solids processing 2020-2021

Solids Processing

Thomas Rodgers

2020-2021

UG Notes

Contents

List of Figures v

List of Tables vii

Nomenclature ix

Course Information xi

1 Particles 11.1 Chapter 1 ILOs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Why measure particle properties? . . . . . . . . . . . . . . . . . 31.3 Particle Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3.1 Sphericity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.2 Specific Surface Area . . . . . . . . . . . . . . . . . . . . . . . . 51.3.3 Equivalent Spherical Diameter . . . . . . . . . . . . . . . . . . . 5

1.4 Particle Size Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 61.4.1 Weighted Distributions . . . . . . . . . . . . . . . . . . . . . . . 61.4.2 Frequency and Cumulative Distributions . . . . . . . . . . . . . . 61.4.3 Particle Size Distribution Functions . . . . . . . . . . . . . . . . 81.4.4 Means and Percentiles . . . . . . . . . . . . . . . . . . . . . . . 9

1.5 Particle Measurement Methods . . . . . . . . . . . . . . . . . . . . . . . 101.5.1 Sieving Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5.2 Sedimentation Methods . . . . . . . . . . . . . . . . . . . . . . 101.5.3 Particle Counting . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5.4 Light Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5.5 Comparison of Methods . . . . . . . . . . . . . . . . . . . . . . 13

1.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Single Particle Motion in Fluids 212.1 Chapter 2 ILOs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3 Flow Past a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.1 Stokes’ Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4 Drag Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4.1 Drag Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . 252.5 Terminal Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

i

2.5.1 Stokes’ Terminal Velocity . . . . . . . . . . . . . . . . . . . . . 282.5.2 General Terminal Velocity . . . . . . . . . . . . . . . . . . . . . 28

2.6 Non-spherical Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.7 Transient Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Settling and Settlers 413.1 Chapter 3 ILOs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3 Settling Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3.1 Type I - Discrete Particle Settling . . . . . . . . . . . . . . . . . 443.3.2 Type II - Flocculent Particles . . . . . . . . . . . . . . . . . . . . 443.3.3 Type III - Hindered Settling . . . . . . . . . . . . . . . . . . . . 443.3.4 Type IV - Compression . . . . . . . . . . . . . . . . . . . . . . . 45

3.4 Gravity Separators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.5 Design of Gravity Separators . . . . . . . . . . . . . . . . . . . . . . . . 47

3.5.1 Limit for Low Solids Concentration - Rectangular . . . . . . . . 473.5.2 Limit for Low Solids Concentration - Circular . . . . . . . . . . . 483.5.3 Mass Balance of Solids . . . . . . . . . . . . . . . . . . . . . . . 49

3.6 Accelerating Settling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.6.1 Flocculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.7 Settler Rules of Thumb . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4 Centrifuges 634.1 Chapter 4 ILOs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.3 Forces on a Sphere in a Centrifuge . . . . . . . . . . . . . . . . . . . . . 65

4.3.1 Centrifugal Force . . . . . . . . . . . . . . . . . . . . . . . . . . 664.3.2 Terminal Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.4 Laboratory Centrifuges . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.5 Continuous Centrifuges . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.5.1 Settling Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.5.2 Throughput . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.5.3 Cut-size Diameter . . . . . . . . . . . . . . . . . . . . . . . . . 694.5.4 Sigma Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.6 Types of Centrifuges . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.6.1 Tubular Bowl Centrifuge . . . . . . . . . . . . . . . . . . . . . . 704.6.2 Disc Bowl Centrifuge . . . . . . . . . . . . . . . . . . . . . . . 714.6.3 Scroll Centrifuge . . . . . . . . . . . . . . . . . . . . . . . . . . 71


5 Cyclones 795.1 Chapter 5 ILOs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.3 Cyclone Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.4 Dimentionless Number Design . . . . . . . . . . . . . . . . . . . . . . . 835.5 Cyclone Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

ii

5.6 Stairmand’s Design Procedure . . . . . . . . . . . . . . . . . . . . . . . 875.7 Multiple Cyclones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.7.1 Series Overflow . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.7.2 Series Underflow . . . . . . . . . . . . . . . . . . . . . . . . . . 915.7.3 Partial Overflow Recycle . . . . . . . . . . . . . . . . . . . . . . 925.7.4 Underflow Recycle . . . . . . . . . . . . . . . . . . . . . . . . . 92


iii

List of Figures

1.1 Sizes of typical powder products . . . . . . . . . . . . . . . . . . . . . . 31.2 Weighted distributions for the same sample . . . . . . . . . . . . . . . . 71.3 Comparison of frequency and cumulative distributions . . . . . . . . . . 71.4 Weighted number distribution with associated means . . . . . . . . . . . 91.5 Particle size collection from a seive . . . . . . . . . . . . . . . . . . . . . 111.6 Comparison of particle size measurement methods for a sample of soil. . 13

2.1 Flow around a sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2 Flow of a fluid over a surface against a pressure gradient . . . . . . . . . 232.3 Flow of a fluid over a sphere . . . . . . . . . . . . . . . . . . . . . . . . 242.4 Variation of CD with Rep. . . . . . . . . . . . . . . . . . . . . . . . . . . 262.5 Forces on a falling sphere. . . . . . . . . . . . . . . . . . . . . . . . . . 272.6 Variation of Rep with Ar. . . . . . . . . . . . . . . . . . . . . . . . . . . 292.7 Drag coefficients for non-spherical particles. . . . . . . . . . . . . . . . . 302.8 Variation of u/ut,St with Fo. . . . . . . . . . . . . . . . . . . . . . . . . 32

3.1 Settling Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2 Gravity Settlers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.1 Forces on a sphere in a centrifugal field . . . . . . . . . . . . . . . . . . 654.2 Centrifugal force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.3 Typical Tubular Bowl Centrifuge . . . . . . . . . . . . . . . . . . . . . . 704.4 Typical Disc Bowl Centrifuge . . . . . . . . . . . . . . . . . . . . . . . 714.5 Typical Scroll Centrifuge . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.1 Cyclone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.2 St versus Re for cyclones of various geometries. . . . . . . . . . . . . . . 845.3 Cyclone performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.4 Simple theory separation versus actual separation . . . . . . . . . . . . . 865.5 Grade Efficiency parameters . . . . . . . . . . . . . . . . . . . . . . . . 875.6 Standard cyclone dimensions. . . . . . . . . . . . . . . . . . . . . . . . 885.7 Performance curves, standard conditions. . . . . . . . . . . . . . . . . . 885.8 Cyclone pressure drop factor. . . . . . . . . . . . . . . . . . . . . . . . . 895.9 Performance curves, standard conditions. . . . . . . . . . . . . . . . . . 91

v

List of Tables

1.1 Example values of sphericity. . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Typically used particle size means . . . . . . . . . . . . . . . . . . . . . 91.3 Typically used particle size means . . . . . . . . . . . . . . . . . . . . . 10

5.1 Parameters for Stairmand’s standard designs. . . . . . . . . . . . . . . . 89

vii

Nomenclature

RomanA Area m2

Ap Particle surface area m2

asm Specific mass surface area m2 kg−1

asv Specific volume surface area m−1

C Solids concentration m3-solids m−3

CD Drag cooefficient −da,b Average particle size mdsv Surface-volume ratio equivalent spherical diameter mds Surface equivalent spherical diameter mdv Volume equivalent spherical diameter mET Total efficiency −F (...) Cumulative distribution −f (...) Frequency distribution −Fc Centrifugal force NFD Drag force NG G-force −G Solids flux kg s−1

G (d) Grade efficiency −G′ (d) Reduced grade efficiency −B Breath mH Height mh Vertical fall hL Length mMc Coarse (Underflow) mass kgMf Fine (Overflow) mass kgMp Particle mass kgQ Volumetric flow rate m3 s−1

r Radius mRf Underflow to throughput ratio −RPM Rotations per minute min−1

R′ Force per unit projected area N m−2

RPS Centrifugal force s−1 or HzS Svedberg Coefficient −u Velocity m s−1

uc Overflow rate m s−1

uR Radial velocity m s−1

ix

us Hindered settling velocity m s−1

uθ Tangental velocity m s−1

Vbed Total volume of a packed bed m3

Vp Particle volume m3

Greekε Porosity −η Collection efficiency −ω Angular velocity c s−1

ϕ Sphericity −φp Concentration Depends on fitρ Density kg m−3

Σ Centrifuge factor m2

σ Standard deviation −Θ Hydraulic retention time sDimentionless VariablesAr Archimedes number d3p(ρp − ρf )ρfg/µ2

Eu Euler’s Number 2∆P/ρfu2f

Fo Fourier number µt/ρpd2p

Rep Particle Reynolds number = udρf/µSt Stokes’ Number d250ρpuf/18µ

x

Course Information

The unit aims to develop an understanding of various industrial solid processing units,their applications, and the fundamental principles where solid-fluid interactions are im-portant in these units.

This handbook includes the material covered in the first half of the course, which includesthe topics of:

Topic 1. Particle characterization and the motion of a single particle in a fluid: in thistopic the methods used to describe particle size and shape are introduced and linkedto how these particles move in a fluid. The focus will be on how to describe andevaluate the motion of particles through a fluid.

Topic 2. Sedimentation – settlers and thickeners: in this topic the concepts of particlessettling under gravity are explored. The focus will be on how to design industrialsettlers and thickeners using knowledge of particle motion in a fluid.

Topic 3. Centrifuges and Cyclones: in this topic the methods to enhance particle settlingare explored. The focus will be on how to design centrifuges and cyclones forindustrial processes using key design methodologies including Sigma theory andStairmand’s design procedure.

By the end of this first half of the course you will be able to:

ILO 1. Use methods to characterise solid particle sizes and shapes.

ILO 2. Describe the motion of solid particles in a fluid-solid system.

ILO 3. Design solid separating units such as settlers, thickeners, centrifuges, and cy-clones.

The first half of the course is built around this handbook, and the content within is sup-ported by the online material, tutorial questions, past exam papers, key concept videos,and a selection of online formative questions.

For discussion of specific topics regarding this course with your fellow students or tocommunicate with the module leader, please use the Discussion Board and post yourthreads. The module leader will look on the board regularly and answer those unclearedquestions, or come to the drop in session.

xi

Chapter 1Particles

Contents1.1 Chapter 1 ILOs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Why measure particle properties? . . . . . . . . . . . . . . . . 3

1.3 Particle Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3.1 Sphericity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3.2 Specific Surface Area . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.3 Equivalent Spherical Diameter . . . . . . . . . . . . . . . . . . 5

1.4 Particle Size Distributions . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4.1 Weighted Distributions . . . . . . . . . . . . . . . . . . . . . . 6

1.4.2 Frequency and Cumulative Distributions . . . . . . . . . . . . . 6

1.4.3 Particle Size Distribution Functions . . . . . . . . . . . . . . . 8

1.4.4 Means and Percentiles . . . . . . . . . . . . . . . . . . . . . . 9

1.5 Particle Measurement Methods . . . . . . . . . . . . . . . . . . . . . 10

1.5.1 Sieving Methods . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5.2 Sedimentation Methods . . . . . . . . . . . . . . . . . . . . . 10

1.5.3 Particle Counting . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5.4 Light Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5.5 Comparison of Methods . . . . . . . . . . . . . . . . . . . . . 13

1.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1

CHAPTER 1. PARTICLES

1.1 Chapter 1 ILOs

.

ILO 1.1. Discussing about the concept of specific surface area.

ILO 1.2. Introducing the concept of the equivalent spherical diameter.

ILO 1.3. Introducing the methds used to determine particle size distributions.

ILO 1.4. Types of average particle sizes.

2 c©T.L. Rodgers 2020

1.2. INTRODUCTION

1.2 Introduction

A particle can be defined as being a discrete sub-portion of a substance. For the purposesof this course we shall narrow this definition to include only solid particles (though thereis no reason why the same analysis cannot be undertaken with liquid droplets or gasbubbles) with physical dimensions ranging from sub-nanometers to several millimeters insize. The most common types of material consisting of these solid particles are powdersand granules, e.g. pigments, cement, and pharmaceutical ingredients. These powders andgranules come in a range of sizes such as Figure 1.1.

Particle size / µm105

104

103

102

101

100

10−1

10−2

Pelleted products

Crystalline industrial chemicals

Granular fertilisers, herbicides, fungicides

DetergentsGranulated sugars

Spray dried products

Powdered chemicalsPowdered sugar

FlourTonersPowder metalsCeramics

Electronic materialsPhotographic emulsionsMagnetic and other pigments

Organic pigments

Fumed silicaMetal catalystsCarbon black

Figure 1.1: Sizes of typical powder products [1].

Not all solid particles are the same size in a sample and generally, we deal with collectionsof particles, with a distribution of properties,

• Size - Affects the settling rate and surface area to volume ratio.

• Composition - Determines density, conductivity, porosity etc.

• Shape - Can be regular (crystals, spheres) or irregular

1.2.1 Why measure particle properties?

The distribution in particle properties affect,

• strength and load-bearing properties of rocks and soils

• reactivity of solids participating in chemical reactions

• solubility of particles in fluids

• powder flow and handling

c©T.L. Rodgers 2020 3


• suspension stability

• taste, texture, and feel in foods

Therefore there are two key reasons why industries employ particle characterisation,

1. Better control of product quality to

• charge higher premium for products

• reduce customer rejection rates

• demonstrate compliance in regulated markets

2. Better understanding of products, ingredients, and processes to

• improve product performance

• troubleshoot manufacturing issues

• optimise the efficiency of or design manufacturing processes

1.3 Particle Shape

Characterising a particle by size can be done relatively easy when all particles are uni-form in shape and can be described with just one dimension as it is the case with spheres.Irregular-shaped particles cannot be simply described as there may be different dimen-sions to consider. However, there are some key metrics that can be used.

1.3.1 Sphericity

Sphericity, ϕ, is the measure of how closely the shape of an object approaches that ofa mathematically perfect sphere [7]. This is defined as the ratio of the surface area ofa sphere with the same volume as the particle (Vp) to the surface area of the particle(Ap),

ϕ =π1/3 (6Vp)

2/3

AP≤ 1 (1.3.1)

Table 1.1 provides values for the sphericity of some example solid particles.

