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Hydrometallurgy: Practice
Hydrometallurgy: Practice
Hydrometallurgy: Practice
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Hydrometallurgy: Practice

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Hydrometallurgy: Practice provides the necessary fundamental background to the multidisciplinary field of hydrometallurgy and provides the tools to be able to utilize the theory to quantitatively describe, model and control the unit operations used in hydrometallurgical plants. The book describes the development and operation of processes utilizing hydrometallurgical operations. It is a valuable resource and reference for researchers, academics, students and industry professionals. The book focuses on quantitative problem solving with many worked examples and focused problems based on Nicol’s many years’ experience in the teaching of hydrometallurgy to students, researchers and industry professionals.
  • Helps to master detailed chemistry and chemical engineering fundamentals required to fully engage in the field of hydrometallurgy
  • Provides a ready reference for the students, academic and practicing professionals when confronted by a particular problem or opportunity in hydrometallurgy
  • Features many worked problems and appropriate workshops providing the necessary skills to tackle quantitative problems in hydrometallurgy
LanguageEnglish
Release dateAug 13, 2022
ISBN9780323995740
Hydrometallurgy: Practice
Author

Michael Nicol

Michael J. Nicol is Emeritus Professor of Extractive Metallurgy, Murdoch University, Australia. He spent 24 years in Mintek, South Africa in various roles largely in the field of hydrometallurgy. He taught hydrometallurgy at Wits University in South Africa for 4 years and subsequently at Murdoch University in Perth for 15 years. He has supervised many postgraduate students. In addition to many international conference presentations, he is the author or co-author of over 200 refereed publications in international journals and is the inventor or co-inventor of 10 patents. Over 300 hundred professionals from the industry and research organizations have attended various graduate courses on various aspects of hydrometallurgy delivered by Prof Nicol in Australia, South Africa, USA, Canada, Chile, Brazil and South Korea.

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    Hydrometallurgy - Michael Nicol

    Hydrometallurgy

    Practice

    Michael Nicol

    Murdoch University, Murdoch, WA, Australia

    with contributions by

    Nicholas Welham

    Gamini Senanayake

    Table of Contents

    Cover image

    Title page

    Copyright

    1. Leaching practice

    1.1. Leaching methods

    1.2. Typical leaching processes

    1.3. Batch leaching kinetics

    1.4. Continuous leaching—micro- and macrofluids

    1.5. Counter-current leaching

    1.6. Bacterial oxidation and leaching

    1.7. Pressure leaching

    1.8. Heap leaching

    1.9. In situ leaching

    1.10. Leaching of gold and silver

    1.11. Alternative lixiviants for gold

    1.12. Summary

    Problems—leaching

    Case study

    Appendix

    2. Solid–liquid separation

    2.1. Introduction

    2.2. Sedimentation

    2.3. Thickeners

    2.4. Filtration

    2.5. Summary

    3. Precipitation and crystallization

    3.1. Introduction

    3.2. Thermodynamics of precipitation

    3.3. Iron removal from zinc sulfate solutions

    3.4. Kinetics of precipitation

    3.5. Dissolution–precipitation processes

    3.6. Summary

    4. Solvent extraction

    4.1. A typical SX process

    4.2. Chemistry of SX processes

    4.3. Extraction methods

    4.4. Common SX contactors

    4.5. Some SX processes

    4.6. Summary

    Problems

    Case study

    5. Adsorption and ion exchange

    5.1. Ion-exchange resins

    5.2. Speciality resins and adsorbents

    5.3. Ion-exchange equilibria

    5.4. Ion-exchange kinetics

    5.5. Ion-exchange processes and equipment

    5.6. Process examples

    5.7. Quantitative description of fixed-bed processes

    5.8. Mass transport parameters

    5.9. Modeling of breakthrough

    5.10. The resin (carbon)-in-pulp process

    5.11. Summary

    Problems

    Case study

    Appendix 1

    Appendix 2

    Appendix 3

    Appendix 4

    6. Cementation and reduction

    6.1. Cementation

    6.2. Reduction by dissolved gas

    6.3. Summary

    Problems

    7. Electrowinning and electrorefining of metals

    7.1. General considerations

    7.2. Mass transfer at vertical electrodes

    7.3. Electrocrystallization

    7.4. Current distribution in cells

    7.5. Materials for cells and electrodes

    7.6. Tankhouse current distribution

    7.7. Energy consumption

    7.8. Electrorefining of metals

    7.9. Electrowinning of copper

    7.10. Electrowinning of zinc

    7.11. Electrowinning of nickel and cobalt

    7.12. Electrowinning of manganese metal and dioxide

    7.13. Electrowinning in novel cells

    7.14. Economic optimization

    7.15. Summary

    Problems

    Case study

    Appendix

    8. Process selection

    8.1. Process selection

    Index

    Copyright

    Elsevier

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    Notices

    Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary.

