, T=30 °C, pH=1.8–1.9, FeT=0.05–0.36 M

Em−Eð Þ��E0h

�� Electrochemical cell, T=30 °C, pH=1.8, FeT=0.05–1g L−1

23T.V. Ojumu et al. / Hydrometallurgy 83 (2006) 21–28

both consolidated and comprehensive, and whichincludes substrate utilization, biomass synthesis andbiomass maintenance.

2. Development of fundamental rate equations

The rate equations for microbial ferrous-iron oxida-tion have been reviewed previously [1–3]. Some of thepublished rate equations are shown in Table 1. Thisshows that they have developed from the simple equationfor specific growth rate based on substrate utilisation viaa Michaelis–Menten mechanism, as suggested by Laceyand Lawson [4] and MacDonald and Clark [5], whichcorresponds to a basic Monod type model:

l ¼ rXcX

¼ lmax

1þ KFe2þ

½Fe2þ�ð3Þ

This has been extended to include the effects of ferricinhibition on microbial growth, as proposed by Jonesand Kelly [6]:

l ¼ lmax

1þ KFe2þ

½Fe2þ� þKFe2þ

KFe3þ

½Fe3þ�½Fe2þ�

ð4Þ

Braddock et al. [7] has further extended this approachthrough the inclusion of threshold values to account forthe fact that no growth will occur at low ferrousconcentrations:

l ¼ lmax

1þ KFe2þ

ð½Fe2þ�−½Fe2þ�tÞð5Þ

Above equations are formulated in terms of specificgrowth rate. However, following the work of Boon andco-workers [2,14], formulation in terms of specificferrous utilisation was found to be more useful in termsof experimental measurement as well as engineeringapplication. This approach acknowledges the correlationbetween ferrous oxidation rate, microbial growth andmaintenance via the Pirt equation [17]:

rFe2þ ¼ rXYmaxFeX

þ mFe2þcX ð6Þ

A simplified version of ferric inhibited growth model, asgiven in Eq. (4), expressed in terms of specific ferrousutilisation rate was presented by Hansford [3]:

qFe2þ ¼ −rFe2þcX

¼ qmaxFe2þ

1þ K½Fe3þ�½Fe2þ�

ð7Þ

Boon and co-workers [14] combined the ferricinhibition model and Braddock's threshold approach,expressed in terms of ferrous utilisation rate:

qFe2þ ¼ qmaxFe2þ

1þ KFe2þ

½Fe2þ�−½Fe2þ�tþ KFe2þ

KFe3þ

½Fe3þ�½Fe2þ�−½Fe2þ�t

ð8ÞCrundwell [12] formulated a model based on the

fundamental chemiosmotic theory of the electron/protontransport mechanism of At. ferrooxidans after Ingledew.In recognizing that this mechanism serves as the energygenerator for all the cell's metabolic activity, the ferrousoxidation process is modelled analogous to an electro-chemical fuel cell:

−rFe2þ ¼ k½Fe2þ�=½Hþ�

KFe þ ½Fe2þ�=½Hþ� þ Ki½Fe3þ�� 0:5

� ½O2�KO þ ½O2�

� 0:5

ð9Þ

Meruane and co-workers [15] derived a modelcombining electrochemical theory with Michaelis–Menten dynamics for the ferrous–ferric oxidationreaction:

qFe2þ ¼qmax−K3

½Fe3þ�½Fe2þ�

1þ K1

½Fe2þ� þ K2½Fe3þ�½Fe2þ�

ð10Þ

Eq. (10) is almost identical to (4) except theadditional term in the numerator, which results fromthe assumption that the electron transfer step in the cellmembrane is fully reversible, i.e. there exists a ferric/ferrous iron ratio at which the rates of electron transferfrom and to ferrous adsorbed onto the membrane areequal and no net oxidation occurs.

