1 4. Particulates and Coagulation, and Flocculation Main Topics : 4.1 Particulates in water 4.2 Coagulation process 4.3 Reaction kinetics 4.4 Mixing theory 4.5 Flocculation process References : 1. Crittenden, Water Treatment Principles and Design, 2005, Chapter 9. 2. Davis & Cornwell, Introduction to Environmental Engineering, 3 rd ed. 1998, McGraw- Hill. 4.1 Particulates in water Particulates of concern in water treatment are those solids larger than molecules covering a size range from 10 nm (0.01 µm or 100 Å) to 100 µm. Dissolved constituents (molecules) are considered to be smaller than 0.001 µm. The principal sources of particulates in natural water are soil erosion processes and biological activities. Clays are the main component of particulates produced by weathering. Algae, bacteria, and other microorganisms are the predominant types of particles produced biologically. Another source of particulates is from industrial activities e.g. food processing, paints, and minerals processing. These activities produce undesirable particulate residues which may enter the surface waters through either direct discharge or atmospheric deposition. Because of their fine size, particulates exhibit a large surface area which serves as a potential adsorption sink for accumulation of toxic substances, such as heavy metals and chlorinated hydrocarbons. Ingestion of the particulates may cause acute or chronic toxic effects. The large particulate surface area also causes strong scattering of incident light resulting in a high turbidity. The design of water treatment facilities depends on the characteristics of the colloidal particles. Fine particles (< 10 µm) settle slowly, and efficient removal by sedimentation requires a long time (see Table 2) or a large size facility. And for effective disinfection of water, these particulates have to be first removed as they could shield pathogens. Particulates present in natural waters are diverse and exhibit a wide range of sizes, shapes, densities, and surface chemical properties. Size Characteristics of Particulates A typical particulate suspension will consist of colloids and suspended solids. Colloids are defined as substances consisting of approximately 100 or more molecules, which cover a size range from roughly 50 Å (0.005 µm) to 1 µm. They include organic macro-molecules present in water, biologically produced debris, viruses, bacteria, clays, and inorganic precipitants. Natural organic matter (NOM) which comprises colloidal organic material smaller than 0.45
Transcript
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4. Particulates and Coagulation, and Flocculation Main Topics: 4.1 Particulates in water 4.2 Coagulation process 4.3 Reaction kinetics 4.4 Mixing theory 4.5 Flocculation process References: 1. Crittenden, Water Treatment Principles and Design, 2005, Chapter 9. 2. Davis & Cornwell, Introduction to Environmental Engineering, 3rd ed. 1998, McGraw-
Hill. 4.1 Particulates in water Particulates of concern in water treatment are those solids larger than molecules covering a size range from 10 nm (0.01 µm or 100 Å) to 100 µm. Dissolved constituents (molecules) are considered to be smaller than 0.001 µm. The principal sources of particulates in natural water are soil erosion processes and biological activities. Clays are the main component of particulates produced by weathering. Algae, bacteria, and other microorganisms are the predominant types of particles produced biologically. Another source of particulates is from industrial activities e.g. food processing, paints, and minerals processing. These activities produce undesirable particulate residues which may enter the surface waters through either direct discharge or atmospheric deposition. Because of their fine size, particulates exhibit a large surface area which serves as a potential adsorption sink for accumulation of toxic substances, such as heavy metals and chlorinated hydrocarbons. Ingestion of the particulates may cause acute or chronic toxic effects. The large particulate surface area also causes strong scattering of incident light resulting in a high turbidity. The design of water treatment facilities depends on the characteristics of the colloidal particles. Fine particles (< 10 µm) settle slowly, and efficient removal by sedimentation requires a long time (see Table 2) or a large size facility. And for effective disinfection of water, these particulates have to be first removed as they could shield pathogens. Particulates present in natural waters are diverse and exhibit a wide range of sizes, shapes, densities, and surface chemical properties. Size Characteristics of Particulates A typical particulate suspension will consist of colloids and suspended solids. Colloids are defined as substances consisting of approximately 100 or more molecules, which cover a size range from roughly 50 Å (0.005 µm) to 1 µm. They include organic macro-molecules present in water, biologically produced debris, viruses, bacteria, clays, and inorganic precipitants. Natural organic matter (NOM) which comprises colloidal organic material smaller than 0.45
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µm is referred to as dissolved organic constituents. Suspended solids are defined as the material, which can be collected on a 0.45 µm membrane filter. They cover the size range from 0.5 to 500 µm. Particulate’s physical characteristics of interest include particle length, surface area, volume, mass, and surface charge. Traditionally, particulates are characterized by light-scattering techniques e.g. Secchi disc used in limnological investigations and turbidity in water treatment. Another collective parameter to describe particulate concentration in water is total suspended solids (TSS). It is useful in the size range above 0.5 µm and is greater than 10 mg solids/L in concentration. Below these levels, measurements are not accurate and require large-volume sampling. There are various types of particulate counters which are now available that can provide particle size distribution.
Particle size The size of a spherical particle is uniquely defined by its diameter. All other irregularly shaped particles, however, may exhibit a variety of sizes depending upon the method used for size measurement (Table 1)
Table 1 Definitions of particle size.
Symbol Name Definition Formula
dv Volume diameter
Diameter of a sphere having the same volume as the particle
V = (π dv3)/6
dS Surface diameter
Diameter of a sphere having the same surface as the particle
S = π ds2
da Projected area diameter
Diameter of a circle having the same area as the projected area of the particle resting in a stable position
A = (π da2)/4
dA Sieve diameter
The width of the minimum square aperture through which the particle will pass
-
Table 2 Effect of decreasing size of spheres on settling time.
Particle diameter Order of size Total surface area* Time required to settle**
10 mm Gravel 0.000314 m2 0.3 sec 1 mm Coarse sand 0.003141 m2 3 sec
0.1 mm Fine sand 0.031419 m2 38 sec 10 µm Silt 0.31419 m2 33 min 1 µm Bacteria 3.1419 m2 55 hr
0.1 µm Colloidal particles 31.419 m2 230 days 0.01 µm Colloidal particles 314.19 m2 6.3 years 0.001 µm Colloidal particles 3141.9 m2 63 years
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* area for particles of indicated size produced from a particle 10 mm in diameter with a specific gravity of 2.65.
** based on a sphere with a specific gravity of 2.65 to settle 300 mm. Particle shape Particulates can have shapes ranging from nearly spherical, semi-spherical to elongated shape. The shape of particles will affect the electrical properties, particle-particle interactions, and particle-solvent (water) interactions. The sphericity is defined as the ratio of surface area of a sphere having the same volume as the particle to its actual area. The sphericity and porosity of a variety of different shaped materials are given in Table 3.
Table 3 Sphericity and typical porosity of granular materials.
Specific surface For a sphere, the ratio of surface area to volume is equal to:
3m
2m d6
6/3d
2d VS
Volumearea Surface
=π
π== (1)
Hence, for a spherical particle of 1 µm in diameter, its S/V ratio is 6 x 106 m2/m3. If the particle has a specific gravity of 1, i.e. a density of 1 x 106 g/m3, then its surface area to mass ratio is:
g
2m 6 6101
16101
61d61
VS
Massarea Surface
=×
×−×
=ρ
×=ρ
×= (2)
For a given volume, the sphere has the smallest surface area compared with other non-spherical particles. Hence, the S/V ratio of spheres will also be smaller. The sphericity factor Ψ is used to account for difference in the S/V ratio between non-spherical and spherical particles.