Table 1.1: Example values of sphericity

Shape Sphericity Example

Sphere 1.00 Glass beadsRounded 0.82 Water worn solids, atomised dropsCubic 0.81 Sugar, CalciteAngular 0.66 Cruched mineralsFlaky 0.54 Gypsum, TalcPlatelet 0.22 Clays, Mica, Graphite


1.3. PARTICLE SHAPE

1.3.2 Specific Surface Area

A large interfacial area per unit volume gives rise to increases fluid-solid interactions. Infact, sometimes particles are designed to increase this area, for example in adsorption andheterogeneous catalysis.

The specific surface area of particles may be defined as either the total surface area perunit mass of particles, asm, or the total surface area per unit volume of particles, asv.

asm =ApMp

(1.3.2)

asv =ApVp

(1.3.3)

This means that for a spherical particle of radius r the specific surface area can be givenby,

asm =ApMp

=ApρpVp

=4πr2

ρp(4/3)πr3=

3

ρpr(1.3.4)

asv =ApVp

=4πr2

(4/3)πr3=

3

r(1.3.5)

For a circular disk of radius r and thickness x the specific surface area can be givenby,

asm =ApρpVp

=2πr2 + 2πrx

ρpπr2x=

2 (r + x)

ρprx(1.3.6)

When particles are collected in a packed bed, the void fraction, ε, (or porosity) can begiven by,

ε =Vbed − VsVbed

= 1− VsVbed

(1.3.7)

Where the void fraction is a measure of how much empty space, Vbed − Vs, is in the bedvolume, Vbed. This means that the specific surface area of a packed bed can be calculatedfrom the void fraction and the specific surface area of the particles,

abed = as (1− ε) (1.3.8)

1.3.3 Equivalent Spherical Diameter

Normally an equivalent spherical diameter is defined for irregular-shaped particles. It isdefined as the diameter of a spherical particle which will give identical geometric, optical,electrical, or aerodynamic behaviour to that of the non-spherical particle being examined[3]. As the diameter of the spherical particle would be different for different propertiesit is therefore important to specify which is being used. Although more complex options



can be used it is typical to use the diameter referring to a sphere with, the same volume,the same surface area, or the same surface area to volume ratio.

dv =

(6Vpπ

)1/3

(1.3.9)

ds =

(Apπ

)1/2

(1.3.10)

dsv =6

asv(1.3.11)

1.4 Particle Size Distributions

Unless the sample of particles is perfectly mono-disperse, i.e. every single particle hasexactly the same dimensions, it will consist of a statistical distribution of particles ofdifferent sizes. It is common practice to represent this distribution in the form of either afrequency distribution curve or a cumulative (undersize) distribution curve.

1.4.1 Weighted Distributions

Particle size distributions can be represented in different ways with respect to the weight-ing of the individual particles,

Number weighted distributions have each particle given an equal weighting irrespectiveof its size. This is most useful where knowing the absolute number of particles isimportant, e.g. in foreign particle detection.

Area weighted distributions have a contribution of each particle related to the surfacearea of that particle.

Volume weighted distributions have a contribution of each particle related to the volumeof that particle (equivalent to mass if the density is uniform). This is commonlyused as it means the distribution represents the composition of the sample in termsof the volume/mass and therefore its potential value.

When comparing particle size data for the same sample weighted by different methods,it is important to realise that the distributions can look very different. This is clearlyillustrated in Figure 1.2 where a sample consisting of equal numbers of particles withnominal diameters of 5 and 10. The number weighted distribution gives equal weightingto both sizes of particle, whereas the volume weighted distribution gives eight times theweighting to the larger particles.

It is possible to convert particle size data from one type of distribution to another; however,this requires assumptions to be made about the particle shape, e.g. spherical.

1.4.2 Frequency and Cumulative Distributions

Particle size distributions can also be represented as either a frequency distribution or asa cumulative distribution. The frequency distribution represents the frequency or percent-


1.4. PARTICLE SIZE DISTRIBUTIONS

2.5 5 7.5 10 12.5

d

Rel

ativ

e%

incl

ass

(a) Number: 1 to 1

2.5 5 7.5 10 12.5

d

Rel

ativ

e%

incl

ass

(b) Area: 1 to 4

2.5 5 7.5 10 12.5

d

Rel

ativ

e%

incl

ass

(c) Volume: 1 to 8

Figure 1.2: Example of number, area, and volume weighted particle size distributionsfor the same sample. Numbers in the sub-captions relate the the area under each of thecurves.

age of the particles which are at that size, while the cumulative distribution represents thepercentage of particles smaller than the size (hence often called undersize).

Often particle size distributions are given in terms of the cumulative distribution as forseparations it is useful to know what fraction of the particles are above or below a crit-ical size. It is possible to convert between the frequency, f (d), and cumulative, F (d),distributions as,

F (d) =

∫f (d) d d (1.4.1)

Figure 1.3 shows the frequency distribution and cumulative distribution of the same sam-ple.

0

1

2

3

4

d

f(d

)=

dF

(d)/

dd

(a) Frequency Distribution

0

0.25

0.5

0.75

1

d

F(d

)

(b) Cumulative Distribution

Figure 1.3: Comparison of frequency and cumulative distribution for the same sample.



1.4.3 Particle Size Distribution Functions

There are many options for particle size distribution functions, some of the commonlyused functions for solid particles are:

Normal distribution, this is symmetric about its mean value and is a solution to the aver-age of many samples of a random variable with finite mean and variance. Therefore,physical quantities that are expected to be the sum of many independent processesoften have distributions that are nearly normal [4]. The value of the normal distri-bution is practically zero when the value lies more than a few standard deviationsaway from the mean (e.g. a spread of three standard deviations covers all but 0.27%of the total distribution). Therefore, it may not be an appropriate model when oneexpects a significant fraction of outliers.

F (d) =1

2

[1 + erf

(d− d50√

2σ

)](1.4.2)

Log-normal distribution, this is similar to the normal distribution; however, due to thelog scale the distribution can vary over several orders of magnitude. This distribu-tion has been shown to fit experimental results across a wide range of applicationsproducing particle size distributions.

F (d) =1

2

[1 + erf

(1√2σ

ln

(d

d50

))](1.4.3)

Rosin-Rammler distribution, this is an empirical correlation typically used in crushing,milling, and grinding of solid particles. It was developed in 1933 by Rosin andRammler [5], and because of its relatively good fit with shredded refuse has becomewidely used in the resource recovery field.

F (d) = 1− exp

(− d

d63.2

)n(1.4.4)

Gaudin-Schumann distribution, this is an empirical correlation that is typically used inthe same applications as the Rosin-Rammler distribution. However this is generallyused across the metalliferous mining industry [6].

F (d) =

(d

dmax

)n(1.4.5)

Harris distribution, is similar to the Rosin-Rammler distribution apart from it has beendeveloped for situations were the Rosin-Rammler distribution did not fit the exper-imental results with enough accuracy [2].

F (d) = 1−(

1− exp

(d

dmax

)n)m(1.4.6)


1.4. PARTICLE SIZE DISTRIBUTIONS

1.4.4 Means and Percentiles

There are many different means that can be defined depending upon how the distributionis analysed. The general expression for any mean can be given by,

da,b =

∫f (d) dad d∫f (d) dbd d

1/(a−b)

a 6= b

exp

∫f (d) da ln d d d∫f (d) dad d

a = b

(1.4.7)

Typically used means can be seen in Table 1.2 and displayed for the number distributionin Figure 1.2(a) on Figure 1.4. Means dependent on the volume and surface area are morestrongly weighted towards the higher particle diameters as these have more surface areaand volume. The Sauter mean diameter is the specific surface area mean, so is relevantfor processes such as bioavilability, reactivity, and dissolution. The de Broukere meanreflects the size of the particles which constitute the bulk of the sample volume.

Table 1.2: Typically used particle size means.

Mean Description

d0,0 Geometric meand1,0 Arithmetic or number meand2,0 Surface meand3,0 Volume meand3,2 Volume/surface mean or Sauter meand4,3 de Broukere mean

2.5 5 7.5 10 12.5

d1,0

d2,0

d3,0

d3,2

d4,3

d

Rel

ativ

e%

incl

ass

Figure 1.4: Weighted number distribution from Figure 1.2(a) with means from Table 1.2shown.



It is also useful to know parameters based on the fraction of particles above or belowcertain values. The most common of these are d10, 10% of the particles are smaller thanthis diameter, d50, 50% of the particles are smaller or larger than this diameter i.e. themedian, and d90, 10% of the particles are larger than this diameter.

1.5 Particle Measurement Methods

A number of particle measurement methods and their measurement size ranges are givenin Table 1.3. The correct method must be selected for the application of interest.

Table 1.3: Typically used particle size means.

0.1 nm 1 nm 10 nm 100 nm 1µm 10µm 100µm 1 mm 10 mm

SievingSedimentationParticle countingLaser DiffractionDynamic light scattering

1.5.1 Sieving Methods

A sieve analysis is a procedure used to assess the particle size distribution of a granu-lar material by allowing the material to pass through a series of sieves of progressivelysmaller mesh size and weighing the amount of material that is stopped by each sieve as afraction of the whole mass. A sieve analysis can be performed on any type of non-organicor organic granular materials including sands, crushed rock, clays, granite, feldspars, coal,soil, a wide range of manufactured powders, grain and seeds, down to a minimum sizedepending on the exact method. Being such a simple technique of particle sizing, it isprobably the most common.

The column of sieves are typically placed in a mechanical shaker, Figure 1.5(a). Theshaker shakes the column, usually for some fixed amount of time. The results of thistest are provided in graphical form to identify the type of distribution of the particles,Figure 1.5(b). The average particle size for each section is the geometric mean of thesieve sizes. As particles may not be spherical then the diameters are effective diameters.Sieves are typically given a Mesh number (Mesh = number of holes per inch).

Although this technique is typically used in a lab as a batch technique to measure theparticle distribution of samples it is sometimes used as an industrial separation technique,for example separation of titanium from alloy powders and tea leaves from waste.

1.5.2 Sedimentation Methods

Gravitational sedimentation (originally called the pipette method) measures the settlingrate of particles in liquid medium and relates this rate to the particle mass (hence size) byuse of the Stokes law, see Chapter 2. Pipette withdrawals of a precise volume are takenfrom a suspension column at known depths and times after stirring ceases. The times and


1.5. PARTICLE MEASUREMENT METHODS

0µm25/2

25µm

√50× 25

50µm

√100× 50

100µm100×

√2

(a)

0 25 50 75 100 125 1500

20

40

60

80

100

Size / µm

%of

Mas

s

(b)

Figure 1.5: Particle size collection from a seive, (a) the seive system and (b) the resultingfrequency distribution (bars) and cumulative distribution (points).

depths are generally predetermined to sample specific maximum sizes. In the solution ofStokes Law, elapsed time and depth determine the largest particle diameter represented bya particular pipette fraction: particles finer than that diameter will be present but coarserparticles will have settled below the depth of withdrawal.

From the dry weight of each fraction (collected in the pipette), the cumulative weightpercentage is derived for the corresponding diameter. Currently, the concentration duringsedimentation analysis is determined by measuring x-ray/light transmission in the liquidat specific heights and time intervals. This allows high-resolution measurements within arelatively short time frame.

For smaller particles this technique can be coupled with centrifuging to speed up thesedimentation.

1.5.3 Particle Counting

Originally images of particles would be counted and measured by hand, now this processis often automated by computer. Individual particle images are captured from dispersedsamples and analyzed to determine their particle size, particle shape and other physicalproperties. Statistically representative distributions can be constructed by measuring tensto hundreds of thousands of particles per measurement. Static imaging systems require astationary dispersed sample whereas in dynamic imaging systems the sample flows pastthe image capture optics. The technique is often used in conjunction with ensemble basedparticle sizing methods such as laser diffraction, to gain a deeper understanding of thesample or to validate the ensemble based measurements.

Different size particles can be analysed using different imaging techniques, although typ-ically used with a microscope, it can be used with Scanning Electron Microscopy.



1.5.4 Light Scattering

There are several different techniques that use light; however, two most commonly usedoptions are laser diffraction and dynamic light scattering.

Laser Diffraction

Laser diffraction is a widely used particle sizing technique for materials ranging fromhundreds of nanometers up to several millimeters in size. The main reasons for its successare:

• wide dynamic range - from submicron to the millimetre size range

• rapid measurements - results generated in less than a minute

• repeatability - large numbers of particles are sampled in each measurement

• instant feedback - monitor and control the particle dispersion process

• high sample throughput - hundreds of measurements per day

Laser diffraction measures particle size distributions by measuring the angular variation inintensity of light scattered as a laser beam passes through a dispersed particulate sample.Large particles scatter light at small angles relative to the laser beam and small particlesscatter light at large angles. The angular scattering intensity data is then analyzed tocalculate the size of the particles responsible for creating the scattering pattern, usingthe Mie theory of light scattering. The particle size is typically reported as a volumeequivalent sphere diameter.

Dynamic Light Scattering

Dynamic light scattering, sometimes referred to as Photon Correlation Spectroscopy, isa non-invasive, well established technique for measuring the size of particles and macro-molecules typically in the submicron region down to below 1 nanometre. It can be usedto measure samples which consist of particles suspended in a liquid e.g. proteins, poly-mers, micelles, carbohydrates, nanoparticles, colloidal dispersions, and emulsions. Keyadvantages include:

• particle size range ideal for nano and biomaterials

• small quantity of sample required

• fast analysis and high throughput

• non-invasive allowing complete sample recovery.

Particles in suspension undergo Brownian motion caused by thermally induced collisionsbetween the suspended particles and solvent molecules. If the particles are illuminatedwith a laser, the intensity of the scattered light fluctuates over very short timescales at arate that is dependent upon the size of the particles; smaller particles are displaced furtherby the solvent molecules and move more rapidly. Analysis of these intensity fluctuationsyields the velocity of the Brownian motion and hence the particle size using the Stokes-Einstein relationship.


1.6. REFERENCES

The diameter measured in Dynamic Light Scattering is called the hydrodynamic diameterand refers to the way a particle diffuses within a fluid. The diameter obtained by thistechnique is that of a sphere that has the same translational diffusion coefficient as theparticle being measured. The translational diffusion coefficient will depend not only onthe size of the particle ’core’, but also on any surface structure, as well as the concentrationand type of ions in the medium.