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    1: Leaching practice

    Abstract

    The most important operation in a hydrometallurgical application is the leaching of the valuable metal. There are a number of different methods for conducting the leaching step that varies from the simple such as heap leaching to the complex that involves the use of pressure autoclaves. This chapter will review each of these with examples taken from actual industrial processes. The kinetics of leaching using batch and continuous systems is described for cocurrent and counter-current processes. Detailed information is presented for some important leaching processes that include pressure leaching, bioleaching, and heap leaching. Considerable attention is focused on the most widely applied process, namely the cyanidation of gold and silver. Factors that should be taken into account in devising alternatives to the use of cyanide are discussed as an example of the development of novel processing options. This chapter should provide a useful practical guide as a complement to the theoretical aspects dealt with in Chapter 6.

    Keywords

    Batch and continuous; Bioleaching; Copper; Counter-current; Cyanidation; Heap leaching; Leaching; Leaching systems; Methods of leaching; Pressure leaching

    The chemistry of processes aimed at the dissolution of valuable metals from various feed materials was covered in Chapter 6 in Volume 1. In this chapter, the various methods employed to carry out the leaching reactions will be outlined

    The choice of a particular leach process and the equipment to be used depends on the performance that can be achieved by the various options. The main factors to be taken into account in assessing leach performance are the following.

    • Degree of dissolution of desired species.

    • Selectivity of leaching process with respect to the desired species.

    • Leaching time required to achieve the desired extraction.

    • Operating cost (lixiviants, power).

    • Capital cost.

    There is generally an economic optimum that determines the most appropriate strategy for leaching. Thus, for the leaching of gold ores, the objective should be to maximize extraction given that operating costs are generally not high relative to the value of the product. On the other hand, in the case of the leaching of low-grade copper ores, the operating costs are the most important consideration and heap leaching is the only viable option. Even in this case, high acid consumption can rule out heap leaching for some ores.

    1.1. Leaching methods

    The techniques shown next are applied in the leaching of ores and concentrates.

    In situ leaching: Lixiviant pumped directly into fractured ore-body and pregnant solution recovered.

    Heap (or dump) leaching: Lixiviant sprinkled over heaps of mined (crushed) ore in heaps built over the impervious base.

    Vat leaching: Crushed ore fills a large vat that is then filled with lixiviant and left to leach. A batch process that is not very common.

    Agitation leaching: Milled ore contacted with lixiviant in agitated tanks (mechanical or air sparging). Batch or multistage continuous reactors.

    Pressure leaching: Milled ore or concentrate contacted with lixiviant in high-pressure, high-temperature reactors (autoclaves) that are generally operated in continuous mode.

    Bacterial leaching: A variation of heap or agitation leaching in which bacteria assist in the leaching reactions.

    Some of the more important practical methods used to dissolve or leach valuable components of an ore, concentrate, or other intermediate product are summarized in Fig. 1.1. The actual method to be adopted depends largely on the value of the material to be treated with low-grade materials requiring methods to the left and high grade materials to the right of the diagram.

    1.2. Typical leaching processes

    The dissolution of a solid species is a chemical reaction that can be classified into one of several types such as acid leaching and oxidative leaching as outlined in Chapter 6 in Volume 1. The chemistry that is possible in the dissolution of a solid is often extensive and varied and depends on the ingenuity of the researcher. However, practical considerations determined largely by the cost of the lixiviant have restricted the choice of the leaching process. Table 1.1 summarizes some of the more important leaching processes that are in operation in various parts of the world.

    Figure 1.1  Selection of leaching method with grade of feed material.

    The design and operation of leaching processes are critically dependent on an understanding and application of the kinetics of the reactions taking place in leach reactors. In the following sections, we will show how kinetic information can be used to describe how we can design and operate a leaching process.