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Consider an irregular “angular” shaped particle with a Ψ value of 0.78. The S/V and S/M ratios for this angular particle are:
d6.8
d 78.06
d 6
VS
==ψ
= (3)
sd6.8
s d 6
MS
ρ×=
ρψ= (4)
For an equivalent spherical diameter d = 1 µm and specific gravity = 2.5, S/V ratio of the angular particle = 8.6 x 106 m2/m3 and S/M ratio = 3.4 m2/g. Surface charge As indicated in Table 2, surface area of colloids is very large (relative to its mass) and as a result, surface properties of colloids, such as electrostatic charges, become more important. The charge may vary in magnitude, and it may be positive or negative, according to the nature of the colloid. Metallic oxides (e.g. aluminum, iron) are mostly positively charged, while non-metallic oxides, metallic sulfides, clay, organic color and most proteins are usually negatively charged in water. Particles of like charge repel one another, whereas oppositely charged particles attract one another. The pH corresponding to a surface charge of zero is defined as the zero point of charge (ZPC). Above the ZPC, the surface charge will be negative (anionic), and below the ZPC the charge will be positive (cationic). The subscripts “am” refer to amorphous (i.e. having no fixed form or shape) solids.
Table 4 ZPC of inorganic and organic particulate (Crittenden, 2005).
Besides surface charge, colloidal particles can be hydrophobic (water-hating) or hydrophilic (water-loving). Colloids usually encountered in water treatment are mostly hydrophobic. Soap, detergents, soluble starch, and soluble proteins are examples of hydrophilic colloids. Stability of these colloids (hydrophobic and hydrophilic) depends upon the hydration and the electric charge on their surface.
Stability of Colloids In the case of hydrophilic colloids, they are stabilized by the formation of adherent thick layers of oriented water molecules around the particle. For hydrophobic colloids, they are stabilized by an electrostatic repulsion between the particles arising from ions that are attracted to the surface from bulk solution or dissolved out of the solid’s surface (Fig. 4a). Colloids found in natural waters typically acquire negative charges on their surfaces either by ionization of surface functional groups or by adsorbing negatively charged organic molecules. When a charged particle is immersed in water, and the bulk of the solution is neutral electrically, a layer of oppositely charged ions, from the bulk solution or from the particle itself, is attracted to the particle surface. This is called the “Stern layer”, which is about 5 nm thick (Fig. 1). A second layer of diffuse cloud of ions, also predominantly of charge opposite to that of the particle surface, extends several hundred nm into the solution surrounding the Stern layer. These two layers of counter-ions surrounding the particle are called the electrical double layer. A gradient of counter-ions – from high concentration in the inner Stern layer to low concentration in the outer bulk solution is thus formed. When the particle moves, the bound water moves with it giving rise to a shear plane. The electrical potential between the shear plane and the bulk solution is called zeta potential, which is responsible for the repulsive force between colloidal particles. A suspension of particles held apart by electrostatic repulsion is said to be stable: the particles will not combine to form large aggregates even if brought into contact by vigorous mixing. We can reduce the repulsive force or destabilizing the particle suspension by changing the solution chemistry.
Fig. 1 Diffuse double layer.
Fig. 2 Forces acting on a colloid
(Qasim et al.).
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Instability of Colloids Instability of colloidal particles is caused by Brownian motion and van der Waals forces of attraction. Due to Brownian motion, particles come close to each other, so that the influence of interacting forces may become effective. The van der Waals force is significant only when particles are very close to one another; otherwise it is small compared to the repulsive force. Thus, whether particles will repel or attract one another depends on the resultant of the electrostatic repelling and the van der Waals forces. If the electrostatic repelling force is substantial, it gives rise to an energy barrier that must be overcome before the particle can approach closely enough to adhere to one another (see Fig. 2). The zeta potential can be lowered by coagulation through the addition of ions of opposite charge to the particle surface. The absolute value of the zeta potential has to be reduced to lower than 20 mV before flocculation can occur.
4.2 Coagulation Process Coagulation is the process that involves the addition of chemical coagulant(s) to reduce the repulsion forces between particles or the neutralization of the charges on the particles. Traditionally, metal salts such as aluminum sulfate or alum, ferric sulfate, ferric chloride, and ferrous sulfate have been utilized as coagulants. In recent years, polymers have been used in conjunction with, or in lieu of, metal salts to enhance the coagulation process. When a metal salt is added to water that has sufficient alkalinity, it will hydrolyze into complex metal hydroxides of the form Meq(OH)p (Me = metallic ion). The actual hydroxide formed depends on the pH of water and the coagulant dosage as shown in Fig. 3 for aluminum hydroxide. Aluminum hydroxide is precipitated within the shaded area while polynuclear and polymeric species are formed outside the shaded areas at higher and lower pH values. The rectangle box in the figure indicates the dosage rates and pH ranges typically used in the coagulation process in water treatment.
Fig. 3 Equilibrium solubility domain of aluminum hydroxide in water (Reynolds).
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Mechanisms of destabilization The effective removal of colloidal particles from water depends on a reduction in particle stability. Several mechanisms contribute to the destabilization of particles: compression of the electrical double layer, charge neutralization, interparticle bridging, and enmeshment or “sweep coagulation”. Double-layer compression: This is accomplished by adding a coagulant having a positive charge to counteract the negatively charge colloids. The counterions then enter into the double layer. As its concentration in the solution increases, it causes the net charge in the diffused layer to neutralize and result in the compression of this layer (Fig. 4b), and the zeta potentials will decrease. This effect increases greatly as the valence of the counterion is bigger. The order of effectiveness is Me3+ > Me2+ > Me+. The van der Waals force thus becomes predominant when the colloids are close enough, and allow them to agglomerate into a floc. Although coagulants such as aluminum and ferric salts used in water treatment ionize, at the concentration commonly used, they would not increase the ionic concentration sufficiently to affect ion layer compression.
Fig. 4 Coagulation of colloids (a) stable suspension of particles where repulsive force greater than attraction force (b) addition of coagulant suppress double layer charge (c) agglomeration
of destabilized particles by coagulant and polymer bridging.
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Fig. 5 Forces acting on a colloid after compression of double layer (Qasim et al.).
Counter-ions adsorption for charge neutralization: The counter-ions from the coagulant can also be adsorbed onto the surface of the colloidal particles (Fig. 6d). The repulsive charges on the particle surface may be fully neutralized. The destabilized particles can adhere to each other to form a floc. However, the net charge on the particle may be reversed by the adsorption of an excess of counter-ions.
Fig. 6 (a) zeta potential (b) to (c) reduction in zeta potential due to compression of ion layer (d) adsorption and charge neutralization.