1.5.5 Comparison of Methods

Due to the different properties used to measure particle sizes by different methods, thedistribution measured can be different. Also if there are a particles over a wide range ofsizes, then multiple methods may need to be used to generate the full distribution. Forexample, Figure 1.6 shows how several different techniques can be used to find the fullparticle size distribution for a sample of soil where the sizes range over five decades ofsize.

10−6 10−5 10−4 10−3 10−2 10−10

0.25

0.5

0.75

1

r / cm

F(r

)

SievePipetteGravitational sedimentationCentrifugational sedimentationStatic light scatteringDynamic light scattering

Figure 1.6: Comparison of particle size measurement methods for a sample of soil,adapted from [8].

1.6 References

[1] Allen, T. [1997], Particle Size Measurement, Vol. 1 and 2, 5th edn, Chapman andHall, London.

[2] Harris, C. C. [1960], ‘Alternative forms of some exponential and power functionsfor graphical representation’, Journal of Applied Physics 31, 215.

[3] IUPAC, ed. [1997], Compendium of Chemical Terminology (the "Gold Book"), 2nded edn.

[4] Lyon, A. [2014], ‘Why are normal ddistribution normal?’, The British Journal forthe Philosophy of Science 65, 621–649.



[5] Rosin, P. and Rammler, E. [1933], ‘Laws governing the fineness of powdered coal’,Journal of the Institute of Fuel 7, 29–36.

[6] Schuhmann, R. [1940], Technical report publication no. 1189, Technical report,American Institute of Mining and Metallurgical Engineers, New York.

[7] Wadell, H. [1935], ‘Volume, shape and roundness of quartz particles’, he Journal ofGeology 43, 250–280.

[8] Wu, Q., Borkovec, M. and Sticher, H. [1993], ‘On particle-size distributions insoils’, Soil Science Society of America Journal 57, 883–890.


1.7. PROBLEMS

1.7 Problems

1.1 Calculate the sphericity of a cube of edge length of a, and a circular cylinder with adiameter of d and the height h (d = 1.5h)?



1.2 A packed bed is composed of cylindrical particles having a diameter of 0.02 m anda length equal the diameter. The density of the solid cylinders is 1600 kg m−3.Calculate:

(a) the specific surface are of the particle (asv)

(b) the specific surface are of the bed, assuming a porosity of 0.4


1.7. PROBLEMS

1.3 In the plastic recycling industries, after different types of plastics have been meltedand mixed, small cylindrical pellets are made of the new plastic mixture to deter-mine the characteristics of the new plastic. Plastic pellets are typically 1 mm inradius and 2 mm in length. What will be the equivalent diameters of a typical pelletin this application? Calculate and compare the values of the equivalent sphericaldiameters:

(a) volume (dv)

(b) surface (ds)

(c) surface to volume ratio (dsv)



1.4 Consider a cuboid particle 5× 3× 1 mm3. Calculate for this particle the followingdiameters:

(a) volume (dv)

(b) surface (ds)

(c) surface to volume ratio (dsv)


1.7. PROBLEMS

1.5 A sieve analysis of a catalyst gives the mass distribution below. Calculate and plotthe frequency and cumulative particle size distribution.

Channel size / µm Mass / g

10 50.5015 47.2120 29.9030 18.5045 4.3055 1.0075 0.40

100 0.00



1.6 Based on the data below, what is the mean size by number and by mass?

Size range / µm < 5 5− 7 7− 10 10− 15 15− 20 20− 30 30− 40 40− 50 > 50

Number in Range 0 50 150 200 55 45 20 5 0Relative number 0 0.0952Cumulative no undersize 0 0.0952 1.000Relative no. per µm 0 0.0476(mid point)3× frequency 0 10353Relative mass 0 0.0038Cumulative mass undersize 0 0.0038 1.000


Chapter 2Single Particle Motion in Fluids


2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3 Flow Past a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.1 Stokes’ Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4 Drag Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4.1 Drag Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5 Terminal Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5.1 Stokes’ Terminal Velocity . . . . . . . . . . . . . . . . . . . . 28

2.5.2 General Terminal Velocity . . . . . . . . . . . . . . . . . . . . 28

2.6 Non-spherical Particles . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.7 Transient Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

21

CHAPTER 2. SINGLE PARTICLE MOTION IN FLUIDS

2.1 Chapter 2 ILOs

.

ILO 2.1. Describing the motion of a particle of regular shape in a fluid moving under theeffect of gravity in different flow regimes.

ILO 2.2. Determining terminal settling velocities of particles falling under different flowregimes.

ILO 2.3. Calculate the time until terminal velocity.


2.2. INTRODUCTION

2.2 Introduction

Processes for the separation of particles of various sizes and shapes often depend on thevariation in the behaviour of the particles when they are subjected to the action of a mov-ing fluid or have to move within a fluid subjected to an external force e.g. gravity.

2.3 Flow Past a Sphere

For a non-viscous fluid flowing past a sphere, as shown in Figure 2.1, the velocity anddirection of flow varies around the circumference. Thus at A the fluid is brought to restand at B and C the velocity is at a maximum. The pressure falls from A to B and fromA to C and rises again from B to D and C to D. The pressure A and D are the same andthus there is no net force on the sphere. Although the predicted pressure variation for anon-viscous fluid agrees well with the results obtained for a viscous fluid over the frontface (B-A-C), there are large differences over the rear face (B-D-C).

A

B

D

C

Figure 2.1: Flow around a sphere.

When a viscous fluid flows over a surface, the fluid is retarded in the boundary layerwhich is formed near the surface and that the boundary layer increases in thickness withincrease in distance from the leading edge. If the pressure is falling in the direction offlow, the retardation of the fluid is less and the boundary layer is thinner in consequence.If the pressure is rising there will be greater retardation and the thickness of the boundarylayer increases more rapidly. The force acting on the fluid in the boundary layer may thenbe sufficient to bring it to rest or cause flow in the reverse direction with the result thatan eddy current is set up. A region of reverse flow then exists near the surface where theboundary layer has separated as shown in Figure 2.2.

u u u u u u

Forward Flow

Reversed Flow

Direction of increasing pressure

Figure 2.2: Flow of a fluid over a surface against a pressure gradient. The dotted line isthe start of the separated zone and has as velocity parallel to the surface of 0.



For the flow of a viscous fluid past the sphere, the pressure decreases from A to B and Ato C so that the boundary layer is thin. From B to D and C to D the pressure is increasingand therefore the boundary layer rapidly thickens with the result tht it tend to separatefrom the surface. If separation occures, eddies are formed in the wake of the sphere andenergy is dissipared and an aditional force, known as form drag, develops.

All items immersed in a fluid are subject to a buoyancy force and typically a gravity force.In a flowing fluid, or an item moving in a fluid, there is an addition drag force which ismade up of two components, a skin friction (of viscous drag) and the form drag. At lowvelocities no separation of the boundary layer takes place. As the velocity is increasedthe separation occurs and the skin friction forms a gradual decreasing proportion of thetotal drag force. If the velocity is very high then the flow within the boundary layer willchange from streamline (laminar) to turbulent before this separation takes place. Since therate of momentum transfer through a fluid in the turbulent regime is much greater thanthat in laminar flow, separation is less likely to occur, as the faster flowing outside fluidis able to keep the fluid within the boundary layer moving in the forward direction. Ifseparation does occur then this takes place much closer to D and the resulting eddies aresmaller so the total drag force is reduced. This can be seen in Figure 2.3. The tendencyfor separation, and hence the magnitude of the form drag, are also dependent on the shapeof the body.

(a) Rep = 0.1 (b) Rep = 1

(c) Rep = 500 (d) Rep = 20 000

Figure 2.3: Photos of flow of a fluid over a sphere at different particle Reynolds numbers[5].

As with flow in a pipe the flow can be characterised by the Reynolds number, thoughbased on the particle,

Rep =udρfµ

(2.3.1)


2.4. DRAG FORCE

2.3.1 Stokes’ Law

For the case of creeping flow, i.e. very low velocities, the drag force, FD, on a sphericalparticle has been calculated by solving the hydrodynamic equations of motion, the Navier-Stokes equation, to give [6],

FD = 3πµdpu (2.3.2)

Equation 2.3.2 is know as Stokes’ Law and is applicable at very low values of the particleReynolds number. In this regime the skin friction constitutes two-thirds of the total dragon the particle and the form drag the remainder. As the Rep increases the skin frictionbecomes proportionally less and at values greater than about 20 flow separation occurswith the formation of vortices progressively increasing.

2.4 Drag Force

The best method to represent the relationship between the drag force and the velocityinvolves the use of two dimensionless groups, similar to those used for correlating thepressure drop for the flow of fluids in pipes.

The first group is that as discussed above, the particle Reynolds number. The second isthe group R′/ρu2, where R′ is the force per unit projected area of the particle in a planeperpendicular to the direction of motion,

R′ =FDA⊥

(2.4.1)

This means that for a sphere,

R′

ρfu2=

FDA⊥ρfu2

=4FD

πd2pρfu2

(2.4.2)

R′/ρu2 is a form of drag coefficient, often denoted at C ′D. Frequently, a drag coefficientCD is defined as the ratio of R′ to the dynamic pressure (1/2ρfu2),

CD = 2C ′D =2R′

ρfu2=

8FDπd2pρfu

2(2.4.3)

CD is therefore analogous to the Fanning fraction factor, f , for pipe flow. This means thatthe drag force can be written in terms of the drag coefficient as,

FD = CD

(π4d2p

) ρfu22

(2.4.4)

2.4.1 Drag Coefficient

The value of the drag coefficient is dependant on the particle Reynolds number due to thechanging proportions of the contribution of the skin friction and the form drag.



At low Reynolds number we have seen that the drag force can be given by Stokes’ law,Equation 2.3.2. Substituting this into Equation 2.4.4, the drag coefficient can be seen tobe given by,

FD = CD

(π4d2p

) ρfu22

= 3πµdpu

CD =24µ

ρfudp=

24

Rep(2.4.5)

This relationship works up until a particle Reynolds number of about 0.3.

The drag coefficient variation with particle Reynolds number has been measured experi-mentally and can be seen in Figure 2.4. A large number of equations have been proposedto allow calculation of the drag coefficient from the particle Reynolds number [2]. How-ever, a useful set of equations for ease of calculation can be given by,

CD =

24

RepRep < 0.3

24

Rep

(1 + 0.15Re0.687p

)0.3 ≤ Rep < 500

0.44 500 ≤ Rep < 4× 105

0.19 Rep ≥ 4× 105

(2.4.6)

It should be noted that the boundary between the region called the Newton regime (500 ≤Rep < 4 × 105 and the fully turbulent regime Rep ≥ 4 × 105 for a sphere is determinedby the roughness of the sphere surface. The rougher the sphere surface the lower thetransition particle Reynolds number.

10−4 10−2 100 102 104 106 108 101010−2

100

102

104

106

Rep

CD

Figure 2.4: Variation of CD with Rep for spherical particles with regions shown as inequation 2.4.6.

2.5 Terminal Velocity

If a spherical particle is allowed to settle in a fluid under gravity, its velocity will increaseuntil the accelerating force is exactly balanced by the resistance force. At this point the


2.5. TERMINAL VELOCITY

particle falls with its terminal velocity. The forces on the falling particle, Figure 2.5, arethe accelerating gravity force,

d

FDFb

Fg

Figure 2.5: Forces on a falling sphere.

Fg = mg = ρpVpg = ρpπd3p6g (2.5.1)

the buoyancy force,

Fb = mfg = ρfVpg = ρfπd3p6g (2.5.2)

and the drag force, equation 2.4.4. At the terminal velocity the forces on the particle arebalanced so that, ∑

i

Fi = 0 = Fg − Fb − FD (2.5.3)

This means that,

Fg − Fb = FD

ρpπd3p6g − ρf

πd3

6g = CD

(π4d2p

) ρfu2t2

πd3p6g (ρp − ρf ) = CD

(π4d2p

) ρfu2t2

(2.5.4)

So the drag coefficient can be given in terms of the terminal velocity,

CD =4dp (ρp − ρf ) g

3ρfu2t(2.5.5)

This expression is true under the following assumptions:

• the settling is not affected by the presence of other particles in the fluid. Thiscondition is known as “free settling”.

• The walls of the settling vessel do not exert an appreciable retarding effect.

• The fluid can be considered as a continuous medium, that is the particle is largecompared with the mean free path of the molecules of the fluid.



2.5.1 Stokes’ Terminal Velocity

At low particle Reynolds number, then it is known that the force is due to Stokes’ law,thus the drag coefficient is given by Equation 2.4.5. Substituting this relation into Equa-tion 2.5.5 means that,

CD =4dp (ρp − ρf ) g

3ρfu2t=

24

Rep=

24µ

ρfutdp(2.5.6)

Which can be rearranged for the Stokes’ terminal velocity,

ut =d2p (ρp − ρf ) g

18µ(2.5.7)

2.5.2 General Terminal Velocity

The above analysis to find the terminal velocity is true only for the Stokes’ law regime;however, any expression for the drag coefficient from Equation 2.4.6 can be taken de-pending on the falling regime. This means that the regime must be known before pickingthe correct expression for the drag coefficient, or an iterative procedure can be used.

This problem is most effectively solved by defining a new dimensionless number whichis independent of the particle velocity, this can be done by combining the drag coefficientand the Reynolds number as,

CDRe2p =4dp (ρp − ρf ) g

3ρfu2

(ρfudpµ

)2

CDRe2p =4

3

d3p (ρp − ρf ) ρfgµ2

(2.5.8)

The group on the right hand side of Equation 2.5.8 is dimensionless and can be defined asthe Archimedes number (sometimes referred to as the Galileo number, Ga),

Ar =d3p (ρp − ρf ) ρfg

µ2(2.5.9)

Substituting Equation 2.5.9 into 2.5.8 then produces the general terminal velocity expres-sion,

CDRe2p =4

3Ar (2.5.10)

For example for Stokes’ law, using the drag coefficient from Equation 2.4.5, Equation 2.5.10can be simplified to be,

CDRe2p =24

RepRe2p =

4

3Ar

Ar = 18Rep (2.5.11)


2.6. NON-SPHERICAL PARTICLES

Using the drag coefficients from Equation 2.4.6, expressions for the Archimedes numberin terms of the Reynolds number can be created, as seen in Figure 2.6,

Ar =

18Rep Ar < 3.6

18Rep + 2.7Re1.687p 3.6 ≤ Ar < 1× 105

1

3Re2p Ar ≥ 1× 105

(2.5.12)

10−4 10−2 100 102 104 106 108 1010

10−4

10−2

100

102

104

106

Ar

Re p

Figure 2.6: Variation of Rep with Ar for spherical particles showing the 3 key regions asin equation 2.5.12.