3. Comparison of fundamental rate equations

The fundamental rate models presented above haveemerged over some 30 years of research and in each casewere calibrated against a set of experimental datagenerated by the individual authors using widelyvarying experimental techniques, as indicated in Table1. It is worth comparing some of these models if all arecalibrated against the same set of experimental data.This has been done for a set of data points presented byBoon [14] and is presented in Fig. 1.

The models compared were those given in Eqs. (3),(4), (7)–(10). For each model the relevant model

0

2

4

6

8

10

0.1 1 10 100 1000

[Fe3+]/[Fe2+]

Sp

ecif

ic ir

on

uti

lisat

ion

rat

e

(mo

lFe2+

.mo

lC-1

.hr-1

)Data (Boon)

Basic Monod (3)

Ferric Inhibition (4)

Hansford (7)

Boon (8)

Crundwell (9)

Meruane (10)

Fig. 1. Comparison of various rate equations calibrated to the same set of data for Acidithiobacillus ferroxidans by Boon [14]. The numbers in roundbrackets refer to the relevant equations in the text.

24 T.V. Ojumu et al. / Hydrometallurgy 83 (2006) 21–28

parameters have been obtained through the Excel Solverroutine, minimising the sum of square errors betweenmeasured and predicted values. The μ-based modelsbecome directly comparable to the q- or rate basemodels by assuming the maintenance coefficient in thePirt Equation (Eq. (6)) is small.

The results indicate that in the case of At.ferrooxidans the differences between the models basedon substrate utilisation and ferric inhibition ((3), (4) and(7)) are effectively negligible, and that none fits the dataparticularly well beyond a Fe3+/Fe2+ ratio of about 20.Crundwell's model (9) gives a significantly worse fit,but follows the same trends as the previous models. It ispostulated here, therefore, that the effects of ferricinhibition on microbial growth and utilisation rates arenegligible and that it is in fact the limitation of ferroussubstrate that effectively drives microbial ferrousoxidation kinetics.

The models by Boon and Meruane (Eqs. (8) and (10)respectively), on the other hand, fit the data reasonablywell over the entire range. Both of these models allowfor a subtractive term—a threshold ferrous concentra-tion in Boon's model and a term accounting for thereversibility of electron transfer in the cell membrane inMeruane's. In this latter case it could be either relativelack of ferrous iron or relative abundance of ferric ironthat reduces the overall rate of ferrous iron utilisation.

Both of these two would predict negative ferrous ironoxidation rates at a Fe3+/Fe2+ ratio above about 150–200, which appears contradictory. In Boon's model thiscould be seen as a cut-off point, beyond which themodel is no longer valid. By contrast, Meruane's model,

which is based on an assumption of reversibility, shouldremain valid with the cell reducing ferric—a scenariothat remains unproven.

In the absence of data measured at Fe3+/Fe2+ ratioshigher than 100, however, any model extrapolation mustbe treated with caution. In reality it is likely thatmicrobial growth does indeed cease below a certainminimum concentration of ferrous iron, while ferrousutilisation for cell maintenance may still continue. Asindicated above, maintenance was considered negligiblefor the purpose of this evaluation. A meaningfuldescription of maintenance ferrous utilisation wouldrequire more data at substantially higher ferrous to ferricratios.

4. Model compensation for factors other than ironspecies

While extensive study has been dedicated to the effectsof ferrous and ferric iron concentrations on microbial ironoxidation kinetics, other factors that may have asignificant effect on this have been much less extensivelystudied. A brief overview is given in the following.

4.1. Effect of pH

Changes in pH have not been found to have asignificant effect on either the growth or the ironoxidation kinetics of iron oxidizing microbes a narrowrange around their optimum pH. Breed and Hansford [18]have modelled the effect of pH by letting constant K inEq. (7) increase linearly with increasing pH within the