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Enmeshment in precipitated solids: The dosage of metal salts used in coagulation is usually slightly in excess of the amount required for reduction of the zeta potential. The excess metal salts hydrolyze into hydroxides which are extremely insoluble in water. As the hydroxide precipitate forms (Al(OH)3(s), Fe(OH)3(s), and accumulates, small colloidal particles are entrapped or enmeshed in the sticky flocs settle. This process by which colloids are swept from suspension is known as sweep coagulation. Interparticle bridging: This occurs when a large polymer molecule is adsorbed onto the surfaces of separate particles. The resulting structure grows into a single particle several times larger than the individual colloids (Fig. 4c). An excess dosage of polymer may cause restabilization of the destabilized particles. Chemical agents for coagulation The coagulants most widely used for removing turbidity, color, taste, odor, bacteria and surface charge of particles in water treatment are compounds of iron and aluminum and polymers: Alum (hydrated aluminum sulfate) Ferric (or ferrous) chloride or sulfate Organic polymers Aluminum sulfate, the most commonly used coagulation in water treatment, is most effective between pH ranges of 5 and 7.5. Ferric chloride is effective down to pH 4.5, and ferrous sulfate, effective only above pH 9.5, are sometimes used. Although they generally produce a dense, rapid-settling floc, they can cause color problems if the precipitate is not removed completely. Alum dosages may range from 5 mg/L to 50 mg/L, depending upon the turbidity and nature of the water. At low turbidity and high dosage, Al(OH)3 is almost certain to form so that the predominant turbidity-removal mechanism is sweep coagulation. At high turbidity and lower dosages, adsorption and charge neutralization will be the predominant mechanism. Reactions of alum and iron coagulants in water Aluminum Sulfate: The ionization of alum (hydrated aluminum sulfate Al2(SO4)3 .x H2O where x = 14) in water produces sulfate anions and aluminum cations. The sulfate ions may remain in this form or combine with other cations. Al2(SO4) 3 . 14H2O ⇔ 2Al3+ + 3SO4
2- + 14H2O (reactions very fast, ms) (5)
However, the Al3+ cations react immediately with water to form a variety of soluble mononuclear (one aluminum ion) and polynuclear (several aluminum ions) species. Examples of mononuclear species are Al(OH)2+, Al(OH)2
+.
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The predominant solid product (minimum solubility at pH 5.5) is Al(OH)3 (s):
Al3+ + 3H2O ⇔ Al(OH)3 (s) + 3H+ (reactions slower, 1 – 7 sec) (6) The typical dosage of aluminum salts in coagulation (10 – 150 mg/L) will result in an aluminum precipitate well within the solid region of Al(OH)3 (s) at pH values of 6 – 8 (Fig. 3). Aluminum is least soluble within a pH range of 5.7 to 6.2. Below pH 5.7, some of the polynuclear species is quite soluble. For Al(OH)3 (s) to occur, alum dosages must be greater than 10-6 M and pH between 6 and 8. The hydrogen ions released (Eqn. 6) will react with the alkalinity in the water to yield: MW: 594 3 x 100 CaCO3 2 x 78 3 x 136 6 x 44 14 x 18 Al2(SO4)3.14H2O + 3Ca(HCO3)2 → 2Al(OH)3↓ + 3CaSO4 + 6CO2 + 14H2O (7) Aluminum Calcium Aluminum Calcium Carbon Water sulfate bicarbonate hydroxide sulfate dioxide Equation 7 predicts that each g of alum will consume approximately 0.5 g (as CaCO3) of alkalinity and produce 0.26 g of hydroxide sludge and 0.44 g of carbon dioxide. Note that non-carbonate hardness (calcium sulfate) is also produced in the water. If the alkalinity is not sufficient to react with the alum and buffer the pH, it is necessary to add alkalinity to the water in the form of lime, sodium bicarbonate, soda ash, or other similar chemical. Reactions of alum with calcium hydroxide (hydrated lime) as a source of alkalinity:
Al2(SO4)3.14H2O + 3Ca(OH)2 → 2Al(OH)3↓ + 3CaSO4 + 14H2O (8) Slaked lime (milk of lime), Ca(OH)2 is produced by reacting quicklime CaO with water in lime-slaking equipment. Quicklime is available in the dry form as granules or lumps and usually contains 70 to 96% CaO. Reactions of alum with soda ash as a source of alkalinity: Al2(SO4)3.14H2O + 3Na2CO3 + 3H2O → 2Al(OH)3↓ + 3Na2SO4 + 3CO2 + 14H2O (9) Ferrous Sulfate: It also requires alkalinity in the form as hydroxide ion (e.g. hydrated lime) in order to produce a rapid reaction. This reaction is an oxidation-reduction reaction – dissolved oxygen in the water is reduced:
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2FeSO4.7H2O + 2Ca(OH)2 + 0.5O2 → 2Fe(OH)3↓ + 2CaSO4 + 13H2O (10) For this reaction to occur, the pH must be raised to about 9.5 where the ferrous ions are precipitated as ferric hydroxide. Ferrous sulfate and lime coagulation is usually more expensive than alum but the ferric hydroxide precipitate formed is a dense and quick-settling floc. Ferric Sulfate: The simplified reaction of ferric sulfate with natural bicarbonate alkalinity to form ferric hydroxide is given in Eqn. 11. This reaction also produces a dense and rapid-settling floc. Slaked or hydrated lime may be used if natural alkalinity is insufficient. The optimum pH range for ferric sulfate is from about 4 to 12 since ferric hydroxide is relatively insoluble within this range.
Fe2(SO4)3 + 3Ca(HCO3)2 → 2Fe(OH)3↓ + 3CaSO4 + 6CO2 (11) Ferric Chloride: The simplified reaction of ferric chloride with natural bicarbonate alkalinity to form ferric hydroxide is given in Eqn. 12 and that with slaked lime is given in Eqn. 13. The optimum pH range for ferric chloride is also from about 4 to 12 and the floc formed is generally a dense and rapid-settling floc.
Pre-hydrolyzed metal salts It is difficult to control the metal species formation when using alum or iron salts especially at low dosages. This led to the development of prehydrolyzed metal salts – prepared by reacting alum or ferric with various salts (e.g. chloride, sulfate) and water, and hydroxide. One such commercial prehydrolyzed alum salts is PACl which can form tougher and denser floc and is less temperature dependent than unmodified alum salts. Organic Polymers These are long-chain molecules consisting of repeating chemical units and are often called polyelectrolytes. Organic polymers are often used as a flocculant aid to promote the formation of larger and more shear-resistant flocs after the particles have been destabilized to some extent with metal coagulants by forming a bridge between particles. Although they can destabilize particles through charge neutralization, they are not generally used as primary coagulants for the destabilization of particles. They are also used a filter aid to promote large floc formation in direct filtration (without coagulation and sedimentation). The typical dosage is 1 to 10 mg/L for sedimentation and 0.005 to 0.05 mg/L before granular filtration in the case of direct filtration. Polymers can be from natural or synthetic origin - but synthetic organic polymers used more often because it is cheaper. Cationic organic polymers used in water treatment are generally water soluble to adsorb readily with particulates. Coagulant aids The coagulation process can be enhanced by adding insoluble particulate materials such as clay (bentonite, kaolinite), sodium silicate, diatomite, powdered activated carbon, and fine sand. This is especially so for waters that contains low concentration of particles. Their addition helps to form nucleating sites with inorganic coagulants, organic polyelectrolyte, or both. Since the density of these materials is heavier than most floc particles, they can settle faster. Flocculation aids Flocculation aids improve the flocculation process by making the floc stronger. Organic polymers and the inorganic activated silica are the two common flocculant aids added to stabilized particles. Effects of NOM on coagulation processes Natural organic matter is a complex matrix of organics originating from natural sources (e.g. low molecular weight acids, amino acids proteins and polysaccharides, fulvic and humic acids) having a wide range of molecular weights. NOM is usually measured in terms of TOC (total organic carbon) , DOC (dissolved organic carbon), UV254 absorption, and specific UV absorbance (SUVA). NOM can react or bind with metal ion coagulants. The most effective metal salts for the removal of NOM, in order of increased effectiveness, are iron, alum, and PACl.