2.6 Non-spherical Particles

There are two difficulties when trying to use the above equations on real particles. Thefirst is that an infinite number of non-spherical shapes exist, and the second is that each ofthese shapes is associated with an infinite number of orientations which the particle is freeto take up in the fluid, and the orientation may oscillate during the course of settling.

To allow calculation of the drag coefficient for non-spherical particles some parametersto assess the particle shape need to be picked. Based on experimental data the mostappropriate particle diameter is the equivalent volume sphere diameter, Equation , and theextent of departure from a spherical shape is represented by the sphericity, Equation 1.3.1[1].

On of the key equations developed for non-spherical particles is Equation 2.6.1 [3], whichhas good prediction for experimental data as seen in Figure 2.7.

CD =24

Rep

[1 + exp

(2.3288− 6.4581ϕ+ 2.4486ϕ2

)Re0.0964+0.5565ϕ

p

]+

Rep exp (4.9050− 13.8944ϕ+ 18.4222ϕ2 − 10.2599ϕ3)

Rep + exp (1.4681 + 12.2584ϕ− 20.7322ϕ2 + 15.8855ϕ3)(2.6.1)



10−2 10−1 100 101 102 103 104 105 10610−1

100

101

102

103

104

105

Rep

CD

ϕ = 0.026ϕ = 0.043ϕ = 0.123ϕ = 0.230ϕ = 0.670ϕ = 0.806ϕ = 0.846ϕ = 0.906ϕ = 1.000

Figure 2.7: Reported drag coefficients for spherical particles and non-spherical particlesfrom [3]. Hollow shapes for isometric particles and filled shapes for disc like particles.Lines are taken from equation 2.6.1.

A more complicated expression for the drag coefficient for non-spherical particles is givenas Equation 2.6.2, where the parallel (‖) and perpendicular (⊥) sphericity are factored inas separate variables to take into account the direction of flow [4].

CD =8

Rep

1√ϕ‖

+16

Rep

1√ϕ

+3√Rep

1

ϕ3/4+ 0.42100.4(− logϕ)0.2 1

ϕ⊥(2.6.2)

2.7 Transient Effects

Although for most particle systems that need to be separated the time taken for the parti-cles to get to their terminal velocity is short and thus can be ignored, it is useful to be ableto check that particles are at their terminal velocity. In the case of a particle accelerating toits terminal velocity for forces on the particle, Figure 2.5, are not balanced as thus,∑

i

Fi = Fg − Fb − FD = mpdu

d t(2.7.1)

Substituting the forces into Equation 2.7.1 from Equation 2.5.1, 2.5.2, and 2.4.4 gives,

ρpπd3p6

du

d t=πd3p6g (ρp − ρf )− CD

(π4d2p

) ρfu22

(2.7.2)

If the particle starts from a velocity of 0, then the initial drag coefficient can be given beStokes’ law (also for most separation systems the particles are in Stokes’ regime), thismeans that the drag coefficient can be given by Equation 2.4.5, so that,

ρpπd3p6

du

d t=πd3p6g (ρp − ρf )− 3πµudp (2.7.3)


2.7. TRANSIENT EFFECTS

This can be rearranged to give,

d2pρp

18µ

du

d t=d2p (ρp − ρf )

18µg − u = ut,St − u (2.7.4)

where ut,St is the terminal velocity in the Stokes’ regime. This equation can now beintergrated between u = 0 at t = 0 and u = u at t = t,∫ t

0

18µ

ρpd2d t =

∫ u

0

du

ut,St − u18µt

ρpd2= [− ln (ut,St − u)]u0

18µt

ρpd2= ln

(ut,St

ut,St − u

)(2.7.5)

At this point it is interesting to note that the Fourier number (Fo), ratio of the transportrate to the storage rate, is given by,

Fo =µt

ρpd2p(2.7.6)

This means that substituting Equation 2.7.6 into Equation 2.7.5 gives,

18Fo = − ln

(1− u

ut,St

)(2.7.7)

Which can be rearranged for the ratio of the actual velocity to the terminal velocityas,

u

ut,St= 1− exp (−18Fo) (2.7.8)

If the Fourier number is greater than 0.25 the velocity is greater than 0.99ut,St thereforecan be considered to be the terminal velocity, as in Figure 2.8.

If the velocity of the particle makes the particle Reynolds number exceed 0.3, then thedrag coefficient substituted into Equation 2.7.2 will have to change meaning that the timeto the terminal velocity will increase 1.

1The expression for the time changes to,

18Fo =

∫ u

0

1

ut,St −CD

CD,Studu

whereCD/CD,St is ratio of the actual drag coefficient to the Stokes’ law drag coefficient, and is a functionof the particle viscosity



10−5 10−4 10−3 10−2 10−1 10010−4

10−3

10−2

10−1

Fo

u/u

t,St

Figure 2.8: Variation ofu

ut,Stwith Fo for spherical particles.

2.8 References

[1] Chhabra, R. P., L..Agarwal and K.Sinha, N. [1999], ‘Drag on non-spherical parti-cles: an evaluation of available methods’, Powder Technology 101, 288–295.

[2] Cliff, R., Grace, J. R. and Weber, M. E. [1978], Bubbles, Droplets and Particles,Academic Press.

[3] Haider, A. and Levenspiel, O. [1989], ‘Drag coefficient and terminal velocity ofspherical and nonspherical particles’, Powder Technology 58, 63–70.

[4] Hölzer, A. and Sommerfeld, M. [2008], ‘New simple correlation formula for thedrag coefficient of non-spherical particles’, Powder Technology 184, 361–365.

[5] Milne-Thomson, L. M. [1968], Theoretical Hydrodynamics, fifth edition edn,MacMillan & Co Ltd.

[6] Stokes, G. G. [1851], ‘On the effect of internal friction of fluids on the motion ofpendulums’, Transactions of the Cambridge Philosophical Society 9(Part ii), 8–106.


2.9. PROBLEMS

2.9 Problems

2.1 Use Stokes’ law to calculate the time needed for the following particles to settle adepth of 0.2 m in an aqueous suspension at 25 ◦C.

(a) particles of diameter 50 microns

(b) particles of diameter 2 micron

µ = 0.89 mPa s, ρp = 2650 kg m−3, and ρf = 1000 kg m−3



2.2 Use Stokes’ law to calculate the terminal velocity of particles with diameter 40 mm,density 2000 kg m−3 falling through water with density 998 kg m−3 and viscosity1 mPa s. Is Stokes’ Law the correct equation to use in this case?


2.9. PROBLEMS

2.3 Calculate the upper limit of particle diameter dmax as a function of particle densityρp for sedimentation in the Stokes’ law regime. Assume the particle is sphericaland settling in air with density 1.2 kg m−3 and viscosity 1.84 × 10−5 Pa s and thatStokes’ law holds for Re < 0.3.

Dust particles have a density of 1490 kg m−3, what is the largest dust particle thatobeys Stokes’ law?



2.4 Calculate the terminal velocity of a steel ball (density 7870 kg m−3) of 2 mm diam-eter settling in oil (density 900 kg m−3, viscosity 50 mPa s).


2.9. PROBLEMS

2.5 The drag coefficient CD of a sphere of diameter d due to its motion with velocity utthrough a fluid of density ρf and viscosity µ varies with Reynolds number as givenbelow

log Re 2 2.5 3 3.5 4

CD 1.050 0.630 0.441 0.385 0.390

Find the mass of a sphere of 0.008 m diameter which falls with a steady velocityof 0.6 m s−1 in a large deep tank of water of density 1000 kg m−3 and viscosity0.0015 Pa s.



2.6 A sphere of density 2500 kg m−3 falls freely under gravity in a fluid of density700 kg m−3 and viscosity 5 × 10−4 Pa s. Given that the terminal velocity of thesphere is 0.15 m s−1, calculate its diameter.

10−4 10−2 100 102 104 106 108 101010−2

100

102

104

106

Rep

CD


2.9. PROBLEMS

2.7 A particle of diameter 5 × 10−5 m with density 3000 kg m−3 settles in a fluid ofviscosity 0.001 Pa s and density 1000 kg m−3. How long does it take the particle toreach a velocity of 0.002 m s−1?



2.8 A particle of diameter 1 mm with density 2500 kg m−3 settles in a fluid of viscosity0.01 Pa s for 1 second. Has this reached its terminal velocity?


Chapter 3Settling and Settlers


3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3 Settling Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3.1 Type I - Discrete Particle Settling . . . . . . . . . . . . . . . . 44

3.3.2 Type II - Flocculent Particles . . . . . . . . . . . . . . . . . . . 44

3.3.3 Type III - Hindered Settling . . . . . . . . . . . . . . . . . . . 44

3.3.4 Type IV - Compression . . . . . . . . . . . . . . . . . . . . . . 45

3.4 Gravity Separators . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.5 Design of Gravity Separators . . . . . . . . . . . . . . . . . . . . . . 47

3.5.1 Limit for Low Solids Concentration - Rectangular . . . . . . . 47

3.5.2 Limit for Low Solids Concentration - Circular . . . . . . . . . . 48

3.5.3 Mass Balance of Solids . . . . . . . . . . . . . . . . . . . . . . 49

3.6 Accelerating Settling . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.6.1 Flocculation . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.7 Settler Rules of Thumb . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

41

CHAPTER 3. SETTLING AND SETTLERS

3.1 Chapter 3 ILOs

.

ILO 3.1. Examine the settling process.

ILO 3.2. Design settling tanks of varying concentrations.


3.2. INTRODUCTION

3.2 Introduction

In Chapter 2 equations were developed for the motion of an isolated particle moving in afluid. If the particle is settling in a gravitational field, it rapidly reaches its terminal ve-locity. In practice, the concentrations of suspensions used in industry will usually be highenough for there to be significant interaction between the particles. This could mean thatthe drag force is effectively increased thus slowing the settling, or particles join togethereffectively increasing the size and thus the the settling.

The generally settling process can be seen in Figure 3.1. Initially there is a brief accelera-tion period in the interface between the clear liquid (supernatant) and the suspension (A),then the suspension moves downwards at a constant rate and a layer of sediment buildsup at the bottom of the container (B and C). At the critical sedimentation point this inter-face meets the sediment layer (D). From then on the level only changes due to the furthercompacting of the sediment layer with liquid being forced upwards around the solids arethen forming a bed in which the particles are in contact with one another (E).

•Critical Sedementation point

Clear supernatant

Suspension

Sediment Compacted Sediment

A B C D E

Time

Inte

rfac

ehe

ight

A B C D E

Figure 3.1: Variation of the interface heights during the settling process. Black repre-sents the compacted sediment, dark grey represents the sediment, light grey representsthe suspension, and white is the clear supernatant.



3.3 Settling Types

Depending on the concentration of solids and the tendency of particles to interact fourtypes of settling may occur.

3.3.1 Type I - Discrete Particle Settling

In discrete settling, particles settle as individual entities, and there is no significant inter-action with neighbouring particles. Discrete particles have little tendency to flocculate orcoalesce upon contact with each other and hence they do not change their size, shape ormass during settling. Discrete settling refers to the sedimentation of particles in a suspen-sion of low solids concentration and has a settling velocity which can be represented byStokes’ law, Equation 2.5.7.

Grit in sewage behave like discrete particles and hence their settling in grit chamberscorresponds to discrete settling.

3.3.2 Type II - Flocculent Particles

In flocculent settling, particles flocculate or coalesce during settling. By flocculation orcoalescing, the particles increase in mass and thus settle at a faster rate. Flocculent settlingrefers to the sedimentation of particles in a rather dilute suspension with concentration ofsolids usually less than 1000 mg l−1.

The degree of flocculation depends on the contact opportunities which in turn are affectedby the surface overflow rate, the depth of the basin, the concentration of the particles, therange of particle sizes and the velocity gradient in the system. Often flocculation is bene-ficial to the settling process as it speeds it up and this is discussed in Section 3.6.1.

The removal of organic suspended solids from raw or untreated sewage in primary settlingtanks, settling of chemical floes in settling tanks and of bioflocs in the upper portion ofsecondary settling tanks are examples of flocculent settling.

3.3.3 Type III - Hindered Settling

When concentration of particles is in an intermediate range, they are close enough to-gether so that inter-particle forces are sufficient to hinder the settling of neighbouringparticles resulting in hindered settling. The particles maintain their relative positions withrespect to each other and the whole mass of particles settles as a unit or zone, especiallyif the particles are of a similar size (size range < 6 : 1)

This type of settling is applicable to concentrated suspensions such as are found in sec-ondary settling tanks used in conjunction with biological treatment units such as tricklingfilters and activated sludge units. In the hindered settling zone, the concentration of parti-cles increases from top to bottom leading to thickening of sludge.

A simple modification to Stokes’ law, Equation 2.5.7, has been suggested where the den-sity and viscosity of the suspension are taken rather than the properties of the fluid [2],such that,


3.3. SETTLING TYPES

us =d2p (ρp − ρs) g

18µs(3.3.1)

The density difference can then be given by,

(ρp − ρs) = ρp − (ρp (1− ε) + ρfε) = ε (ρp − ρf ) (3.3.2)

where ε is the voidage of the suspension, and the concentrated suspension viscosity canbe approximated by [3],

µs = µ exp

(k1φp

1− k2φp

)(3.3.3)

This means that the ratio of the hindered settling velocity to the free settling velocity canbe written as,

usut

= ε exp

(− k1φp

1− k2φp

)(3.3.4)

Another option that is commonly used is a simplified empirical version [1] of,

usut

= k (1− φp)n (3.3.5)

where k is a dimensionless multiplier in the range 0.8-1 and n is the model exponenttypically found in the range of 2-5 for hard spheres, but can be much higher for aggregatedmineral suspensions > 100.

A third correlation is popular in the waste-water treatment industry [4],

usut

= k exp (−nφp) (3.3.6)

The key in this region is that due to complexity, experimental settling data is really neededand the above correlations can be fitted to that data to help in modelling systems.