25T.V. Ojumu et al. / Hydrometallurgy 83 (2006) 21–28

range studied (1.1–1.7). No significant effect on themaximum specific ferrous iron and oxygen utilizationrates in this range was shown. However, other sourcesreport significant inhibitions below pH 1.5 and above 3.5[1]. Although the resistance of iron oxidizing microbes tolow pH as been attributed to the composition of the cellwall, at very low pH the cell might require more energy tomaintain the proton gradient, since the cell cytoplasmmust be maintained at or near neutral values. Thus cellmaintenance will be at the expense of cell growth.Inhibition at high pH, on the other hand, could beexplained by the fact that protons are required as asubstrate in reaction (2), and also by the fact that theproton gradient is the driving force for the synthesis ofATP, as described by Ingledew [19]. Crundwell's modelincorporates the effect of pH in terms of protonconcentration in his model (Eq. (9)), postulating thatpH affects the speciation of the ferrous ion, and that infact it is the Fe(OH)+ complex that adsorbs to themicrobe, the predominance of which is strongly pHdependent [12]. It was reported recently that microbialferrous-iron oxidation rate was not affected at pH 0.9[20], although the increased tolerance at lower pH mighthave been due to adaptation of the microbial species.

4.2. Effect of temperature

Microorganisms are classified in terms of thetemperature range in which they survive, with optimumtemperatures in the 30–40 °C range for mesophiles,around 50 °C for moderately thermophiles and above65 °C for extreme thermophiles. At temperatures belowthe optimum the microbes become inactive, and theybecome rapidly destroyed at temperatures above it. Themodels proposed by Hinshelwood [21] and Ratkowskyet al. [22] are most commonly used to show thedependency of bacterial growth on temperature.

Hinshelwood model: lmax ¼ K1e−EaRT−K2e

−EbRT

ð11Þ

Ratkowsky model: lmax ¼ bðT−TminÞf1−ecðT−TmaxÞgð12Þ

where K1, K2, b and c are constants; Tmin and Tmax arethe minimum and maximum growth temperatures. Inboth cases a dependence of the maximum specificgrowth rate on temperatures is proposed. Optimum andmaximum growth temperatures are usually close, aswas shown by Gomez and Cantero [23]. Nemati andWebb [13] and Breed and Hansford [18] also showedthe dependence of maximum specific substrate utiliza-

tion rate on temperature, and included an Arrheniusterm in their respective kinetic models. The optimumtemperature has also been reported to be pH dependent,decreasing with decreasing pH [1].

4.3. Effect of oxygen and concentration

Oxygen is the electron acceptor in microbial ferrousiron oxidation (Eq. (2)). Its reduction occurs at the insideof the cytoplasmic membrane, and oxygen needs to betransported from the solution across the cell membraneto participate in the reaction. To this end, it interacts inan enzyme–substrate interaction mechanism, verysimilar to that for ferrous iron, and can thus be modelledby Michaelis–Menten kinetics. A simple Monod termaccounting for oxygen concentration would hence lookas follows:

l ¼ lmaxtFe2þb

KFe þ ½Fe2þ� d½O2�

KO þ ½O2� ð13Þ

This formulation of the oxygen term has been usedby Huberts [11], except that oxygen concentration hasbeen replaced by the partial pressure of oxygen in thegas phase, which is assumed proportional to thedissolved concentration, but more easily measured.Crundwell [12] also incorporates a Monod type oxygenterm into his model, but raised to the half power, asshown in Eq. (9). In this case the oxygen term occurs asa consequence of evaluating the entire electron–protoncircuit as a fuel cell, with oxygen reduction as thecathodic half reaction.

The use of oxygen partial pressure as measure ofdissolved oxygen concentration is dubious. Dissolvedoxygen concentrations in solution are always low due toits low solubility in water. In a rapidly operating bio-oxidation system the adsorption of oxygen may be gas–liquid mass transfer limited and thus govern the overallrate rather than microbial oxidation kinetics.