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Selecting optimum conditions for coagulation The chemical parameters involved in the coagulation process are: choice of coagulation, dosage, and pH value of water. Physical parameters include reaction time and mixing intensity of the chemicals. The chemical reactions give an idea as to how the reactions proceed, but the equations cannot be used to predict the actual amount of chemicals required owing to the complexity of natural particle surfaces and the complexity of coagulant chemistry. Therefore, selection and optimum dosages of coagulants are determined experimentally by the “Jar test” instead of quantitatively by equations. The important factors to be evaluated from jar tests are floc strength, size, and settling rate. The test is performed with several (usually six) 1-L beakers and a mechanical stirring device. Each of the six jars is filled with 1-L raw water whose color, turbidity, pH, and alkalinity have been predetermined. One jar is used as a control and the remaining five are dosed with different amounts of coagulant. The water is mixed rapidly for 1 minute to ensure complete dispersion of the chemicals, and then slowly mixed for 15 to 20 min to aid in the formation of flocs. The mixture is allowed to settle and the supernatant is tested for the quality parameters. The optimum dose for maximum removal of color and turbidity is selected. The pH may also vary and the optimum value for effective coagulation determined by repeating these steps.
Fig. 8 Typical jar test results using alum and ferric sulfate.
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Measure of effectiveness: CF/Co = Final turbidity (or SS concentration)/Turbidity (or SS conc.) before treatment A typical result of CF/Co versus coagulant dose is shown in Fig. 9: Zone I: stable suspension Zone II: destabilized suspension Zone III: destabilized suspension (charge reversal) Zone IV: enmeshment
Fig. 9 CF/Co versus coagulant dose and corresponding EPM values. Surface charge of particles can be measured indirectly by observing the velocity of particle movement in an electric field. The measurement is expressed EPM and is proportional to surface charge at the shear plane:
Electrophoretic Mobility, EPM = velocity/(electric field strength) = [µm/s]/[Volts/cm]
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4.3 Reaction Kinetics Many reactions take place in water treatment: removal of organic matter from water, gas transfer into and out of water, and disinfection of water. They do not reach equilibrium quickly and the study of how these reactions proceed is called reaction kinetics. The rate of reaction, r, represents the rate of formation or disappearance of a compound. Reactions that take place in a single phase (i.e. in liquid, gas, or solid) are called homogeneous reactions. Those reactions that take place at surfaces between phases are called heterogeneous reactions.
For homogeneous reactions: )(unit time ume)(unit vol
milligramsor moles r =
For heterogeneous reactions: )(unit time surface)(unit
milligramsor moles r =
A positive r infers the production of a compound and a negative r implies the disappearance of a substance. Reactions rates depend on temperature, pressure, and the concentration of reactants. In general, the stoichiometric form of reaction is as follows:
aA + bB → cC (14) where a, b, and c are the proportionality coefficients for the reactants A, B, and C. The change in concentration of compound A is equal to the reaction rate equation for compound A:
γβα =−== k[C] [B] ]A[kr dt
]A[dA (15)
where [A], [B], and [C] are the concentrations of the reactants, and α, β, and γ are empirically determined exponents. The proportionality term k is called the reaction rate constant. k is not a constant but is dependant on temperature and pressure. Since A and B are disappearing, the sign of the reaction rate equation is negative. C is positive because it is being formed. The order of reaction is defined as the sum of the exponents in the reaction rate equation. The exponents may be either integers or fractions. Examples of reaction order are: Reaction order Rate equation Zero rA = − k First rA = − k[A] Second rA = − k[A]2 Second rA = − k[A] [B]
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Reactors Physical, chemical, and biochemical reactions take place in tanks called reactors. Reactors are classified based on their flow characteristics and their mixing conditions. In batch reactors, materials are added in the tank and mixed for sufficient time to allow reaction to occur and then drained. Although the reactor is well mixed, and the contents are uniform at any instant of time, the composition within the tank changes with time as the reaction progresses. Thus, a batch reaction is unsteady (i.e. changes with time). In flow reactors, material flows into, through, and out of the reactor at all times. They are further classified based on mixing conditions as completely mixed flow reactors where mixing conditions are uniform throughout the tank, and plug-flow reactors in which the content passes through the tank in sequence. For time-dependent reactions, the time that a fluid particle remains in the reactor affects the reaction complete time. In ideal reactors, the time in the reactor (detention time or retention time) is defined as:
QV to = (16)
where to = theoretical detention time, s V = volume of fluid in basin, m3
Q = flow rate into basin, m3/s In actual reactors, they do not behave as ideal reactors because of density differences due to temperature or density, short circuiting because of uneven inlet or outlet condition, and local turbulence or dead spots in the tank corners. As a result, the detention time in real tanks is generally lower than the theoretical detention time using Eqn. 16. Reactor Design Equations 1. Complete–mix batch reactor (CMB)
• Closed system: No input or output for t > 0 • Reactants added at t = 0 • Reaction proceeds until reactants are depleted or equilibrium is reached • Ci varies with time, but is uniform throughout the reactor
Mass balance:
V, Ci
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[In] – [Out] + [Rate of Reaction] = [Rate of Accumulation] V. r(Ci) = V. (dCi/dt) [In] = [Out] = 0 (no flow in or out during reaction)
V = reactor volume; Ci = concentration of reactant in reactor at t
∫ ==∫t
0
EC
IC i
i tdtrCdC CE = final Ci in reactor; CI = initial Ci
(a) First order reaction (irreversible):
– r(Ci) = k Ci
ii kC
dtdC
−= CE = CI e–k t
Time required to achieve a given value of CE : ⎟⎟⎠
⎞⎜⎜⎝
⎛=
E
ICCln
k1)t(