3.3.4 Type IV - Compression

This refers to settling in which the concentration of particles is so high that particles arein physical contact with each other resulting in the formation of a structure with lowerlayers supporting the weight of upper layers. Consequently further settling occurs dueto compression of the whole structure of particles and accompanied by squeezing out ofwater from the pores between the solid particles.

Compression takes place from the weight of particles which are constantly being added tothe structure by sedimentation from the supernatant liquid. Compression settling usuallyoccurs in the lower layers of a deep sludge mass, such as in the bottom of secondary set-tling tanks following biological treatment by trickling filters and activated sludge process,and in tanks used for thickening of sludge.



3.4 Gravity Separators

Sedimentation tanks, also called settling tanks or clarifiers, are a component of a modernwater supply systems or wastewater treatment, plus others. A sedimentation tank allowssuspended particles to settle out of the liquid as it flows slowly through the tank, therebyproviding some degree of purification. A layer of accumulated solids, called sludge, formsat the bottom of the tank and is periodically removed.

These tanks may function either intermittently (batch) or continuously. The intermittenttanks are those which store water for a certain period and keep it in complete rest. In acontinuous flow type tank, the flow velocity is only reduced and the water is not broughtto complete rest as is done in an intermittent type. Settling basins may be either longrectangular or circular in plan; however, the long rectangular basins are hydraulicallymore stable.

The bottom is slightly sloped to facilitate sludge scraping and a slow moving mechanicalsludge scraper continuously pulls the settled material into a sludge hopper.

Figure 3.2 shows a simple rectangular settling tank which is divided into the four keyzones:

Zone 1, Inlet zone. In this region the flow is uniformly distributed over the cross sectionsuch that the flow through settling zone follows a horizontal path.

Zone 2, Settling zone. In this region settling occurs under the low flow rate of liquid.The size of this region is designed to allow the particles to settle as in Section 3.5.

Zone 3, Outlet zone. In this region the clarified effluent is collected and discharge throughoutlet weir.

Zone 4, Sludge zone. In this region the sludge settles, this is often slopped so that thesludge can be scrapped out.


3.5. DESIGN OF GRAVITY SEPARATORS

L

H

Q

Sludge

Clarified water1 2 3

4

(a) Side view

L

B

Q

Sludge

Clarified water1 2 3

(b) Plan view

Figure 3.2: Schematic of a general rectangular gravity settler.

3.5 Design of Gravity Separators

3.5.1 Limit for Low Solids Concentration - Rectangular

For Type I settling it can be assumed that the key parameters for the settling are thevelocity of the fluid and the settling velocity of the particles. The average fluid velocityacross the vessel can be calculated from,

uf =Q

HB(3.5.1)

This means that the hydraulic retention time (i.e. the time taken for the fluid to cross thelength of the settler) is,

Θ =L

uf=LHB

Q=V

Q(3.5.2)

If a particle is settling with a vertical velocity equal to it’s terminal velocity, ut, then itsvertical fall, h, over the length of the settler is,

h = utΘ =utL

uf=utLHB

Q(3.5.3)

The terminal velocity of the slowest particle should be selected as the limiting designcriterion, generally the smallest particles (but check density if in a mixture). If h > H



then the particle hits the bottom before the end of the tank and is collected, if h < H ,then the particle may not hit the bottom and may escape with the outflow. This leads to adefinition of the critical settling velocity, uc, where h = H ,

uc =H

Θ= H

Q

LHB=

Q

LB(3.5.4)

This critical settling velocity is also known as the overflow rate. It is important to notehere that this definition is the inlet flow rate, Q, divided by the horizontal area of the tank,BL, (the footprint).

We can define a collection efficiency for particles with settling velocity lower than thecritical velocity, using Equations 3.5.3 and 3.5.4,

η =h

H=

utL

ufH=utuc

< 1 (3.5.5)

If the particle settling velocity is is faster than the critical velocity then the collectionefficiency is 1.

When designing a gravity settler the footprint area needed is therefore given by,

A =Q

ut(3.5.6)

3.5.2 Limit for Low Solids Concentration - Circular

The circular settling basin for Type I can be designed in the very similar manner to therectangular settler; however the fluid velocity changes as it moves to the outside of thebasin. The suspension enters the settler at the centre and moves to the outside, the cross-sectional area for the fluid flow therefore increase as it is dependant on the radius, suchthat,

uf (r) =Q

2πrH=

d r

d t(3.5.7)

The settling velocity of the particles can be written as the rate of change of their heightwith time as,

ut =dh

d t(3.5.8)

Combining Equations 3.5.7 and 3.5.8 gives the variation of the particle height with ra-dius,

dh

d t· d t

d r=

dh

d r= ut ·

2πrH

Q(3.5.9)

Equation 3.5.9 can be integrated to find the height fallen by the particle in a given radius.At r = 0 h = 0 until r = r where h = h∫ h

0

dh = ut ·2πH

Q

∫ r

0

rd r

h = ut2πH

Q

r2

2(3.5.10)


3.5. DESIGN OF GRAVITY SEPARATORS

As with the rectangular settler the critical settling velocity is given if the height fallen isequal to the settler height,H , at the radius of the settler, R, such that,

uc =Q

πR2(3.5.11)

The collection efficiency can be given by the same equation as the rectangular settler,Equation 3.5.5. The design process is also the same as the rectangular settler, and thesettler footprint area needed is therefore given by,

A = πR2 =Q

ut(3.5.12)

3.5.3 Mass Balance of Solids

When settler has Type III settling the settling velocity can vary with height in the settlerdue to the high concentration of particles and the fact that the concentration can vary. Suchclarifiers have to be designed on the basis of solids loading or solid flux and checked forsurface overflow rate.

Assuming that there are no solids in the overflow, then the flux of solids must be constant(under steady state operation) through the settler to the underflow.

The inlet flux, G, can be given by,

G = QC0ρp (3.5.13)

where Q is the inlet slurry flow rate, C0 is the inlet concentration of solids, ρp is thedensity of the particles. The flux exiting the settler, the underflow, is given by,

G = AuuCuρp (3.5.14)

where uu is the underflow velocity. The flux of solids in the settler is given related to thesettling velocity of the particles, us, but is also sped up due to the underflow velocity (i.e.they are dragged down slightly faster), as,

G = A (us + uu)Cρp (3.5.15)

Equation 3.5.14 can be rearranged for the underflow velocity,

uu =G

ACuρp(3.5.16)

which can then be substituted into Equation 3.5.15,

G = A

(us +

G

ACuρp

)Cρp (3.5.17)



The solids flux can then be taken from Equation 3.5.13 and rearranged for the settlerarea,

QC0ρp = A

(us +

QC0ρpACuρp

)Cρp

QC0ρp =ACρpACuρp

(ACuρpus +QC0ρp)

QC0ρp = ACρpus +QC0ρpC

Cu

ACρpus = QC0ρp −QC0ρpC

Cu

ACρpus = QC0ρp

(1− C

Cu

)A =

Q

us

C0

C

(1− C

Cu

)(3.5.18)

This now takes into account the change in concentration which varies from C0 to Cu1. Asus is a function of the concentration, the area has to be calculated for different concentra-tions and then the area needed is the largest value found.

The overflow rate of the liquid, Q′, is the difference between the inlet and underflow ratesas,

Q′ = Q0 (1− C0)− (Q0 −Q′) (1− Cu) (3.5.19)

Rearranging this for the overflow rate gives,

Q′ = Q0

(1− C0

Cu

)(3.5.20)

3.6 Accelerating Settling

Often when we have a settler the settling rate of the particles is low, therefore if it is pos-sible to accelerate the settling we want to, there are a number of options to do this:

• Modify the particle shape. For example crystals can be spherical or needle shapedwhich will give different settling velocities. Normally this isn’t possible as theparticle shape are fixed either by having no control over it (e.g. silt in dirty water)or a particular shape being needed for the function (e.g. Pharmaceuticals for bio-activity).

• Modify the fluid viscosity and density. For example a change in temperature couldchange the viscosity and density; however, this is not practical in most cases,but changing the density is used to separate diamonds in a process called “densemedium separation”.

• Raking or stirring. This creates free channels for particles to settle in but you haveto be careful not to cause re-suspension of the particles with the agitation of thefluid.

1 This expression does reproduce that for the low concentration assumption if it is assumed thatC = C0, i.e. settling concentration is low and doesn’t change, and that Cu >> C0, i.e. the slurry isvery concentrated compared to the solution.


3.7. SETTLER RULES OF THUMB

• Flocculation. This increases the particle’s size by coagulating particles, see Sec-tion 3.6.1.

3.6.1 Flocculation

Small particles (around < 40 mm) and some biologically active particles will take un-reasonably long times to settle, if at all. This means that if we want the to be separatedsomething needs to be done. One common option is to cause the particles to flocculatedwhich causes them to cluster together and thus settle at higher rates.

Due to this it is almost impossible to predict the shape and hence the settling rate, soexperimental data has to be used to find the settling rate. This technique is often used inclarifiers where a clear supernatant is desired.

When designing settlers flocculation can be “included” with the sedimentation step byadding an initial zone onto the settler where the flocculation can occur.

3.7 Settler Rules of Thumb

When designing settlers we need to take into account several rules of thumb which havebeen developed based on practical knowledge and experience to make sure the settlerswork efficiently:

• Surface over flow rate (SOR) of around 40 m3 per day per m2 for primary units

• Secondary units as low as 12 up to 30 m3 per day per m2

• Solids collection of 50 to 120 kg per day per square meter

• Depth of sedimentation tanks is 3 to 5 m (typically 3.5 m)

• Circular sedimentation, minimum diameter of 6.0 m

• Rectangular sedimentation, lengths up to 90 m (typically 25-40 m)

• Length range from 2 to 5 times the width

• Gravity sedimentation tanks normally provide for 2 hour retention of solids, basedon average flow

• Longer times for light solids, or in winter times

• Organic solids generally will not compact to more than 5 to 10 %

• Inorganic solids will compact up to 20 or 30 %

• We have to design sludge pumps to remove the solids: high concentration solidsrequire diaphragm pumps



3.8 References

[1] Richardson, J. F. and Zaki, W. N. [1954], ‘The sedimentation of a suspension ofuniform spheres under conditions of viscous flow’, Chemical Engineering Science3, 65–73.

[2] Robinson, C. D. [1926], ‘Some factors influencing sedimentation’, Industrial & En-gineering Chemistry 18, 869–871.

[3] Vand, V. [1948], ‘Viscosity of solutions and suspensions’, The Journal of Physicaland Colloid Chemistry 52, 277.

[4] Vesilind, P. A. [1968], ‘Theoretical considerations: Design of prototype thickenersfrom batch settling tests’, Water and Sewage Works 115, 302–307.


3.9. PROBLEMS

3.9 Problems

3.1 A particle of diameter 5 × 10−5 m with density 3000 kg m−3 settles in a fluid ofviscosity 0.001 Pa s and density 1000 kg m−3. If the concentration of particles is0.2 and the Richardson and Zaki model can be used with k = 0.9 and n = 5, whatis the settling velocity?



3.2 A gravity settler is to be used to separate particles larger than 75µm from a gas offlow rate 1.6 m3 s−1. The density of the particles is 2100 kg m−3. The gas density is1.18 kg m−3 and the viscosity is 1.85× 10−5 kg m−1 s−1. Estimate the dimensionsof the settling chamber assuming a rectangular cross section with length to be twicethat of the breadth, and that the particles fall via Stokes’ law.


3.9. PROBLEMS

3.3 A gravity settler is to be used to separate particles of 50µm diameter from a waterof flow rate 6372 m3 hr−1. The density of the particles is 2300 kg m−3. The waterdensity is 998 kg m−3 and the viscosity is 0.89 mPa s. If the settling chamber is arectangular cross section with length 40 m and width 20 m what is the efficient ofthe separation?



3.4 A gravity settler is to be used to separate particles of 1×10−5 m diameter of density2000 kg m−3. The water density is 998 kg m−3 and the viscosity is 1 mPa s. If thesettling chamber is a rectangular cross section with length 60 m and width 20 mwhat is the maximum flow rate to ensure full settling?


3.9. PROBLEMS

3.5 Sketch the particle trajectory in a rectangular and circular settler.



3.6 A sample of material was settled in a graduated lab cylinder 300 mm tall (to the500 ml mark). The interface dropped from 500 ml to 215 ml on the graduationsduring a 4 minute period.

Give a preliminary estimate of the clarifier diameter required to treat a waste streamof 2100 l min−1. Over-design by a factor of 2, based on the settling rate, and accountfor 7 m2 of entry area used to eliminate turbulence in the entering stream.

If the feed concentration is 1.2 kg m−3 feed, what is the loading rate? Is it withinthe typical thickener range?


3.9. PROBLEMS

3.7 Calculate the minimum area of a settler to treat 720 m3 hr−1 of slurry containing65µm particles of silica, whose density is about 2600 kg m−3. Use an over-designfactor of 1.5 on the settling velocity.



3.8 For the system in Q3.3, if the inlet concentration of solids is 0.03 m3-solids m−3

what is the underflow concentration, assuming the settling concentration can beapproximated to that of the inlet?


3.9. PROBLEMS

3.9 A settler needs to be designed to separate a slurry of concentration 200 kg m−3 at aflow rate of 0.0333 m3 s−1. The desired underflow concentration is 1200 kg m3. If ithas been found that the settling velocity of the solids varies with the concentrationas in the table below, what area is needed for the settler?

C / kg m−3 us / m s−1

200 2.233E-04225 1.793E-04257 1.433E-04300 1.100E-04360 8.167E-05450 5.333E-05600 3.000E-05692 2.017E-05720 1.850E-05818 1.333E-05900 1.000E-05

1000 6.667E-06


Chapter 4Centrifuges


4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3 Forces on a Sphere in a Centrifuge . . . . . . . . . . . . . . . . . . . 65

4.3.1 Centrifugal Force . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.3.2 Terminal Velocity . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.4 Laboratory Centrifuges . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.5 Continuous Centrifuges . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.5.1 Settling Time . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.5.2 Throughput . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.5.3 Cut-size Diameter . . . . . . . . . . . . . . . . . . . . . . . . 69

4.5.4 Sigma Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.6 Types of Centrifuges . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.6.1 Tubular Bowl Centrifuge . . . . . . . . . . . . . . . . . . . . . 70

4.6.2 Disc Bowl Centrifuge . . . . . . . . . . . . . . . . . . . . . . 71

4.6.3 Scroll Centrifuge . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

63

CHAPTER 4. CENTRIFUGES

4.1 Chapter 4 ILOs

.