4.4. Effect of dissolved metal ions

Inhibitory concentrations of dissolved metals, suchas arsenic, copper, mercury, nickel, uranium, etc., onmicrobial growth and ferrous oxidation have beenreported by several authors [24–26]. However, there isa limited number of publications where an inhibitionterm is incorporated into a rate equation. The effects ofarsenic have had by far the most attention [27]. It is alsonoteworthy that, while the catalytic effect of Ag+ inchalcopyrite bioleaching was reported [26], this ion isalso highly inhibitory at concentration greater than 1 μM[25]. The fact that these metals have different inhibitory

26 T.V. Ojumu et al. / Hydrometallurgy 83 (2006) 21–28

concentration means they have different effects on thephysiology of the microbes. This can in part beexplained in terms of Ingledew's chemiosmotic model[19]—these dissolved metals or salts exert differentosmotic pressures on the microbial cell thus resulting indifferent tolerance levels. Also, the observed effects canbe due to direct inhibition of the cells' metabolism bythese ions. Blight and Ralph [28] report that increasedionic strength significantly increases bacterial doublingtime and reduces microbial ferrous oxidation rate. Theinhibitory effect of these metals depend on theirconcentration, such that the bacteria, if not completelypoisoned, will have a prolonged lag phase until theybetter adapt to the prevailing conditions.

4.5. Synergistic effects

The combined/synergistic effects of all the factorsmentioned above on microbial ferrous-iron oxidationkinetics have not been adequately studied. Combinationof rate terms accounting for all three substrates of theferrous oxidation process–ferrous ions, oxygen andacid–has been incorporated only in Crundwell's Eq. (9).Other rate equations rarely combine terms for more thantwo effects at the same time. The microbial optimumtemperature was reported to be pH dependent [1], and sowere the various kinetic constants, for example asshown by Breed and Hansford [18].

It has been suggested that the microbial specificgrowth rate should be the product of the Monod terms ofall essential substrates of the microbe [29]. Likewise, itcould be argued, should limiting terms be multiplied intoa given rate equation. From a purely mathematical pointof view this would appear illogical as the overallcontribution of all the terms to specific growth ratediminishes. More logical would be that the one substratethat is most growth limiting at a particular point in time(or the factor most inhibiting) should govern the rateequation at that moment. However, little systematicstudy into such synergistic effects and how to modelthem has been conducted.

Table 2Analysed composition of a heap bioleach PLS

Element Concentration [mg L−1] Element Concentratio

Al 12200 P 221Ca 467 K 29.0Co 16.2 Na 1670Cu 2000 Zn 376Fe 2460 Cl− 1300Mg 10100 F− 80.1Mn 669 NO2

− 28.1

5. Validity of rate equations in the context ofindustrial bioleach processes

Industrial application of bioleaching falls into twocategories, tank and heap bioleaching. Almost allapplication of rate equations has been in the context oftank bioleaching. This is not surprising considering thatthese are engineered processes operated under con-trolled conditions at which the overall rate of bioleach-ing is near optimal. This implies that parameters such astemperature and pH are maintained close to optimumvalues, whereas the ferric to ferrous ratio prevailing inthe system is governed primarily by the interplaybetween micro-organisms, mineral concentrate andtank residence times. It is interesting to note thatLeptospirillum species have been found to dominateover At. ferrooxidans in pyrite based tank bioleachprocesses precisely because it offers a more favourablebalance in this regard [30].

Heap bioleach processes, on the other hand, offerno control over the prevailing operating conditions.Moreover, parameters such as temperature and pHcan vary widely over time and location within theheap. While pyrite heaps are known to reachconsiderable temperatures, often into the thermophilerange, heaps of copper sulphide minerals remainlargely cold, often at temperatures well below thosecommonly studied in the laboratory. The relativelylarge quantity of gangue compared to valuableminerals within a typical heap, continuous recycleof the solution inventory, and the protracted times ofexposure can result in the release of considerableconcentrations of gangue cations into the heap leachsolution—to the point where they exceed limitscommonly considered toxic to iron-oxidising micro-organisms. Table 2 represents the solution analysis ofa pregnant leach solution (PLS) from a Chileanchalcocite based heap operation. The values indicatethat in every respect the solution conditions are farfrom what would be considered optimal in a typicaltank bioleach operation.

n [mg L−1] Element Concentration [mg L−1]