    1.3. Batch leaching kinetics

    In Chapter 6 in Volume 1, we dealt with the expected profiles for the leaching of particles for several cases in which the rate-determining step is either mass transport or chemical reaction. In many real cases (see later) involving ores and, to a lesser extent, concentrates, the curve of fraction leached versus time does not conform to any of the theoretical forms. Under these conditions, we have to resort to empirical rate equations.

    Table 1.1

    Consider the leaching of a typical gold ore for which the following rate equation has often been found to describe the rate of dissolution of gold from cyanide pulp,

    (1.1)

    in which [Au] is the concentration (mass gold/unit mass of pulp) at any time, t, and [Au]f is the corresponding concentration after an infinite time, that is, it is the ultimate achievable barren concentration (mainly locked gold not accessible to the lixiviant). k is a rate constant.

    This equation can be integrated to give,

    (1.2)

    where [Au]o is the initial concentration. This equation can be written in the form

    (1.3)

    in which k′=k([Au]o −[Au]f) and X=([Au]−[Au]f)/([Au]o −[Au]f) is the fraction of gold leached.

    The parameters [Au]f and k can be obtained from a small-scale batch leach carried out under typical envisaged plant conditions of pulp density, cyanide concentration, and pH. For example, the time required to achieve a 50% dissolution of gold from a batch of pulp for which [Au]o =5g/t, [Au]f =0.2g/t, and k=0.05/h, will be given by

    In a practical situation, the time required to fill and empty the batch reactor will have to be added to the actual leaching time. This generally is most efficient with large reactors.

    Thus, for a batch leach plant to treat 100t/h of the above pulp in a tank that can hold 950 t (for comparison with a later section—see Appendix) of pulp, the leach time will be 4.5h, which will leave 4.5h for charging and discharging the tank, that is, an average pulp flow-rate of 420t/h during these operations.

    Notice that in the case of a batch reactor, all of the ore particles are exposed to the lixiviant for the same leaching period.

    Batch leaching is seldom used in practice except for relatively small-scale operations such as, for example, the dissolution of a precious metal concentrate (gold and/or platinum group metals) in, for example, a chlorine/chloride system. The requirement for accurate accounting of the metal in various stages of processing is also considerably simplified in batch processing. The formation of metastable species in solution (such as hydrated silica) may also be controlled more appropriately in a batch rather than in a continuous reactor.

    1.4. Continuous leaching—micro- and macrofluids

    In continuous leaching, the ore, concentrate, or other material containing the metal or mineral to be leached (generally as a slurry) and the lixiviant are fed continuously into a stirred tank reactor (CSTR) and the leached slurry discharged continuously generally into another reactor in series.

    The analysis of the performance of this type of reactor depends on the nature of the fluid. Microfluids consist of solutions such that one cannot distinguish one molecule of a solute, such as an ion or solvent molecule from another. Thus, in a CSTR treating a microfluid, all reactants are at their exit (i.e., low) concentrations, reactants entering are immediately diluted to exit concentrations by perfect mixing, and the reactions, therefore, take place at a relatively low rate, as dictated by the reactant concentrations. The flow of a microfluid is often characterized as nonsegregated.

    On the other hand, as shown in Fig. 1.2, macrofluids consist of suspensions of particles each of which is distinguishable and each of which is an aggregate (crystal in some cases) of a large number of atoms or molecules. Thus, for example, a particle of ore suspended in a slurry will behave in a CSTR in the same way as a batch reactor. Thus, the reactant concentrations (in the solid phase) do not immediately drop to a low value but decrease as they would in a batch reactor and the extent of reaction in each of the ore particles in the reactor depends only on the length of stay (residence time) in the reactor. This is equally true for any particle in the exit stream. Thus, the fractional conversion in the exit stream is determined by summing the conversions of all the particles (Population Balance Method). This is an example of segregated flow.

    Thus, in a CSTR treating a slurry, the solution phase would behave as a microfluid and the solid phase as a macrofluid. Thus, in an ideal CSTR in which all particles of the above gold ore have the same residence time tR (= Vol. of reactor/Volumetric flowrate of slurry), the performance would be identical to that of the batch reactor discussed earlier.

    Figure 1.2  Schematic to distinguish between microfluid (left) and macrofluid (right).