ILO 4.1. Eaplain particle motion in centrifuges

ILO 4.2. Design centrifuges for the seperation of solids from liquids

ILO 4.3. Review types of industrial centrifuges


4.2. INTRODUCTION

4.2 Introduction

A centrifuge is a piece of equipment that puts an object in rotation around a fixed axis(spins it in a circle), applying a force perpendicular to the axis of spin (outward) thatcan be very strong. The centrifuge works using the sedimentation principle, where thecentrifugal acceleration causes denser substances and particles to move outward in theradial direction. At the same time, objects that are less dense are displaced and move tothe center.

There can be benefits to using a centrifuge when gravity (which is freely available) is notfast enough. This can be caused by particles being very small (< 10µm), the liquid beingvery viscous, or the density difference between particles and liquid being very small, i.e.slow settling velocity. This means that for the same system the separation time will bereduced and the separation factor can be increased. Due to the fact that the settling timeis reduced the equipment used for the settling can be much smaller than with gravitysettling.

The centrifugal force can be much larger than the force due to gravity, this means thatit is possible to achieve separations not possible by just gravity. Due to the fact thatthese increased forces can overcome Brownian limits, overcome convection currents, andovercome stabilizing forces that hold an emulsion together.

A centrifuge can be used in batch or continuous operation and is typically used to separatesystems such as,

• Cream from milk (milk is an emulsion)

• Clarification: juice, beer (yeast removal), essential oils

• Widely used in bioseparations: blood, viruses, proteins

• Remove sand and water from heavy oils

When using centrifuging there are some key terms used; suspension, the mixed materialadded into the centrifuge; pellet or precipitate, hard-packed concentration of particlesafter centrifugation; and supernatant, clarified liquid above the precipitate.

4.3 Forces on a Sphere in a Centrifuge

Similar to a particle settling under gravity, Figure 2.5, there are three key force on aparticle in a rotating space, Figure 4.1. These are the buoyancy force, Fb, the drag force,FD, and the centrifugal force, Fc.

Fc

r

FDFb

Figure 4.1: Forces on a sphere in a centrifugal field.



4.3.1 Centrifugal Force

The force caused by a centrifugal field is similar to that caused by gravity, equation 2.5.1.The centrifugal force is given by the mass of the particle multiplied by the accelerationdue to the centrifugal force, ω2r,

Fc = ma = ρpVpω2r = ρp

πd3

6ω2r (4.3.1)

where ω is the angular velocity and r is the radius of the circle that the particle is be-ing rotated under. The angular velocity can be related to the rotation rate, either RPM,the number of rotations per minute or RPS, the number of rotations per second (or Hz),as

ω =2πRPM

60= 2πRPS (4.3.2)

The centrifugal force is often represented as the G-force, which is the comparison to theforce due to gravity as,

G =FcFg

=4π2RPM2r

3600g(4.3.3)

4.3.2 Terminal Velocity

As with a particle falling under gravity, the terminal velocity of a sphere can be calculatedfrom a force balance equivalent to equation 2.5.3,∑

i

Fi = 0 = Fc − Fb − FD (4.3.4)

The centrifugal force can be given by equation 4.3.1, the buoyancy force, equation 2.5.2,can be given modified by the centrifugal acceleration,

Fb = mfa = ρfVpω2r = ρf

πd3

6ω2r (4.3.5)

and the drag force can be given based on the assumption of Stokes’ law, equation 2.3.2.

This means that the terminal velocity under centrifugal force can be given by,

FD = Fc − Fbπd3

6ω2r (ρp − ρf ) = 3πµud

ut =d2 (ρp − ρf )ω2r

18µ(4.3.6)


4.4. LABORATORY CENTRIFUGES

4.4 Laboratory Centrifuges

Laboratory centrifuges are typically a batch system where a small sample can be addedto container and then that container span at high speeds. Container is then removed fromthe centrifuge after it has been separated. This smaller laboratory centrifuges are usedbased on knowledge of the systems that need separating. These systems are often given anumber called the Svedberg number, S. The Svedberg unit offers a measure of a particle’ssize based on its sedimentation rate under acceleration (i.e. how fast a particle of givensize and shape settles to the bottom of a solution) [2]. The Svedberg is a measure oftime, defined as exactly 10−13 seconds (100 fs). A particle’s mass, density, and shape willdetermine its S value. It depends on the frictional forces retarding its movement, which,in turn, are related to the average cross-sectional area of the particle. This means thatlarge, heavier particles tend to have larger S values.

The time needed (in minutes) for a separation using a laboratory centrifuge is related tothe Svedberg number, the size of the centrifuge and the rotation speed and can be givenby,

t =2.53× 1011

S

(ln (rmax − rmin)

RPM2max

)(4.4.1)

4.5 Continuous Centrifuges

One of the most common, and simplest, type of continuous centrifuges is the tubular-bowlcentrifuge. In tubular-bowl centrifuges, feed enters from the bottom of the cylindricalbowl. A distributor and baffle assembly accelerates the incoming liquid to rotor speed.Then, a baffle separates the feed into its components (olid and liquid layers for solid/liquidseparations). The outer layer, which consists of the heavier components (generally thesolids), becomes concentrated against the wall, while the inner layer, which consists ofthe lighter components (generally the liquid), floats on top. A simple schematic is givenby Figure 4.2.

Each layer then travels up the side of the bowl as an annulus. Slurry layers are dischargedthrough overflow ports located on the top of the centrifuge or can build up remaining inthe bowl and are recovered manually in a batch wise operation.

The key variables for this type of centrifuge are:

• Rotational speed, ω

• distance from the centre to the surface of the sludge, Ro

• distance from the centre to the inner wall or liquid layer, Ri

4.5.1 Settling Time

The trajectory of the particle in the centrifuge can be seen in Figure 4.2. As it moves upthe centrifuge the velocity towards the wall increases due to the centrifugal force increas-ing.



ω

Ri

Ro

L

Figure 4.2: Centrifugal force on a object.

The particle velocity at a given radius can be given by equation 4.3.6, as the velocity isthe change in radius position with time then we can write,

u =d r

d t=d2 (ρp − ρf )ω2r

18µ(4.5.1)

The time taken for the particle to settle to the outside of the bowl can then be given by theintegral between the start position at the inlet position, Ri, at t = 0, and the end positionat the outside of the bowl, Ro, as,∫ Ro

Ri

d r

r=d2 (ρp − ρf )ω2

18µ

∫ t∗

0

d t

t∗ =18µ

d2 (ρp − ρf )ω2lnRo

Ri

(4.5.2)

4.5.2 Throughput

Once we know how long a particle should be in the centrifuge, t∗, we can calculate a feedflowrate, Q∗. The volume of the centrifuge, V , is,

V = π(R2o −R2

i

)L (4.5.3)

Thus the flow rate through the centrifuge at which the particles just settle on the outsidewall is given by,

Q∗ =V

t∗=d2 (ρp − ρf )ω2

18µ ln (Ro/Ri)π(R2o −R2

i

)L (4.5.4)

Any flow rate lower than this value with mean that the time spent in the centrifuge islonger than the settling time, t∗, thus the particles will separate. Higher flow rates willmeans a time shorter than the settling time thus particles may not fully separate.


4.5. CONTINUOUS CENTRIFUGES

4.5.3 Cut-size Diameter

The time taken in equation 4.5.4 is actually excessive for the particles to separate. Thisis because for the horizontal discharge weir to retain particles they don’t actually needto have reached the outer wall Ro, they just need to be at a radius larger than the weirposition. Therefore, to prevent an excessive over design, we typically use the halfwaymark between Ri and Ro as the key position and then define the tcut, for a particle to settlebased on this cut point,

Qcut =V

tcut=

d2cut (ρp − ρf )ω2

18µ ln [2Ro/ (Ri +Ro)]π(R2o −R2

i

)L (4.5.5)

4.5.4 Sigma Theory

To simplify scale-up calculations for centrifuges it is desirable to have the centrifugeproperties separated from the particle properties. Sigma theory makes this desired char-acteristic a reality by describing a centrifuge’s characteristics in terms of an equivalentsedimentation process that would be conducted with gravity alone [1]. To do this we canexamine the velocity of the liquid in the upward direction, z. This is,

uf =d z

d t=Q

A=

Q

π (R2o −R2

i )(4.5.6)

Dividing equation 4.3.6 by equation 4.5.6 gives an equation for the particle trajectory,dR/d z,

d r

d z=

d r

d t· d t

d z=d2cut (ρp − ρf )ω2r

18µ· π (R2

o −R2i )

Q(4.5.7)

This is the trajectory seen in Figure 4.2. We can now integrating between the entranceR = Ri at z = 0 to the exit Ro at L,∫ Ro

Ri

d r

r=d2cut (ρp − ρf )ω2

18µ· π (R2

o −R2i )

Q

∫ L

0

d z

ln

(Ro

Ri

)=d2cut (ρp − ρf )ω2

18µ· π (R2

o −R2i )

QL (4.5.8)

This expression can be rearranged for the flowrate Q,

Q =

[d2p (ρp − ρf )

18µ

]πLω2 (R2o −R2

i )

ln

(Ro

Ri

)

=

[gd2p (ρp − ρf )

18µ

]πLω2 (R2o −R2

i )

g ln

(Ro

Ri

) = ut,gΣ (4.5.9)

where ut,g is given by Stokes’ law for gravity settling (equation 2.4.5), and Σ with unitsof length2 is a centrifuge sigma factor for the centrifuge.



For scale-up from a small laboratory centrifuge (A) to a large production centrifuge (B),with different dimensions and for operation at a different rotation rate, assuming that ut,gremains the same, gives,

QB = QAΣB

ΣA

(4.5.10)

The assumptions in equation 4.5.10 and sigma theory are,

• The particles are evenly distributed in the continuous liquid and the concentrationsare low, so settling is not hindered.

• Streamline flow at a Reynolds number below 0.3, with the liquid rotating at thesame velocity as the bowl.

• No re-entrainment, displacement of the flow pattern by the deposited material, ornonuniform liquid feed.

The Sigma value is only a function of the centrifuge’s characteristics; not the particle orfluid, and it is equal to the cross-sectional area needed for the equivalent gravity settler.It can be used to for scale-up of the same feed to larger equipment or for changing feedflowrate within the same type of equipment. The Sigma equation is different for differentcentrifuge types.

4.6 Types of Centrifuges

4.6.1 Tubular Bowl Centrifuge

Tubular bowl centrifuge systems are typically used for systems with low concentrationof solids. They can be operated in a batch operation where they are stopped to cleanout the solids and then the process is restarted again. Due to this, and the simple shapecontamination is possible and they are not always suitable for bioseparations.

A high L/D aspect ratio should be used when designing these centrifuges (around 8) as itis more stable to operate. Also the diameter of the centrifuge should be minimised as veryhigh wall stresses are developed at higher diameters. A typical tubular bowl centrifugecan be seen in Figure 4.3.

Figure 4.3: Typical Tubular Bowl Centrifuge.


4.6. TYPES OF CENTRIFUGES

4.6.2 Disc Bowl Centrifuge

The advantage of the disc bowl centrifuge is that the addition of angled discs gives agreater surface area. This allows a greater volume to be treated without increasing thebowl diameter. The discs also reduce any contamination between the phases and thusthey are widely used in bioseparations. They are typically used to treat up to 15% solidsin feed stream and can easily be operated continuously (infrequent cleaning of discs isneeded). A typical disc bowl centrifuge can be seen in Figure 4.4.

Typically the discs are angled, θ, at 35 to 50◦ and there are around 50 to 150 discs perunit, N . They are typically between 0.5 to 0.7 m in radius and are operated at rotationalspeeds of 0 to 12,000 rpm. The Sigma value for a disc bowl centrifuge is given by,

Σ =2πω2N (R3

o −R3i )

3g tan θ(4.6.1)

Figure 4.4: Typical Disc Bowl Centrifuge.

4.6.3 Scroll Centrifuge

The main application of scroll or decanter centrifuges is to separate large amounts ofsolids from liquids on a continuous basis. They are also used to wash and dry varioussolids in industry, such as polystyrene beads; clarify liquids; and concentrate solids. Atypical scroll centrifuge can be seen in Figure 4.5.

The Sigma value for a scroll centrifuge is given by,

Σ =2πω2L

g

(3

4R2o +

1

4R2i

)(4.6.2)



Figure 4.5: Typical Scroll Centrifuge.

4.7 References

[1] Ambler, M. C. [1952], ‘The evaluation of centrifuge performance’, hemical Engi-neering Progress pp. 150–158.

[2] Slonczewski, J. and Foster, J. W. [2009], Microbiology: An Evolving Science, W.W.Norton, New York.


4.8. PROBLEMS

4.8 Problems

4.1 What is the applied centrifugal field at a point equivalent to 5 cm from the centre ofrotation and an angular velocity of 3000 rad s−1?



4.2 A lab scale tubular bowl centrifuge has the following characteristics,Ri = 16.5 mm,Ro = 22.2 mm, L = 115 mm, and operates at 800 RPM.

It is being used to separate bacteria from a fermentation broth, the broth has the fol-lowing properties, ρf = 1010 kg m−3, ρp = 1040 kg m−3, µf = 0.001 kg m−1 s−1,dp,min = 0.7µm.

Calculate both Q∗ and the more realistic Qcut, and verify whether Stokes’ law ap-plies.

What would be the area of the sedimentation vessel that would operate at this Qcut?Hint: recall that A = Q/ut.


4.8. PROBLEMS

4.3 Illustrate the trajectory taken by a particle reaching the cut-point within time tcut.

In the same duration of time, what trajectory will a smaller particle have taken?



4.4 In a test, particles of density 2800 kg m−3 and of size 5µm, equivalent sphericaldiameter, were separated from suspension in water (998 kg m−3 and 1 mPa s) fed ata volumetric throughput rate of 0.25 m3 s−1.

Calculate the value of the capacity factor, Σ


4.8. PROBLEMS

4.5 For the same centrifuge as in Q4.4, what will be the corresponding size cut for asuspension of coal particles in oil fed at the rate of 0.04 kg s−3? The density of coalis 1300 kg m−3 and the density of the oil is 850 kg m−3 and its viscosity is 0.01 Pa s.