NO3− 106

o-PO4 532SO4

2− 116880T (°C) 18–22pH (feed) 1.24pH (PLS) 2.20Eh (mV vs. SHE) 640

27T.V. Ojumu et al. / Hydrometallurgy 83 (2006) 21–28

Modelling of heap bioleaching is complex as it has toincorporate a large number of phenomena, such assolution, gas and heat transport, multi-mineral kinetics,bacterial kinetics, diffusion effects, etc. [31]. Previousstudies have shown that there is clear evidence that heapbioleaching can be rate-limited by the microbialoxidation step in solutions with a composition such asoutlined in Table 2 [32]. Modelling of these effectsremains strictly empirical, however, as none of theexisting rate equations have been confirmed to be validunder such extreme conditions.

6. Conclusions

Rate equations that describe the kinetics of microbialferrous iron oxidation have been formulated for over30 years. A direct comparison of some of the moreprominent models relative to a set of data for At.ferrooxidans can be seen as an indication that rateinhibition by ferric iron is perhaps not as significant aspreviously assumed, but that it is rather the lack offerrous iron as the primary substrate that limits growthand oxidation kinetics. None of the models reviewedappears to describe oxidation kinetics well at high ferricto ferrous ratios, where maintenance effects dominateover growth related ferrous iron utilisation.

The effect of other parameters, such as pH, tempe-rature, concentration of oxygen and other ions, onmicrobial ferrous iron oxidation kinetics have beenstudied to some degree, but usually within fairly limitedranges. No model allows incorporation of all of theseparameters simultaneously, and there is little clarity onhow a comprehensive model should look.

While the available models are of sufficient quality tobe used in the design of tank bioleach processes, whichprogress under fairly optimal and controlled conditions(in terms of temperature, pH, and O2 levels), theirapplicability to heap bioleach processes, which fre-quently operate under highly variable conditions farfrom the optimal, remains questionable. Future researchshould be directed at expanding the ranges of applica-bility of rate equations, as well as incorporating thevariation of all significant parameters within onecomprehensive rate equation.

Nomenclature

Symbol

Definition Units

μ

Bacterial specific growth rate h−1

μmax

Maximum bacterial specificgrowth rate

h−1

[Fe2+]

Concentration of ferrous-iron mM Fe2+

[Fe2+]t

Threshold concentrationof ferrous iron

mM Fe2+

[Fe3+]

Concentration of ferric-iron mM Fe3+

Ks, Km, KFe2+

Model-specific substrate

affinity constants

mM Fe2+

Ki, Kif, Ksi, Ki′,K3⁎, etc.

Model-specific inhibitionconstants

various

qFe2+, qO2

Bacterial specific ferrous-ironutilisation rate

mol (mol C)−1 h−1

qFe2+max , qO2

max

Maximum bacterial specificutilisation rate

mol (mol C)−1 h−1

−rFe2+

Ferrous-iron utilisation rate mMFe2+ h−1

T

Absolute temperature K [X] Cell concentration mMC YSX Bacterial yield on substrate mol C (mol Fe2+)−1

YFe2+Xmax

Maximum bacterial yield

on ferrous-iron

mol C (mol Fe2+)−1

References

[1] Nemati, N., Harrison, S.T.L., Hansford, G.S., Webb, C.,Biochem. Eng. J., 1 (1998), 171–190.

[2] Boon, M., Theoretical and Experimental Methods in theModelling of Bio-oxidation Kinetics of Sulphide Minerals,PhD thesis, Technical University, Delft, The Netherlands, 1996.

[3] Hansford, G.S., in: Rawlings, D.E. (Ed.), Biomining: Theory,Microbes and industrial Processes, Springer, Berlin and LandesBioscience, Austin (1997), 153–176.

[4] Lacey, D.T., Lawson, F., Biotechnol. Bioeng., 12 (1970), 29–50.[5] MacDonald, D.G., Clark, R.H., Can. J. Chem. Eng., 482 (1970),

669–676.[6] Jones, C.A., Kelly, D.P., J. Chem. Tech. Biotechnol., 33B (4)

(1983), 241–261.[7] Braddock, J., Luong, H.V., Brown, E.J., Appl. Environ.