    1.4.1. Residence time distribution in a CSTR

    In an ideal completely mixed reactor vessel, an entering fluid element is instantaneously broken up into tiny fragments that are uniformly distributed throughout the volume of the vessel. Some of these fluid fragments are immediately drawn into the effluent stream while others circulate within the vessel for various lengths of time before finding their way out. Thus, at any instant, the reactor effluent is composed of fluid particles that have spent various lengths of time in the vessel. In contrast, every fluid element entering a plug flow vessel follows the element that entered before it without any intermixing and exits the reactor in exactly the same order. At any instant then, the exit stream is made up of fluid elements, all of which have been resident in the reactor for exactly the same length of time. The time spent in the reactor by a fluid element is called its exit age. The distribution of exit ages of all fluid fragments in the reactor effluent is called the residence time distribution (RTD) and is indicative of the mixing and flow distribution patterns within the reactor.

    In theory, in a perfectly mixed tank, the residence times cover the whole range from zero to infinity, although the average or mean residence time (tR) is the same as for batch treatment, namely the mass of pulp in the tank divided by the mass flowrate. Mixing theory shows that the way to overcome the short-circuiting that occurs in a single tank is to divide the same total volume or mass among a number of tanks in series—the more tanks there are in series, the higher will be the proportion of pulp that will have a residence time close to the mean value.

    For the mean residence time, tR =V/Q, in which V is the volume of fluid in the tank and Q the volumetric flowrate, the actual residence time/mean residence time=t/tR =θ.

    The residence time distribution function, E(θ), is the relative proportion of the discharge pulp having a residence time between θ and θ+dθ. That is, E(θ).dθ is the fraction of the exit stream of age between θ and θ+dθ.

    For an ideal plug flow reactor, the expected RTD function, E(θ) is shown as the vertical arrow in Fig. 1.3.

    On the other hand, for a completely mixed (CSTR) reactor,

    (1.4)

    A graphical representation of the expected RTD function for a completely mixed vessel is shown in Fig. 1.3.

    Consider a number of i mixed tanks in series that are assumed to be completely mixed and to each have the same volume. Thus, if the total vessel volume is V, each tank has volume Vi = (V/N). The RTD function for this situation can be derived and is given by

    Figure 1.3  Residence time distribution for a single CSTR.

    (1.5)

    A family of RTD curves for a number of tanks in series is shown in Fig. 1.4.

    The fraction of fluid that has a residence time between θ1 and θ2 is given by that area under the curve between θ1 and θ2 relative to the total area under the curve. These values are shown in Table 1.2 for several intervals.

    Figure 1.4  Residence time distributions for tanks in series.

    Table 1.2

    It is apparent that as the number of tanks increases, the distribution of residence times becomes sharper with a larger fraction of the pulp having a residence time close to tR.

    For a reaction in the tank that is a first-order process, it can be shown that no error is introduced by assuming that all particles have the same (mean) residence time. This is not true for any other reaction order (see Appendix for details).

    The situation with reactions involving solid particles is further complicated by the fact that a pulp invariably contains solids with a nonuniform particle size distribution, probable different shapes, and even a distribution of reactivities. Thus, Fig. 1.5 shows a mineral liberation analyzer picture of the gold particles (blue) in a typical gold concentrate produced by gravity separation.

    Thus, one could expect to find a distribution of residence times, particle sizes, shapes, and reactivity such as shown in Fig. 1.6.

    It is not difficult to visualize that the larger particles will probably have a longer residence time in the tank and may, if large enough, settle in the bottom of the tank while the fine particles will probably have a shorter mean residence time. This effect will, in part, act to compensate for the normal RTD in that we want the larger particles to have a longer residence time. For this reason, the complications caused by the RTD effect are often ignored.

    Figure 1.5  Distribution of particle sizes and shapes in a gold (blue (dark gray in print)) concentrate. The green particles (light gray in print) are pyrite.

    Figure 1.6  Schematic distribution of various characteristics of ore/concentrate particles.

    In a real agitated tank reactor, however, there is often a degree of short-circuiting of the pulp due to inefficient blending of the incoming pulp with the contents of the tank coupled to inappropriate positioning of the feed and exit points. The settling of larger particles can also often result in a significant fraction of the tank volume being unavailable for reaction. This is, as expected, more prevalent in the first tank in a series and the periodic use of tracer tests to establish the active volume of the tank will enable this problem to be highlighted.