Is Stokes’ law applicable for this case?



4.6 For a tubular bowl centrifuge of diameter 1 m and a length to diameter ratio of 8,what is the equivalent area needed for a gravity settler if the start of the liquid levelis 0.1 m from the centrifuge centre and an angular velocity of 1000 s−1?


Chapter 5Cyclones


5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.3 Cyclone Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.4 Dimentionless Number Design . . . . . . . . . . . . . . . . . . . . . 83

5.5 Cyclone Performance . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.6 Stairmand’s Design Procedure . . . . . . . . . . . . . . . . . . . . . 87

5.7 Multiple Cyclones . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.7.1 Series Overflow . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.7.2 Series Underflow . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.7.3 Partial Overflow Recycle . . . . . . . . . . . . . . . . . . . . . 92

5.7.4 Underflow Recycle . . . . . . . . . . . . . . . . . . . . . . . . 92

5.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

79

CHAPTER 5. CYCLONES

5.1 Chapter 5 ILOs

.

ILO 5.1. Examine cyclone designs and operation.

ILO 5.2. Use the critical cut design method.

ILO 5.3. Design cyclones using Stairmand’s Procedure.

ILO 5.4. Investigate multiple cyclone options.


5.2. INTRODUCTION

5.2 Introduction

Cyclones have found wide application in various fields of technology, such as gas clean-ing, burning, spraying, atomizing, powder classification etc. They are also used for solid-liquid separation; the cyclones specially designed for liquids are typically referred to ashydrocyclones. The basic separation principle employed in cyclones is centrifugal sed-imentation, i.e. the suspended particles are subjected to centrifugal acceleration, whichmakes them separate from the fluid. Unlike centrifuges (which use the very same princi-ple), cyclones have no moving parts and the necessary vortex motion is performed by thefluid itself.

A cyclone consists of a cylindrical section joined to a conical portion, as in Figure 5.1.The suspension of particles in a fluid (liquid/gas) is injected tangentially through the inletopening in the upper part of the cylindrical section and, as a result of the tangential entry, astrong swirling motion is developed within the cyclone. A portion of the fluid containingthe fine fraction of particles is discharged through a cylindrical tube fixed in the centre ofthe top and projecting some distance into the cyclone; the outlet tube is called the overflowpipe or vortex finder. Larger particles go to the wall quickly and move to the bottom ofthe cyclone Smaller particles are separated from the gas near the bottom vortex where thegas reverses direction (low velocity). Therefore the coarse fraction of the material leavesthrough the circular opening at the apex of the cone, called the underflow orifice.

Gas out

Solids out

Feed

Figure 5.1: Cyclone.

As with all separation principles involving particle dynamics, a knowledge of the flowpattern in the cyclone is essential for understanding its function and subsequently for the


CHAPTER 5. CYCLONES

optimum design and evaluation of the particle trajectories, which in turn allow predic-tion of the separation efficiency. This means that most design procedures are based oncorrelations from collected data on operating cyclones.

A well-designed cyclones can separate liquid droplets/particles as small as 10µm from anair stream. Small cyclones are more efficient than large ones and can generate forces 2500times that of gravity. Although for cyclones the effect of the feed and device parametersis complex, and interdependencies are expected, where possible, is is best to consider acyclone before a centrifuge for solid-fluid separations.

The key advantages of cyclones are:

• No moving parts and no consumable components

• Low operating costs, essentially only pay for pressure drop

• Can be operated over a wide rage of temperatures and pressures

• Can be designed as small as 1 cm to over 10 m in diameter

• Very low capital costs compared to other equipment, can be made from many ma-terials

• Even different particle shapes (due to different settling velocities) can be separated

5.3 Cyclone Separation

The settling velocity of a particle in a cyclone can still be given by the same equation asfor a centrifuge, i.e. motion in a centrifugal force, equation 4.3.6,

u =d2 (ρp − ρf )ω2r

18µ

This can be rearranged for the diameter of a particle that is settling at a given speed,

d2 =18µ

(ρp − ρf )u

ω2r(5.3.1)

The tangental velocity, i.e. the velocity in the direction of the swirl, uθ, is equal to ωr, thismeans that,

d2 =18µ

(ρp − ρf )u

u2θr (5.3.2)

As the volumetric flow rate of the fluid doesn’t change in the cyclone in the radial direction(assuming non-compressible flow), i.e. Q = 2πrLu = 2πRLuR, then ur = uRR whereuR is the radial velocity at the cyclone wall. This means that,

d2 =18µ

(ρp − ρf )uRu2θR (5.3.3)

For a confined vortex it has been found that uθr1/2 at all radial positions is constant.Therefore uθr1/2 = uθRR

1/2, thus

u2θ = u2θRR

r(5.3.4)


5.4. DIMENTIONLESS NUMBER DESIGN

Substituting this into equation 5.3.3 gives,

d2 =18µ

(ρp − ρf )uRu2θR

r (5.3.5)

where r is the equilibrium orbit radius for a particle of diameter d. If we assume that allparticles with an equilibrium orbit radius greater than or equal to the cyclone body radiuswill be collected, then we can say that,

d2cut =18µ

(ρp − ρf )uRu2θR

R (5.3.6)

The radial velocity on the wall of the cyclone can be estimated by the volumetric flow rateof fluid divided by the perimeter area of the cyclone,

uR =Q

2πRL=uinAinlet

2πRL(5.3.7)

and the tangential velocity can be approximated to the inlet velocity,

uθR ≈ uin (5.3.8)

This analysis predicts an ideal grade efficiency curve, that is that all particles of diameterdcut and greater are collected as in Figure 5.4.

5.4 Dimentionless Number Design

The cyclone’s cut size, dcut, or d50 can be predicted from the Stokes number. This is theratio of the characteristic time of a particle to the characteristic time of the flow.

St =d250ρpuf18µD

(5.4.1)

This is useful because for a large range of operating conditions the Stokes number is aconstant, as Figure 5.2. This is true when the fluid Re is greater than 2× 104 which is thecase for most cyclones with low viscosity fluids. The fluid velocity can be given by,

uf =4Q

πD2(5.4.2)

This allows scale-up through geometrically similar cyclones.

It is important to also examine the pressure drop through the cyclone to make sure it isnot too high. The Euler number is the ratio of the pressure forces to the inertial forcesand is relatively constant, under different flow conditions, for a given cyclone providedthe solids concentration remains around or below 1 g m−3. The Euler number is givenby,

Eu =2∆P

ρfu2f(5.4.3)

The Euler number can be easily calculated for a given cyclone using just the fluid atambient conditions, but can be approximated from,

Eu = π

(D2

WinletHinlet

)(D

Doverflow

)2

(5.4.4)


CHAPTER 5. CYCLONES

103 104 105 10610−3

10−2

10−1

100

Re

St1/2

[7][8][9][1][2][3][4][5][6]

Figure 5.2: St versus Re for cyclones of various geometries. The key shows the referenceswhere the data is taken from.

5.5 Cyclone Performance

The separation efficiency of a hydrocyclone has a character of probability. This is to dowith the probability of the position of the different particles in the entrance to the cy-clone, their chances of separation into the boundary layer flow and the general probabilitycharacter of turbulent flow. Coarse particles are always more likely to be separated thanfine particles. Effectively, the cyclone processes the feed solids by an efficiency curvecalled the grade efficiency, which is a percentage increasing with particle size. Figure 5.3shows the process schematically; the solids in the feed enter the cyclone and are pro-cessed. There are two products of the separation: the coarse product (i.e. the solids in theunderflow) and the fine product (i.e. the solids in the overflow). Every cyclone is thereforeprimarily a classifier, although we may use it as a separator by setting the cut size as lowas possible to recover as much of the solids as possible into the ’coarse product’.

MFeed

Mf

Mc

Fine

Coarse

Figure 5.3: Cyclone performance.

The total mass of the feed must be equal to the sum of the total masses of the products if


5.5. CYCLONE PERFORMANCE

there is no accumulation of material in the equipment, i.e.

M = Mc +Mf (5.5.1)

The mass balance must also apply to any size fractions present in the feed if there is nochange in particle size of the solids inside the cyclone (i.e. no agglomeration or breakage).By definition, the particle size distribution frequency gives the fraction of particles of sized in the sample. The total mass of particles of size d in the feed for example is therefore thetotal mass of the feedM multiplied by the appropriate fraction f (d) so that equation 5.5.1becomes:

Mf (d) = Mcfc (d) +Mfff (d) (5.5.2)

If a total (or overall) efficiency ET is now defined as simply the ratio of the mass Mc ofall particles separated to the mass M of all solids fed into the separator,

ET =Mc

M= 1− Mf

M(5.5.3)

This means that equation 5.5.2 can be rewritten as,

f (d) = ETfc (d) + (1− ET ) fc (d) (5.5.4)

which relates the particle size distributions of the feed, the coarse product and the fineproduct. This can also be written as,

ET =f (d)− ff (d)

fc (d)− ff (d)(5.5.5)

thus if f (d)− ff (d) is plotted against fc (d)− ff (d) then this should produce a straightline with a slope of ET .

As the performance of most cyclones is highly particle size dependent (and hence differentsizes are separated with different efficiency), the total efficiency ET depends very muchon the size distribution of the feed solids and is, therefore, unsuitable as a general criterionof efficiency. If, however, the mass efficiency is found for every particle size d, a curvereferred to as the gravimetric grade efficiency functionG (d) is obtained which is normallyindependent of the solids size distribution and density and is constant for a particular setof operating conditions, e.g. fluid viscosity, flow rate and often also solids concentration.For particles of a given size the grade efficiency is taken the same as the overall efficiency,similar to equation 5.5.3, as,

G (d)d =(Mc)d(M)d

(5.5.6)

Thus based on the distribution,

G (d) =Mcfc (d)

Mf (d)= ET

fc (d)

f (d)(5.5.7)

Figure 5.4 shows a typical grade efficiency curve for a cyclone compared to the simpletheory perfect cut. The grade efficiency curves are usually S-shaped as particle dynamics


CHAPTER 5. CYCLONES

dcrit0

0.5

1

Simple Theory Actual

d

G(d

)

Figure 5.4: Simple theory separation versus actual separation.

in which the body forces acting on the particles (which are proportional to d3) such ascentrifugal forces are opposed by drag forces (which are proportional to d2).

The shape and position of the grade efficiency curve can be characterized in many ways.We will concentrate on the three commonly used values as depicted in Figure 5.5:

1. The position of the actual cut size d50. This has a direct bearing on the overall solidsrecovery or clarification power of the system. If the cut size reduces, more solidsare recovered from the feed into the underflow and the overflow becomes cleaner(and finer).

2. The amount of ’underflow by-pass’ which is shown as the intercept on the y-axisat x = 0. It is the flow ratio Rf and is a measure of the volumetric amount ofunderflow relative to that of the feed. If a system has a greater Rf it means moreof the fine particles are brought into the underflow. This would be good news insolids recovery applications but disastrous in those classification duties where thecoarse product is required to be free of fines. Where the fine product is of interest,its quality is virtually unaffected by Rf , only the yield is.

3. The sharpness of cut as measured by the sharpness index H ′75/25 derived from thereduced grade efficiency curve,

G′ (d) =G (d)−Rf

1−Rf

(5.5.8)

It is defined as a ratio of particle sizes d′75/d′25. A lower value ofH ′75/25 will indicate

better performance in classification duties, i.e. more ideal. There are two reasonsfor deriving the sharpness index from the reduced grade efficiency curve rather thanfrom the actual one. Firstly, the sharpness of cut is always to some degree affectedby Rf and this way the effect is removed. Secondly, the sharpness index for theactual grade efficiency curve sometimes cannot have a value because the whole ofthe curve may be greater than 25% thus d25 cannot be determined.


5.6. STAIRMAND’S DESIGN PROCEDURE

d′25 d50 d′750

25

50

75

100

Rf

H ′75/25 =d′75d′25

Particle size

Gra

deef

ficie

ncy

/%

Figure 5.5: Grade Efficiency parameters. Solid line is the real grade efficiency, G (d), andthe dashed line is the reduced grade efficiency, G′ (d).

5.6 Stairmand’s Design Procedure

As the effect of feed and device parameters are complex for cyclones if we want a designbased on the full grade curve then this can be difficult. Therefore one of the key designmethods is based on obtaining particle-collection efficiency data for a cyclone of diameterD and establishing geometric ratios that permit scaling up or down. Design methods forsolid-liquid cyclone separators are similar to those for solid-gas or liquid-gas units.

Stairmand’s design procedure [11], is typically used to design cyclone devises and is givenbelow:

1. Pick one of the two standard designs for gas-solid cyclones; a high-efficiency cy-clone, Figure 5.6(a), and a high-throughput design, Figure 5.6(b).

2. Get an estimate of the particle-size distribution in the feed stream to be treated.

3. Estimate the number of cyclones in parallel required.

4. Calculate the cyclone diameter for an inlet velocity of between 9 and 27 m s−1;the optimum inlet velocity has been found to be 15 m s−1. Scale the other cyclonedimensions from Figure 5.6.

5. Obtain a collection-efficiency versus particle-size curve for a feed mixture from theliterature for the standard or a prototype cyclone, as in Figure 5.7.

6. Transform the standard curves to other cyclone sizes and operating conditions byuse of the following scaling equation, for a given separating efficiency,

d = ds

[(D

Ds

)3Qs

Q· ∆ρs

∆ρ· µµs

]1/2(5.6.1)

where ds is the mean diameter of the particle separated at the standard conditions


CHAPTER 5. CYCLONES

0.5Dc

0.375Dc

Dc

0.5Dc × 0.2Dc

0.5Dc

1.5Dc

2.5Dc

(a) High-efficiency cyclone

0.75Dc

0.375Dc

Dc

0.75Dc × 0.375Dc

0.875Dc

1.5Dc

2.5Dc

(b) High-throughput cyclone

Figure 5.6: Standard cyclone dimensions.

0 20 40 600

20

40

60

80

100

Particle size / µm

Gra

deef

ficie

ncy

/%

(a) High-efficiency cyclone

0 20 40 600

20

40

60

80

100

Particle size / µm

Gra

deef

ficie

ncy

/%

(b) High-throughput cyclone

Figure 5.7: Performance curves, standard conditions.

in Table 5.1 and d is the mean diameter of the particles separated in the proposeddesign at the same separation efficiency.