(b) Second order reaction (irreversible):
– r(Ci) = k Ci
2
2i
i kCdt
dC−= ∫−=∫
t
0
EC
IC2
i
i dtkCdC
t kC1 EC
ICi−=⎥
⎦
⎤−
I
IE C t k1
CC+
=
Time required to achieve a given value of CE : ⎟⎟⎠
⎞⎜⎜⎝
⎛−= 1
CC
kC1)t(
E
I
IF
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2. Plug flow reactor (PFR)
Mass balance on a differential element of length ∆x
[In] – [Out] + [Rate of Formation] = [Rate of Accumulation]
QCix – QCix+∆x + r(Ci) ∆V = Vt
Ci ∆∂∂
(a) First order reaction:
CE = CI e– k t CE = concentration of effluent
CI = concentration of influent
Time required to achieve a given value of CE : ⎟⎟⎠
⎞⎜⎜⎝
⎛=
E
ICCln
k1t
(b) Second order reaction:
I
IE C t k1
CC+
=
Time required to achieve a given value of CE : ⎟⎟⎠
⎞⎜⎜⎝
⎛−= 1
CC
kC1t
E
I
I
Q CI
Q CE
∆x
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3. Completely-mixed flow reactor (CMFR)
Mass balance:
[In] – [Out] + [Rate of Reaction] = [Rate of Accumulation]
QCI – QCE + V. r(Ci) = V.dt
dCi = 0 at steady state
For CMFR, Ci = CE (concentration in effluent same as that in the reactor)
(a) First order reaction:
tk1CC I
E +=
Time required to achieve a given value of CE : ⎟⎟⎠
⎞⎜⎜⎝
⎛−= 1
CC
k1t
E
I
(b) Second order reaction:
tk2ktC411
C IE
+±−=
Time required to achieve a given value of CE : ⎟⎟⎠
⎞⎜⎜⎝
⎛−= 1
CC
kC1t
E
I
E
Summary: 1. For CMB and PFR: CE = CI e– k t (for 1st order reaction dC/dt = kC)
% removal = 100 ⎟⎟⎠
⎞⎜⎜⎝
⎛−
I
ECC1 = 100 )e1( kt−−
2. For CMFR or CSTR (completely-mixed stirred tank reactor)
tk1
CC IE += (for 1st order reaction)
% removal = 100 ⎟⎟⎠
⎞⎜⎜⎝
⎛−
I
ECC1 = 100 ⎟
⎠⎞
⎜⎝⎛
+−
kt111
20
4.4 Mixing theory The two different types of mixing applied to water treatment is (1) rapid mixing used to uniformly blend chemicals in the water and (2) slow mixing used in flocculation to promote contacts between particles. When water is mixed, the largest eddies start out at about one-half to one-third the dimension of the mixing device and the size of the smallest eddies is determined by the boundary between the turbulent (macroscale) and viscous (microscale) flow. Mass transfer in the macroscale region mainly takes place by turbulent diffusion and some molecular diffusion. In the microscale, mass transfer is by molecular diffusion only. The size of the smallest eddy η could be estimated from the amount of energy being dissipated in the system:
4/13
⎟⎟⎠
⎞⎜⎜⎝
⎛
ευ
=η (17)
where η = size of smallest eddy, m
υ = kinematic viscosity, m2/s ε = energy dissipated rate at point of interests, J/kg.s
The energy dissipation rate in the mixing vessel or basin is not uniform throughout and the overall average rate of energy dissipation is used:
MP
=ε (18)
where ε = average energy dissipation per unit mass of basin, J/kg.s
P = power of mixing input to entire basin, J/s or W M = mass of water in basin, kg
Another parameter – the root mean square velocity gradient G is used as a design parameter for flocculation:
VPGµ
= (19)
where G = velocity gradient, 1/s
P = power of mixing input to entire basin, J/s or W V= volume of basin, m3 µ = dynamic viscosity, Ns/m2
In rapid mixing, a small flow of chemical solution (e.g. alum solution) is blended with a much larger flow of the water to be treated. The degree of blending can be assessed by (1) the uniformity of the chemical in the water with time and space and (2) the time it takes to accomplish a specified degree of uniformity (mixing).
21
Uniformity of blending The uniformity of the blend can be determined with respect to concentration variations with time: instantaneous concentration at time t, Ct; average concentration over time, C ; and standard deviation of the concentration in the stream with time, σ:
( )1n
CCn1t
2t−
∑ −=σ = (20)
For a specified uniformity of blend, the standard deviation is normalized to the average concentration and is taken as the coefficient of variation COV:
%100C
COV ×⎟⎠⎞
⎜⎝⎛ σ= (21)
where COV = coefficient of variation with time
σ = standard deviation of concentration, mg/L C = average concentration over time, mg/L
Fig. 10 Typical concentration variation at downstream of dosing point
(Crittenden et al., 2005) Further, the degree to which the two streams have been mixed can be described by the intensity of segregation, Is. When Is = 0, the two streams are completely mixed; when Is = 1, the two streams are completely unmixed.
2
u
msI ⎟⎟
⎠
⎞⎜⎜⎝
⎛σσ
= (22)
22
where Is = Danckwerts’ intensity of segregation σm = standard deviation of concentration in mixed stream σu = standard deviation between two streams in unmixed condition
σu can be estimated based on the relative flow rates of the two streams:
aw
aa QQ
QX+
= (23)
and aw X1X −= (24)
σu (vol) ≈ )X1(X aa − (25) where aX = volume fraction of stream containing chemical a in unmixed condition
Qa = flow rate of solution stream of chemical a, m3/s Qw = flow rate of water stream being treated, m3/s
wX = volume fraction of water in unmixed condition σu (vol) = standard deviation of concentrations before mixing (expressed as volume
fraction) In terms of chemical dosage, the mass balance relating flows and concentrations is:
Qa Ca = Qw Cdose (26)
and dosea
dosea CC
CX+
= (27)
where Ca = concentration of chemical a in feed stream, mg/L
Cdose = dose of chemical a to be applied to water stream, mg/L Example 1: A water treatment plant must dose the water with 30 mg/L of alum so that the COV of the blend is ≤ 5%. Estimate the Is value. Assume alum solution in the feed stream is 651,000 mg/L.
dosea
dosea CC
CX+
= = 4.61 x 10-5 )X1(X aau −=σ = 0.00679
%100C
COV ×⎟⎠⎞
⎜⎝⎛ σ= %100
30%5 m ×⎟
⎠⎞
⎜⎝⎛ σ= → σm = 1.5 mg/L
→ σm (vol) = 1.5/651000 = 2.3 x 10-6
2
u
msI ⎟⎟
⎠
⎞⎜⎜⎝
⎛σσ
= = 1.15 x 10-7 ≈ 0 hence alum is completely mixed
23
Time required for blending Besides uniform blending, the time required to accomplish it is equally important. There are two cases where the time required for a particular mixing tm is important (1) when blending must be completed for analysis and control purposes (2) when blending must be completed rapidly to prevent adverse outcomes. When irreversible reactions are involved, it is important to accomplish blending rapidly to avoid adverse reactions/outcomes. Three classes of reactions have been proposed:
1tt
m
k >> (slow reaction) (28)
1tt
m
k ≈ (moderate reaction) (29)
1tt
m
k << (fast reaction) (30)
where tk = time characteristic of reaction of interest e.g. reaction half-life, s
tm = time characteristic of mixing e.g. time required to achieve COV ≤ 5%, s Some of the characteristic times of reactions commonly found in drinking water treatment are given in Table 5:
Table 5 Characteristic times of common reactions (Crittenden, 2005).