    It is apparent that a full treatment of the kinetics of leaching of a real ore or concentrate will require information on the particle size distribution. Given all the previously mentioned complications when dealing with real leach systems, the added complexity of the population balance models cannot generally be justified. The Appendix outlines the application of conventional CSTR theory for the treatment of leaching reactions bearing in mind the previously mentioned problems in particulate systems.

    1.5. Counter-current leaching

    The previously mentioned leaching processes involve cocurrent flow of the ore or concentrate and the lixiviant. While this is often a convenient method of operating, greater efficiencies in terms of overall leaching recovery and maximum utilization of the lixiviant can be accomplished by contacting the lixiviant with the ore in a counter-current fashion. Thus,

    • Maximum extraction is achieved by contacting a leach residue with fresh, concentrated lixiviant.

    • Maximum utilization of the lixiviant can be achieved by contacting fresh, reactive ore or concentrate with lixiviant that has already been used in a prior stage.

    Thus, consider the real example shown in Fig. 1.7 of a three-stage leaching circuit for a zinc calcine that is designed to maximize zinc recovery and utilize all the lixiviant while still minimizing the dissolution of iron by limiting the pH to values above 3 in the first stage leach and providing a solution of the desired composition for purification before electrowinning.

    As this is the first example of a typical hydrometallurgical flowsheet, it is worth spending some time on the details. Ignore in the first instance the operations shown in dashed lines—these are used to bleed iron from the circuit and are not essential in terms of counter-current leaching and we shall return to this section later. In terms of overall leaching in one stage, one is simply reacting the calcine with acid in the spent electrolyte from electrowinning that contains excess acid generated at the anode during electrowinning. However, in this case, in the first leach stage, the calcine is contacted with (a) solution from the second stage leach that contains excess acid and zinc dissolved in the first stage and (b) small amounts of spent electrolyte to control the pH of the solution leaving the first stage. The pulp leaving the first stage is filtered (or settled) to separate the solution from the leach residue. The solution now contains up to 200g/L zinc and is sent to the purification stage of the plant and thereafter to electrowinning.

    Figure 1.7  A three-stage counter-current leaching process for zinc calcine.

    The solid residue from the first stage still contains zinc and is subjected to leaching under more extreme conditions to dissolve most of the residual zinc. The solution used to dissolve this zinc is made up of a solution from the third stage leach and some spent electrolyte. The pulp from this stage is subjected to solid/liquid separation and the solution is routed to the first stage and a small amount is sent for iron removal. The solid from this second stage is the feed to the third stage in which any residual refractory zinc is dissolved at high temperature and acidity using the spent electrolyte. After solid/liquid separation, the solution phase is used in the second stage leach, and the solid is washed and reported as the final residue.

    While this approach is efficient in terms of leaching, the introduction of a solid/liquid step between each counter-current stage introduces an additional unit operation that can be both inefficient and costly for pulps that are difficult to filter or settle. The introduction of additional water for washing the filter cakes further complicates the process. For this reason, counter-current leaching is not applied as widely as may be anticipated.

    1.6. Bacterial oxidation and leaching

    Bioleaching is the extraction of a metal from sulfide ores or concentrates using microorganisms that catalyze the oxidation of sulfide minerals. An associated process is biooxidation in which sulfide minerals associated with but not necessarily part of the mineral of interest is oxidized or dissolved. In biooxidation of refractory gold ores, bacteria are used to solubilize an iron sulfide in which the gold particles are located and thus make the gold available for cyanide leaching. Likewise, in coal desulfurization, bacteria are used to oxidize the pyrite contaminant in the coal thus making the sulfur soluble as ferric sulfate.

    Bioleaching is used today in commercial operations to process ores of copper, nickel, cobalt, zinc, and uranium; whereas, biooxidation is used in gold processing and coal desulfurization. Since bioleaching is a natural process, an undesirable effect is the creation of so-called acid drainage from the slow oxidation of sulfide mineral outcrops and from abandoned tailings dumps.

    Bioleach processing differs depending on the type of resource to be processed.

    Dump leaching—waste rock, low-grade ore, or concentrator tailings (low grade, oxides, and secondary sulfides) are leached from waste dumps.