7. Calculate the cyclone performance and recovery of particles (efficiency). If theresults are unsatisfactory, try a smaller diameter, i.e. more cyclones.

8. Calculate the pressure drop using,

∆P =ρf203

(u21

[1 + 2φ2

(2rtre− 1

)]+ 2u22

)(5.6.2)

where ∆P is the cyclone pressure drop (mbar), ρf is the fluid density (kg m−3, u1


5.6. STAIRMAND’S DESIGN PROCEDURE

Table 5.1: Parameters for Stairmand’s standard designs.

Parameter Standard Value

Ds 203 mmQs High efficiency = 223 m3 hr−1, High throughput = 669 m3 hr−1

∆ ρs 2000 kg m−3

µs 0.0018 mPa s

is the inlet duct velocity (m s−1), u2 is the exit velocity (m s−1). rt is the radiusof circle to which the center-line of the inlet is tangential (m), re radius of exitpipe (m), φ is the factor given in Figure 5.8; Ψ parameter in Figure 5.8 given byΨ = fc (As/Ai) where fc is the friction factor (0.005 for gases), Ai is the area ofinlet duct (m2), and As is the surface area of the cyclone exposed to the spinningfluid, where length equals total height times cross-sectional area of a cylinder withthe same diameter (m2).

Figure 5.8: Cyclone pressure drop factor [10].


CHAPTER 5. CYCLONES

5.7 Multiple Cyclones

Series connections of separators are a common way of improving the performance ofsingle-stage, one-pass installations. They are used with various types of separators, rang-ing from gravity or centrifugal separators to filters. For cyclones in particular, due totheir low capital and running costs, multiple series arrangements are an obvious choice.The particular layout and arrangement should be used depends on the actual applicationneeded.

There are five basic reasons for connecting cyclones into networks, series arrangementsor recycles can be used:

1. to lower the cut size (with the consequence of higher solids recovery and cleaneroverflow),

2. to improve thickening (with dilute feeds where a single stage cannot suffice),

3. to improve the sharpness of cut (beyond what a single stage can achieve, in size ordensity separation),

4. to wash solids in co-current or countercurrent systems,

5. to reduce abrasion (for the same performance, a two-stage system can use lowerinlet velocities than a single-stage).

There are many potential arrangements that can be used, but here we will examine thefour most common options.

5.7.1 Series Overflow

Figure 5.9(a) shows several series arrangements on overflow. These systems may or maynot use intermediate pumps, if they do not, the additional back pressure on the overflowsof the initial stages has to be compensated for by throttling the underflow orifices of thesestages. Otherwise, the underflow rates would be much greater than for the same cyclonesoperated under balanced conditions.

If we assume that all stages have identical grade efficiency curves and Rf ratios, then thesystem grade efficiencies for two-stage and three-stage arrangements would be as shownin Figure 5.9(a). The figure also shows, for comparison, the grade efficiency curve of asingle stage from which the two others were computed.

The simple overflow series reduces the cut size. It does so by repeating the separationprocess and separating a little more every time. Some contribution to the cut size reductionis also due to each stage removing some fluid into the underflow and a correspondingproportion of fines with it, i.e. there being an Rf . It can be seen that there is a law ofdiminishing returns in the cut size reduction and it does not often make much sense touse more than three stages of identical cut size in each stage in series because the furtherstages would separate very little indeed. The usual way to tackle this is to gradually“tighten up” on the cut size in the direction of the flow either by some design changes inthe geometry used or by changes in the operating conditions (e.g. increasing the pressuredrop in each stage).

If the flow ratio Rf is the same in all stages, the overall Rf is much greater, thus theoverflow series can produce very dilute underflows. Technically, it is of course possi-


5.7. MULTIPLE CYCLONES

ble to limit this effect by reducing Rf along the sequence, as less and less solids areseparated in further stages (and correspondingly less and less liquid is needed in eachunderflow).

What happens to the sharpness of cut is not very clear from Figure 5.9(a). For that, itwould be better to plot the reduced grade efficiency and plot this against d/d50. Thesecurves steepen from the single-stage arrangement to the three-stage one, thus the simpleoverflow series sharpens classification. This is mainly beneficial if the fine product isof main interest because the greater amounts of liquid passed into the underflow by thisarrangement pollute the coarse product with fine particles.

The simple overflow series may be used to reclassify the fine product or to lower the cutsize of the system. Its main applications are, therefore, classification of solids, clarifica-tion of liquids and recovery of solids. The main disadvantage is in that it increases theRf , the system value is always greater than that of any one stage in it.

Particle size

Gra

deef

ficie

ncy

/%

1

1 21 2 3

(a) Overflow

Particle size

Gra

deef

ficie

ncy

/%1

1

21

2

3

(b) Underflow

Q/q = 1Q/q = 2

Q/q = 3

Particle size

Gra

deef

ficie

ncy

/%

qQ

(c) Partial overflow recycle

Particle size

Gra

deef

ficie

ncy

/%

1

1 2 3

1 2 31 2 3

(d) Underflow recycle

Figure 5.9: Performance curves, standard conditions.

5.7.2 Series Underflow

In direct analogy with the overflow series, one can also operate separators in series inthe direction of underflow, Figure 5.9(b), but usually for very different reasons. Againassuming that all stages have identical grade efficiency curves and Rf , then the systemgrade efficiencies for two-stage and three-stage arrangements would be as shown in Fig-ure 5.9(b).

The opposite happens to what we could see for the overflow series, the cut size coarsenswith increasing number of stages and the system Rf ratio reduces. This means that par-


CHAPTER 5. CYCLONES

ticle recovery is very poor. The arrangement is very useful; however, when it comes tothickening or co-current washing, i.e. removing the fluid phase.

As in the overflow series this arrangement also sharpens the cut. However, the underflowseries is steeper in the fines region, which is critical to the quality of the coarse prod-uct, whilst it is less steep in the coarse region, and this makes the fine product unclean,compared to the overflow series.

Whether the series connection used, in the direction of the overflow or the underflow,depends very much on which of the two fractions (coarse or fine) represents the product.The product is the material of interest, which should be “clean”, i.e. free of misplacedmaterial. The overflow series in Figure 5.9(a) is used for ’refining’, i.e. for cleaning thefine product whilst the underflow series in Figure 5.9(b) is used to clean or ’wash’ thecoarse product.

5.7.3 Partial Overflow Recycle

A partial overflow recycle can be added to the overflow series arrangement as a meansof lowering the cut size further. A possibility exists to do this in a multi-stage series,but in practice these are typically only applied to a single-stage system as shown Fig-ure 5.9(c). This is quite common in some clarification duties such as cleaning of metal-working coolants.

Cyclones are particularly well suited for this arrangement. Not only the recycle reducesthe cut size through the cyclone being in series with itself, so to speak, but as an addi-tional benefit it dilutes the feed for the cyclone and this also improves the cyclone effi-ciency.

The reduction in the cut size of the whole arrangement, as opposed to that of the single-pass arrangement, is clearly a function of the recycle ratio Q/q: if the cut size of thehydrocyclone remains the same (i.e. greater recycle ratios are matched by a greater num-ber of identical cyclones in parallel), the flow ratio Rf is low (which is often the case inclarification duties) and the geometric standard deviation (i.e. the steepness) of the gradeefficiency curve is 1.8 (a typical value for hydrocyclones), then the rate of decrease in cutsize with the recycle ratio Q/q is as shown in Table 16.3. There is a law of diminishingreturns at play here so it does not make much sense to use Q/q of more than three tofour.

A word of caution is appropriate regarding the use of overflow recycles in particle sepa-ration applications. There is a real danger that fines may accumulate in the recycle andeventually plug the whole system. This is because the recycled stream contains materialappreciably finer than the system feed and the separator may not be able to remove theextra fines at a sufficient rate. Careful mass balance calculations to check this potentialproblem are recommended.

5.7.4 Underflow Recycle

Underflow recycles in conjunction with the overflow series are quite common in practice.However, the recycles are full, not partial, as was the case with the overflow recycle.With three stages, there is a choice of where the recycles are returned to, as shown inFigure 5.9(d).


5.8. REFERENCES

Underflow recycles generally reduce the amount of liquid passed into the system under-flow (if comparing with a simple overflow series where the individual stage Rf ratiosare not suitably reduced). This is at the expense of the system cut size, however, whichcoarsens with the recycles. Both of the recycle options are suitable for thickening andclarification (with the recycle to the start of the series option being a little more efficientin removing solids), with stage 1 being the thickening stage and the following two usedfor clarification.

In industrial practice, these arrangements are used whenever good solids recovery is re-quired coupled with good thickening performance. An example is in washing of wheatstarch where the low density (1500 kg m−3) and fine size (down to 2 microns) of some ofthe starch particles cause appreciable losses of starch in the system overflow.

5.8 References

[1] Beeckmans, J. M. and Kim, C. J. [1977], ‘Analysis of the efficiency of reverse flowcyclones’, The Canadian Journal of Chemical Engineering 55, 640–643.

[2] Dirgo, J. and Leith, D. [1985], ‘Cyclone collection efficiency: Comparison of ex-perimental results with theoretical predictions’, Aerosol Science and Technology4, 401–415.

[3] Iozia, D. L. and Leith, D. [1990], ‘The logistic function and cyclone fractional effi-ciency’, Aerosol Science and Technology 12, 598–606.

[4] John, W. and Reischl, G. [1980], ‘A cyclone for size-selective sampling of ambientair’, Journal of the Air Pollution Control Association 30, 872–876.

[5] Lee, K. W., A.Gieseke, J. and H.Piispanen, W. [1985], ‘Evaluation of cyclone per-formance in different gases’, Atmospheric Environment 19, 847–852.

[6] McCain, J. D., S. S, D. and Farthing, W. E. [1986], Procedures manual for the rec-ommended arb sized chemical sample method (cascade cyclones), sorieas-86-467,pb86-21867 4„ Technical report, Southern Research Institute, Birmingham, AL.

[7] Saltzman, B. E. and Hochstrasser, J. M. [1983], ‘Design and performance of minia-ture cyclones for respirable aerosol sampling’, Environmenatal Science & Technol-ogy 17, 418–424.

[8] Smith, W. B., Wilson, Jr., R. R. and Harris, D. B. [1979], ‘A five-stage cyclonesystem for in situ sampling’, Environmenatal Science & Technology 13, 1387–1392.

[9] Smith, W., Cushing, K. and Wilson, Jr.and D.B. Harris, R. [1982], ‘Cyclone sam-plers for measuring the concentration of inhalable particles in process streams’,Journal of Aerosol Science 13, 259–267.

[10] Stairmand, C. J. [1949], ‘Pressure drop in cyclone separators’, Engineering168, 409.

[11] Stairmand, C. J. [1951], ‘Design and performance of cyclone separators’, Transac-tions of the Institute of Chemical Engineering 29, 356.


5.9. PROBLEMS

5.9 Problems

5.1 A cyclone separator, 0.3 m in diameter and 1.2 m long, has a circular inlet of 25 mmin diameter (dinlet) and an outlet (doutlet) of the same size. If the gas enters at1.5 m s−1 (uin), what is the critical particle size for the theoretical separation? Con-sider that the particles have a density of 2700 kg m−3 and for air µ = 0.018 mPa sand ρ = 1.3 kg m−3 are assumed.


CHAPTER 5. CYCLONES

5.2 What diameter of cyclone do we need to treat 0.177 m3 s−1 of feed, given, µf =1.8 × 10−5 Pa s, ρf = 1.2 kg m−3, ρp = 2500 kg m−3, ∆P = 1650 Pa, d50 desiredis 0.8µm, Eu = 700, St = 6.5× 10−5.

Hint: if we use 1 cyclone, the pressure drop will be too high; so we must split thefeed into multiple, parallel cyclones. So then, how many cyclones, and of whatdiameter should we use?


5.9. PROBLEMS

5.3 Determine the diameter and number of gas cyclones required to treat 2 m3 s−1 ofambient air (viscosity 18.25 × 10−6 Pa s, density 1.2 kg m−3) laden with solids1000 kg m−3 at an optimum pressure drop of 1177 Pa and with a cut size of 4µm.Use a Stairmand HE (high efficiency) cyclone for which Eu = 320 and St = 1.4 ×10−4.


CHAPTER 5. CYCLONES

5.4 The size distribution by mass of dust carried in a gas during wheat processing isgiven in the table below together with the grade efficiency over the size range. Thedust burden at the entrance is 18 g m−3. Assume that the gas flowrate is 0.3 m3 s−1.

Calculate the overall efficiency of the collector.

What is the mass of dust leaving the gas (in kg s−1)?

Determine what is the cut size for this separation.

Size Range / µm 0-5 5-10 10-20 20-40 40-80 80-160

% mass 10 15 35 20 10 10% grade efficiency 20 40 80 90 95 100


5.9. PROBLEMS

5.5 A dryer vent is to be cleaned using a bank of cyclones. The gas flowrate is 60 m3 s−1,density of solids 2700 kg m−3 and the concentration of solids is 10 g m−3. The sizedistribution of the solids is given by:

Particle size / µm 50 40 30 20 10 5 2

% weight less than 90 86 80 70 45 25 10

Calculate the solids removal and the final outlet concentration using the collectionefficiency curve below.

0 10 20 30 40 500

20

40

60

80

100

Particle size / µm

Col

lect

ion

effic

ienc

y/%


CHAPTER 5. CYCLONES

5.6 Air carrying particles of density 2500 kg m−3 enters a cyclone with a linear velocityof 18 m s−1.

Consider the four cases presented in the table below. In each case, estimate whatfraction of the particles will be removed in the cyclone using the collection effi-ciency below in the graph below.

Case Particle Size / µm Cyclone Diameter / m

1 10 0.62 10 13 5 0.64 5 1

100 101 1020

20

40

60

80

100

Particle diameter / µm

Col

lect

ion

effic

ienc

y/%

Q = 0.5 m3 s−1, D = 0.36 mQ = 2.4 m3 s−1, D = 0.8 mQ = 12 m3 s−1, D = 1.8 m

The centrifugal force acting on a particle of mass m entering a cyclone of radius rat a tangential velocity of u is mu2/r. Provide a physically-based explanation forthe results you obtained.


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