Reaction tk (s) Remarks Coagulation with Fe3+ or Al3+ < 0.3 Fast, poorly reversible pH adjustment << 1 Fast, easily reversible HOCl & NOM to DBPs ∼ 90,000 Very slow, not reversible Chlorine hydrolysis 0.06 Fast, easily reversible Chlorine/ammonia (low Cl2/N) < 0.1 – 2000 Fast in early stages, poorly reversible Chlorine/ammonia (high Cl2/N) 104 – 106 Extremely slow, poorly reversible HOCl/coliform ∼ 10 Fast, persistent residual kill HOCl/Giardia ∼ 200 Slow, persistent residual kill HOCl/Cryptosporidium ∼ 90,000 Slow, persistent residual kill Rapid mixing Rapid mixing refers to the operation when water is violently agitated to ensure uniform mixing of the added chemical. Turbine impellers or propellers are commonly used for the rapid mixing and the design parameters are mixing time, t, and velocity gradient, G. Mixing devices are designed to achieve a certain intensity of mixing represented by the G value instead the COV value of ≤ 5%. The earlier formula for G can be used or mechanical or pneumatic agitation:
24
VPGµ
= (19)
The velocity gradient for baffle basins is given by:
th G
µγ
= (31)
where γ = specific weight of water, N/m3
h = head loss due to friction, turbulence etc., m t = detention time, s The rate of particulate collisions is proportional to G and sufficient gradient must be furnished to achieve the desired rate of collisions. However, G is also related to the shear forces in water; large G produces appreciable shear forces that can prevent floc formation. The total number of particle collisions is proportional to the product of G and the detention time, t. Detention times in rapid-mix basins are from 20 to 60 sec although some have t as small as 10 sec or as long as 2 to 5 min. Velocity gradients normally range from 100 to 1000s-1. Typical G values for the respective t are: t = 20 s, G = 1000s-1; t = 30 s, G = 900s-1; t = 40 s, G = 790s-1; t > 50 s, G = 700s-1. Under turbulent flow conditions where Reynold’s number Re > 104, the power imparted to the water by a mixer can be expressed as:
P = Np ρ n3 d5 (32)
Reynolds number µ
=ρnd Re
2 (33)
where Np = power number of impeller
d = impeller diameter, m n = impeller speed, revolutions per sec, rps ρ = mass density of water, kg/m3
The power number depends on the type of impellers, blade number and blade width to diameter ratio.
25
Fig. 11 Rapid-mix systems.
26
4.5 Flocculation Process Flocculation is defined as the aggregation of destabilized particles into larger particles known as flocculent particles or “floc”. The aggregation of colloidal particles takes place in two separate and distinct phases: (1) the repulsion force between particles must be overcome; this requires that the particles be destabilized; and (2) contact between the destabilized particles must be induced so that aggregation can occur. As mentioned earlier, the destabilization step is achieved by addition of chemicals to modify the electrochemistry properties on the particle surfaces. This coagulation process step is virtually instantaneous, in milliseconds to seconds, following addition of the coagulant in rapid mix tanks. The aggregation step on the other hand, requires more time for development of large flocs, by gentle stirring in the flocculation tanks. A quantitative description of this aggregation process will provide a basis for designing coagulation-flocculation facilities, as well as identifying the key process parameters. Kinetics of Particulate Aggregation Quantitative measures: di = diameter of particles of size class i ni = total number of particles in size class i N = total number of particles of all sizes. Consider particle of size di collide with size dj particles forming particles of size dk. At the same time, aggregates of size dk may break up into smaller aggregates due to hydrodynamic shearing forces. The rate of flocculation is given by the general form:
⎥⎦⎤
⎢⎣⎡ βα= jiijij nn21 -
dtdN (34)
where αij = attachment efficiency βij = collision frequency The attachment efficiency αij is defined as the ratio of successful to unsuccessful particle collisions, and has a range of values between 0 ≤ α ≤ 1. Successful particle collisions occur when two particles remain attached after the collision. It depends on the effectiveness of destabilization. (α = 1 for perfectly destabilized particles). The collision efficiency βij is a function of the probability that an individual particle will collide with another in a unit of time. It depends on the transport processes. The sign of Eqn. 34 is negative because each successful event of collision/adhesion reduces the particle number concentration NT. The value of βij can be estimated for different size classes using different transport mechanisms:
27
βij = βPK + βOK + βDS (35) where βPK = collision frequency due to Brownian motion (micro-flocculation or perikinetic
flocculation) βOK = collision frequency due to shear motion (macro-flocculation or orthokinetic flocculation) βDS = collision frequency due to differential settling (or sedimentation)
The volume of floc per unit volume of water i
3i n
6dπ
=Ω (36)
where Ω = floc volume, cm3 of particles/cm3 of water
di = particle diameter, cm ni = number of particles/cm3 or mL
Micro-flocculation or Perikinetic Flocculation, PK Brownian motion affects the movement of colloidal particles of 1 µm or smaller. For a suspension of single-sized spherical particles, the collision frequency function is given by:
⎟⎟⎠
⎞⎜⎜⎝
⎛µ
=βTk
38 B
PK (37)
where βPK = collision efficiency, m3/s
kB = Boltzmann constant = 1.38 x 10-23 J/K T = absolute temperature, K (=273 + oC) µ = dynamic viscosity of water, Ns/m2 or kg/m.s In the following, βij is assumed to be constant, i.e. the particles are uniform in size (N = ni). In reality, particles are non-uniform in size, density etc. Therefore βij must be evaluated separately for each combination of particle classes i,j. Substituting Eqn. 37 into 34 gives the following result for the Perikinetic flocculation (aggregation) rate:
2B N Tk34
dtdN
⎟⎟⎠
⎞⎜⎜⎝
⎛µ
α−= (38)
Perikinetic flocculation works very slowly in most situations.
28
Example 2 A suspension initially contains particles of only one size (i.e. mono-disperse suspension). Estimate the half-life of these particles in a well-mixed batch reactor if the total floc volume fraction is 10-10. Assume water temperature = 15oC and that flocculation proceeds only by Brownian motion. There is no floc break up and α = 1. Compute the half-life for particle diameters of (i) 100 Å (virus), (ii) 0.1 µm, and 1 µm. Note: 1 m = 1010 Å; kB = 1.38 x 10-23 J/K; µ = 1.139 x 10-3 Ns/m2 (i) half-life for 100 Å particles
Particle volume = 3253
10
3m1024.5
A10m1A100
66d −×=⎥
⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛×
π=
π&
&
From Eqn. 36: i
3i n
6dπ
=Ω = 10-10 m3/m3
Number of particles No = 1425
1010908.1
1024.510
×=× −
−particles/m3
From Eqn. 38: 2B N Tk34
dtdN
⎟⎟⎠
⎞⎜⎜⎝
⎛µ
α−=
Separate variables and integrate: ∫µ
α−=∫ t0
BNNo 2 dtTk
34
NdN
Note: O
N
No
N
No 2 N1
N1
N1
NdN
+−=⎥⎦⎤
⎢⎣⎡−=∫ and tdt
t
0=∫
Therefore tN
3Tk41
NNo
Bo
µα
+= or
oB
o
N3
Tk4
1N
N
t
µα
−=
For half-life, N = 0.5 No (initial number of particles) Substituting the values into the equation, half-life t1/2 = 0.313 hours Similarly, compute the half-life for the other 2 particle sizes.