Microbiol., 48 (1984), 48–55.[8] Liu, M.S., Branion, R.M.R., Duncan, D.W., Can. J. Chem. Eng.,

66 (1988), 445–451.[9] Lizama, H.M., Suzuki, I., Appl. Environ. Microbiol., 55 (1989),

2588–2591.[10] Nikolov, L.N., Karamanev, D.G., Biotechnol. Prog., 8 (1992),

252–255.[11] Huberts, R., Modelling of Ferrous Sulphate Oxidation by Iron

Oxidising Bacteria—A Chemiosmotic and ElectrochemicalApproach, PhD thesis, University of the Witwatersrand, SouthAfrica, 1994.

[12] Crundwell, F.K., Bioelectrochem. Bioenerg., 43 (1997), 115–122.[13] Nemati, M., Webb, C., Biotechnol. Lett., 20 (1998), 873–877.[14] Boon, M., Ras, C., Heijnen, J.J., Appl. Microbiol. Biotechnol.,

51 (1999), 813–819.[15] Meruane, G., Sahle, C.C., Wiertz, J., Vargas, T., Biotechnol.

Bioeng., 80 (2002), 280–288.[16] Searby, G.E., Hansford, G. S., in: Tzesos, M., Hatzikioseyian, A.,

Remoundaki, E. (Eds.), Biohydrometallurgy: a sustainabletechnology in evolution—Part II, National Technical Universityof Athens, Greece (2004), 1227–1236.

[17] Pirt, S.J., Arch. Microbiol., 133 (1982), 300–302.[18] Breed, A.W., Hansford, G.S., Biochem. Eng. J., 3 (1999),

193–201.[19] Ingledew, W.J., Biochim. Biophys. Acta, 683 (1982), 89–117.[20] Kinnunen, P., High-rate Ferric Sulphate Generation and Chalco-

pyrite Concentrate Leaching by Acidophilic Microorganisms,

28 T.V. Ojumu et al. / Hydrometallurgy 83 (2006) 21–28

PhD thesis, Tampere University of Technology, Finland (2004),1–55.

[21] Hinshelwood, C.N., The Chemical Kinetics of the Bacterial Cell.Clarendon Press, Oxford, 1946, 254–257.

[22] Ratkowsky, D.A., Lowry, R.K., McMeekin, T.A., Stokes, A.N.,Chandler, R.E., J. Bacteriol., 154 (1983), 1222–1226.

[23] Gomez, J.M., Cantero, D., J. Ferment. Bioeng., 86 (1998),79–83.

[24] Tuovinen, O.H., Niemela, S.I., Gyllenberg, H.G., Antonie vanLeeuwenhoek, 37 (1971), 489–496.

[25] Garcia, O., Silva, L.L., Biotechnol. Lett., 13 (1991), 567–570.[26] Wang, M., Zhang, Y., Deng, T., Wang, K., Miner. Eng., 17

(2004), 943–947.

[27] Harvey, P.I., Crundwell, F.K., Appl. Environ. Microbiol., 63(1997), 2586–2592.

[28] Blight, K.R., Ralph, D.E., Hydrometallurgy, 73 (2004),325–334.

[29] Rossi, G., Biohydrometallurgy, McGraw-Hill, New York, 1990.[30] Rawling, D.E., Microbiology, 145 (1999), 5–13.[31] Dixon, D.G., Petersen, J., in: Riveros, P.A., Dixon, D.G.,

Dreisinger, D.B., Menacho, J., (Eds.), Copper 2003. Hydrometal-lurgy of Copper, CIM, Montreal, Canada, 2003, 493–516.

[32] Petersen, J., Dixon, D.G., in: Tzesos, M., Hatzikioseyian, A.,Remoundaki, E. (Eds.), Biohydrometallurgy: a sustainabletechnology in evolution—Part I, National Technical Universityof Athens, Greece (2004), 65–74.