    Heap leaching—newly mined run-of-the-mine (ROM) material (intermediate grade, oxides, and secondary sulfides) is placed as a heap on an impervious natural surface or a pad and leached. ROM may be leached as mined or may be partially crushed and mixed with acid before depositing on the heap.

    Agitated leaching—concentrates are leached in a tank using mechanical agitation.

    Waste dump leaching uses mesophilic (ambient temperature, 35–45°C) microorganisms, that is, bacteria. Heap leaching of ore may involve mesophiles or moderate thermophiles (high temperature, 50–60°C) microorganisms. Leaching of chalcopyrite and other primary sulfide concentrates requires extreme thermophiles (>70°C). A photograph of thiobacillus ferrooxidans bacteria attached to sulfide minerals is shown in Fig. 1.8.

    In summary, bioleaching involves

    • Oxidative dissolution of sulfides with ferric ions.

    • Reoxidation of ferrous by dissolved oxygen catalyzed by specific bacteria.

    Figure 1.8  Bacteria (yellow (light gray in print)) associated with sulfide minerals.

    • Use of microorganisms of which thiobacillus ferrooxidans and sulphooxidans are most common.

    • Source of CO2, pH about 1.5–2, temperature 35–45°C, nutrients (N, P, K), dissolved O2 greater than 1ppm.

    For example, the following reactions occur during the biooxidation of pyrite,

    (1.6)

    (1.7)

    that is, overall,

    (1.8)

    N.B. In the case of pyrite, acid is produced and must be neutralised.

    In the case of pyrrotite,

    (1.9)

    acid is consumed and must be provided to keep the pH in the optimum region.

    Some of the most important advantages of biooxidation or leaching processes are

    • Rapid oxidation of iron(II) to iron(III)

    • Bacterial oxidation of elemental sulfur layers

    • Lower capital costs for small to medium size plants

    • Relatively simple, low-tech process

    • Environmentally acceptable

    On the other hand, there are some disadvantages such as

    • Slow kinetics (several days under favorable conditions)

    • Sensitivity to process variations (temperature, loss of aeration, poisons such as cyanide and salinity)

    • Limited solids content (<20%)

    • Produces soluble iron(III) that requires removal and disposal.

    • Bioleaching does not recover the precious metals in the ore

    1.6.1. Process parameters for biological oxidation

    The plant size is determined by the ore or concentrate throughput and the rate of oxidation of sulfide sulfur. The relative proportions of each mineral present determine the process acid consumption/production, oxygen demand, and cooling requirements as shown in Table 1.3.

    Major design requirements of reactors are:

    • Agitation to suspend solids and, more importantly, to disperse large volumes of air or oxygen.

    • Cooling coils to dissipate heat generated by exothermic reactions and agitation that is not lost by evaporation, heating of air, and feed pulp.

    Table 1.3

    • Residence time

    • Corrosion resistance of the materials of construction given acidic conditions.

    1.6.2. Biooxidation reactor kinetics and design

    The rate of biological oxidation of a sulfide mineral can often be expressed in terms of the logistic rate equation

    (1.10)

    where v is the rate of oxidation,

    X is fraction oxidized,

    Xm is the max. fraction that can be oxidized, and

    k is a bacterial growth rate constant

    Consider a single-stage CSTR reactor containing a volume V of pulp that is flowing at a rate Q (volume/unit time) through the reactor.

    From the mass balance at steady-state, we obtain the CSTR equation,

    (1.11)

    and, substituting the previous rate equation,

    (1.12)

    in which tR =V/Q is the mean residence time.

    For k=1/tR, X=0, and this is referred to as the bacterial cell wash-out condition that is, the operating point at which the dilution rate, 1/tR, is equal to the maximum rate of growth of the bacterial cells. Thus,

    (a) For three equal reactors in series:

    Cell wash-out will occur for tR =1/k and the system residence time will be 3tR

    (b) For a primary reactor that is twice the size of the secondary reactors cell wash-out will occur for a total residence time=2/k=2tR where tR is the residence time in the primary reactor.

    This is the basis for the common design of two primary reactors in parallel feeding secondary reactors of the same size in series. This ensures that wash-out of the bacteria will not occur at the design flow rates for a single reactor.

    Fig. 1.9 illustrates this phenomenon for a bioleach reactor

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