29
Macro-flocculation or Orthokinetic Flocculation, OK In order to accelerate particle aggregation for colloidal particles and achieve acceptable flocculation rates for coarse-sized particles (> 1 µm), mechanical mixing must be employed. For spherical particles moving under laminar flow conditions, the collision frequency function is given by:
iOK GV8π
=β (39)
where G = average velocity gradient Vi = volume of particles of size di = π d3/6 Substituting Eqn. 39 into 34 gives the following results for the Orthokinetic flocculation (aggregation) rate:
N G 4 dtdN
Ωαπ
−= (40)
where Ω = floc volume fraction = volume of floc/volume of water = No Vp = No 6d3π
For most treatment processes, orthokinetic flocculation predominates for particles > 1 µm. Even in quiescent waters, orthokinetic mechanism also predominates for bigger particles (> 10 µm). Example 3 Calculate the time required to reduce the number of particles by 50% under laminar conditions for macroscale flocculation, assuming first order kinetics in a CMB, without and with the addition of a coagulant. Assume the initial particle concentration before alum addition is 10,000/mL, G =60 s-1, and α = 1. The particle diameter is 1 µm. After the addition of 100 mg/L of coagulant, the floc that is formed has a density of 1 g/mL. Assume 1 mg of coagulant produces 1 mg of floc. Flocculation time without coagulant addition:
Volume fraction of particles i
3i n
6dπ
=Ω = 5.2 x 10-9
N G 4 dtdN
Ωαπ
−= = -kN (first order) where k = (4 Ω G α) /π = 3.973 x 10-7
ln (N/No) = - kt Time needed to achieve 50% particle reduction (half-life) i.e. N = 0.5 No
30
k)N/N5.0ln(t oo
2/1 −= = 20.2 days
With coagulation addition: After addition of 100 mg/L coagulant, floc formed with a density = 1 g/mL or 106 mg/L Volume of floc = 100 mg/106 mg/L = 10-4 L
Floc volume 44
10L1
L10mewater volu
volumefloc −−
===Ω
k = (4 Ω G α) /π = 7.639 x 10-3
k)N/N5.0ln(t Oo
2/1 −= = 90.7 sec
Differential settling (or Sedimentation) The velocity of particles of similar densities settling in a water column is proportional to the size squared. Thus, differential particle motion occurs in heterogeneous suspensions during sedimentation, providing an additional transport mechanism for promoting flocculation.
]dd][)dd[( 72
g ji
3jiDS −+
µαρ∆π
=β (41)
where ∆ρ = difference in density between the particle and water = ρp – ρw
Fig. 12 Collision frequency for different transport mechanisms.
31
Example 4: Perikinetic flocculation in a complete-mix flow reactor (CMFR) NI = number of particle in influent per unit volume NE = number of particle in effluent per unit volume Q = flow rate, m3/s V = volume of reactor, m3
Balance of particle number in reactor: In – Out + “Reaction” = Accumulation
(Q NI – Q NE) ∆t + V dtdN
∆t = 0 (assume no accumulation) → )NN(t1
dtdN
EI −−=
Divide by V and substitute t = V/Q; and substitute 2B N Tk34
dtdN
⎟⎟⎠
⎞⎜⎜⎝
⎛µ
α−= (2nd order)
0N .ttanCons)NN(t1 2
EEI =−− where Constant = Tk 34 B
⎟⎟⎠
⎞⎜⎜⎝
⎛µ×
α
or 2E
EI
N
)NN(ttanCons
1t−
=
Hence, for Perikinetic flocculation in complete-mix flow reactors,
detention time ⎟⎟⎠
⎞⎜⎜⎝
⎛−
×= 1
NN
NttanCons1t
E
I
E compared with: ⎟⎟
⎠
⎞⎜⎜⎝
⎛−= 1
CC
kC1t
AE
AI
AE
If NI = 1012 particles per m3, how long will it take for NE = 1011 particles per m3?
Note: NI = 1012 and NE = 1011 particles per m3, or NE/NI = 0.1 Reduction in particles = 1012 – 1011 = 9 x 1011
% reduction = 9 x 1011/1012 = 0.9 = 90% Example 5: Perikinetic flocculation in a plug-flow reactor (PFR)
Balance of particle number in the plug: In – Out + “Reaction” = Accumulation
Q Ni – Q Ni+1 + ∆V dtdN = 0 (assume no accumulation in reactor)
Ni+1 – Ni = ∆N = ∆t dtdN → 22B N ttanCons N Tk
34
dtdN
×−=⎟⎟⎠
⎞⎜⎜⎝
⎛µ
α−=
∆N = ∆t (– constant x N2)
∫−=∫t
0
EN
IN 2dt.ttanCons
N
dN → tttanConsN1
N1
N1
IE
EN
IN×−=+−=⎥⎦
⎤⎢⎣⎡−
x tttanConsNN
NN
EI
EI =⎟⎟⎠
⎞⎜⎜⎝
⎛ − or ⎟⎟
⎠
⎞⎜⎜⎝
⎛ −=
EI
EINN
NNttanCons
1t
Hence, for Perikinetic flocculation in plug flow reactors,
detention time ⎟⎟⎠
⎞⎜⎜⎝
⎛−
×= 1
NN
NttanCons1t
E
I
I
Considering same conditions as in Example 5: NI = 1012 particles per m3, NE = 1011 particles per m3, Constant = 6.15 x 10-18
days 17 s1046.11010
1010
1015.6
1t 61112
1112
18=×=⎟
⎟⎠
⎞⎜⎜⎝
⎛ −
×=
−
Q (Ni+1)
Q Ni
Q NI
Q NE
∆V
33
Example 6: Orthokinetic flocculation in a complete-mix flow reactor (CMFR)
Balance of particle number in reactor: (Q NI – Q NE) ∆t + V dtdN
∆t = 0
From Eqn 40: N G 4 dtdN
Ωαπ
−= → Q NI – Q NE – V EN G 4Ωα
π = 0
Divide by Q, substitute t = V/Q, and let KF = (4α/π): NI – NE = KF G Ω t NE
(1 + KF G Ω t) = NI /NE compared with : ⎟⎟⎠
⎞⎜⎜⎝
⎛−= 1
CC
k1t
AE
AI
)t G K1(1
NN
FI
E
Ω+=
Rewriting, ⎟⎟⎠
⎞⎜⎜⎝
⎛−= 1
NN
ttanCons1t
E
I where Ωπα
= G 4ttanCons
Assume α = 1 for perfectly destabilized particles, G = 50 s-1, Ω = 10-4, NE/NI = 0.1 t = 1414 s = 23.6 min. for same particle reduction as in perikinetic flocculation in Example 4.
34
Q V p1 p2 D
L
Q
Estimating G values for flocculation (1) Units of G
Pressure drop, ∆p = P1 – P2 Head loss, h = ∆p/ρg Power dissipated, P = Q ρ g h
5.05.0
t h g
V h g Q
VPG ⎥
⎦
⎤⎢⎣
⎡µρ
=⎥⎦
⎤⎢⎣
⎡µρ
=µ
= (45)
35
(4) Diffused air agitation P = QA ∆ρ g D (46) where QA = air flow rate m3/s
∆ρ = (ρW – ρA) = density difference (water – air) D = depth of diffuser below water surface
5.0A
V D g Q
G ⎥⎦
⎤⎢⎣
⎡µρ∆
= (47)
(5) Other cases of agitation • Entrance and exit losses • Flow through a dense suspension of solids • Flow through a packed bed of granular solids
(6) Typical values of G
Table 6.
Condition G (s-1) P/V (W/m3) Flocculation in water treatment 10 – 100 0.1 – 10 Rapid mixing in water treatment 500 – 1000 250 – 1000 Activated sludge process 100 – 250 10 – 60 Natural streams 10 – 100 0.1 – 10
36
Improving flocculation performance The previous flocculation design (Example 6) using the CMFR configuration can be further improved by having a staged operation. The flocculator configuration can consist of a series of stages called “cascade”. The effluent from a particular stage/reactor becomes influent of the next stage. As the number of stage approaching infinity, the cascade performance approaches that of a plug-flow reactor. Normally, the maximum of stages is 3 or 4.
First stage: 1111F
I
1E )t G K1(N
N −Ω+=
Second stage: 1222F
1E
2E )t G K1(NN −Ω+=
Both stages together: 1222F
1111F
1E
2E
I
1E
I
2E )t G K1()t G K1(NN.
NN
NN −− Ω+×Ω+==
For “m” stages have similar conditions in all stages:
∏⎥⎥⎦
⎤
⎢⎢⎣
⎡
Ω+=Ω+=⎟⎟
⎠
⎞⎜⎜⎝
⎛
=
− m
1j jjF
mstageF
seriesI
EtGK1
1)t G K1(NN (48)
tstage = residence time in an individual stage
ttotal = m . tstage (49) Tapered Flocculation It has been shown previously that a “cascade” of tanks in series can improve the removal efficiency of flocs. Further, if KF (=4α/π), G, and Ω are chosen equal for all stages, the optimum design is one in which the detention time t is also equal for all stages. Based on the orthokinetic eqn. (36), the larger is the G value, the larger would be the removal rate of particles. However, a larger G value implies very large power input. Furthermore, extremely high G values would inhibit large floc formation as the intense shear caused will break up large flocs. Hence, an optimum G has to be selected for each stage. With aluminum hydroxide flocs, the optimum G value is between 10 and 100 s-1. The designer may choose a larger G (100 s-1) in the first stage when the flocs are small. Then smaller G in subsequent stages, as the flocs grows. This is known as “tapered flocculation”.
37
Fig. 13 Three-stage flocculation basin.
Flocculation Systems There are two general groups of flocculation units: hydraulic flocculators, and mechanical flocculations. The hydraulic flocculators utilize cross-flow baffles or 180o turns to produce the required turbulence. The main design objective is to achieve gentle, uniform mixing that will not shear the floc. They are only used when flow rate is relatively constant and are rarely used in medium- and large sized water treatment plants. The mechanical flocculators typically used are paddle-wheel mixers, walking beam flocculators, flat-plate turbines, and axial flow propellers or turbines.
Table 7 Typical design criteria for horizontal-shaft paddles and vertical-shaft turbines (Crittenden).
Number of compartments 2 – 6 4 – 6 Variable speed drives Usually Usually
38
Vertical-shaft turbine flocculators These flocculators have impellers attached to a vertical shaft and rotated by an electric motor through a speed reducer. Design considerations are displacement capacity (rate at which the impeller pumps water), power consumption, and pumping head.
53pDN
PNρ
= (50)
3QND
QN = (51)
2H)ND(g HN ∆
= (52)
where P = power requirement, W Np = power number D = diameter of impeller, m NQ = pumping number NH = head number
ρ = water density N = impeller’s rotational speed, s-1
Q = flow rate imparted by impeller, m3/s ∆H = head impeller imparts to impeller flow, m g = acceleration due to gravity, m/s2
The power number of the impeller is dependent on the impeller dimensions and shape. The pumping number affects the circulation pattern (pumping rate) of the water. The circulation time of water is related to the pumping rate by the impeller and the mixing time in the tank required to achieve completely mixed conditions. The head number influences the turbulence, shear, and mixing conditions in the tank. All flocculation impellers operate in the turbulence-flow region i.e Reynolds number ≥ 104.
µρ
=NDRe
2
Other design considerations: - impeller blade to tank diameter should be D/Te > 0.35 (between 0.4 – 0.5)
Equivalent tank diameter π
= plane
A4T
- water depth to tank diameter H/Te between 0.9 – 1.1 - bottom clearance to water depth C/H, between 0.5 – 0.33
39
- N between 10 – 30 rev/min - tip speed between 2 – 3 m/s
Horizontal paddle wheel flocculators They are often employed in conventional treatment plants where a large quantity of solids is to be removed by sedimentation. The power input to the water by horizontal paddles is given by:
2vAC
P3
D ρ= (53)
where P = power imparted to water, W
CD = drag coefficient, depending on length-to-width ratio of paddle blades
(For L/W = 1, CD = 1.16; L/W = 5, CD = 1.2; L/W = 20, CD = 1.5; L/W = ∝, CD = 1.9; L = length and W = width of paddle blades)
A = area of paddles, m2
v = velocity of the paddle relative to the water, m/s (about 0.75 of paddle tip velocity)
The Reynolds number for a paddle flocculator is: µ
Diameter of wheel (m) 3 – 4 Paddle board (blade) width W (mm) 100 – 150 Paddle board (blade) length L (m) 2 – 3.5 Area of paddles/tank section (%) < 20 Paddle tip speed (m/s) 4 (strong floc)
2 (weak floc) Clearance from basin walls (m) 0.7 Minimum basin depth (m) 1 m greater than paddle wheel diameter Minimum clearance between stages (m) 1
Fig. 14 Flocculation basins using paddle-wheel flocculators.
Hydraulic Flocculators There are three groups of hydraulic flocculators: baffled channels, hydraulic-jet flocculators, and coarse-media flocculators. Of these, baffled channels are the most commonly used. Hydraulic flocculators generally can produce good floc and often without much short circuiting. The main disadvantages are inflexible mixing and head loss across the basin.
Table 9 Typical design criteria for hydraulic flocculators (Crittenden). Parameter Value
Average G (s-1) 30 – 40 Channel velocities (m/s) 0.15 – 0.45 Minimum residence time (min) 20 Head loss at 180o turn (of velocity head) 3.2 – 3.5 Head loss through slit or port (of velocity head) 1.5 Minimum distance between baffles (m) 0.75 Minimum water depth (m) 1
where h = head loss through basin, m Head loss in a baffled mixing channel due to turbulence and wall friction on the sides of the channel can be calculated using:
RC Lv h
h2
2
hannelc =
where hchannel = head loss through channel, m L = length of channel, m V = velocity of flow in channel, m/s C = Chezy coefficient Rh = hydraulic radius of channel – flow area/wetted perimeter, m Head loss resulting from each 180o turn can be estimated from:
g2 v kh
2turn =
where k = empirical constant (typically 2.5 – 4) The number of baffles in each stage of a tapered horizontal baffled hydraulic flocculator can be calculated from:
42
31
2
QG L H
f)(1.44 t 2n
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛+ρ
µ= (54)
where: Q = flow rate, m3/s t = flocculation time, seconds ρ = density of water = 1000 kg/m3 µ = absolute or dynamic viscosity of water, Ns/m2 or kg/ms f = roughness coefficient of baffles (for timber = 0.3)
H = depth of water in flocculator, m L = length of flocculator, m G = velocity gradient, s-1 n = number of baffles perpendicular to L
Hydraulic-jet flocculators Turbulence is created by the discharge jet as the flow enters each compartment. As shown, the jet is created by the entrance pip and the water flows in an up/down arrangement in each compartment.
Fig. 16 Hydraulic-jet flocculator (Crittenden) Coarse-media flocculators Energy dissipation is achieved by turbulent flow through a coarse media and flocculation is achieved when floc particles flocculated as the water passes through the voids of the media.
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