The effect of variable yield strength and variable fractal dimension on flocculation of cohesive sediment

water research 43 (2009) 3582–3592

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The effect of variable yield strength and variable fractal dimension on flocculation of cohesive sediment M. Sona,*, T.-J. Hsub a

Department of Civil and Coastal Engineering, University of Florida, Gainesville, FL 32611, USA Center for Applied Coastal Research, Civil and Environmental Engineering, University of Delaware, Newark, DE 19716, USA

b

article info

abstract

Article history:

A new formulation for floc yield strength of cohesive sediment is theoretically derived and

Received 3 February 2009

incorporated into a flocculation model based on variable fractal dimension. The new

Received in revised form

flocculation model is validated with existing data on the temporal evolution of floc size

29 April 2009

measured in the laboratory. Comparing with existing flocculation models using a constant

Accepted 5 May 2009

yield strength, it is found that new flocculation model based on variable yield strength and

Published online 20 May 2009

variable fractal dimension is superior in predicting the temporal evolution of floc size. It is also demonstrated that the present model results are very similar to that using an

Keywords:

empirical formulation of variable yield strength suggested by Sonntag and Russel (1987.

Aggregation

Structure and breakup of floccs subjected to fluid stressses. II. Theory. J. Colloid Interface

Breakup

Sci. 115(2), 378–389) when the empirical coefficient is specified according to our theoretical

Floc

value. Hence, it is concluded that the new variable yield strength formulation derived in

Flocculation

this study and the variable fractal dimension are effective in improving the prediction of

Fractal dimension

flocculation process.

Yield strength

1.

Introduction

1.1.

Purpose of this study

Cohesive sediments, the mixture of fine-grained sediments such as clay, silt, fine sand, organic material and water, have cohesive characteristics due to significant electrochemical or biological–chemical attraction. The physics of cohesive sediment transport is more complicated than non-cohesive

ª 2009 Elsevier Ltd. All rights reserved.

sediment due to flocculation processes (e.g. Dyer, 1989; Winterwerp and van Kesteren, 2004). Cohesive sediments form floc aggregates through binding together of primary particles and smaller flocs (aggregation), and flocs can disaggregate into smaller flocs/particles due to flow shear or collision (breakup or disaggregation) (Dyer, 1989). The properties of floc aggregates constantly change with the fluid condition. The averaged size of cohesive sediment aggregate is determined by flow turbulence, concentration of sediment, biological–chemical

Abbreviations: n, Number of flocs per unit fluid volume; N, Number of primary particles within a floc; Nrup, Number of primary particles in the plane of rupture; Nturb, Rate of collision of particles due to turbulent flow; D,d, Size of floc and primary particle; De, Equilibrium floc size; G, Dissipation parameter (Shear rate); e, Dissipation rate of energy; n, Kinematic viscosity; t, Time; eb,ec,ed, Efficiency parameter; f, Volumetric concentration; fs , Solid volume fraction within the floc; c, Mass concentration; rs ,rf ,rw , Density of primary particle, floc, and water; Drf ,Drs , Immersed density of floc and primary particle; fs,, Shape factor; F, Three-dimensional fractal dimension of floc; a; b,a, p, q, Coefficient; Fc, Characteristic fractal dimension; Dfc , Characteristic size of floc; m, Dynamic viscosity of the fluid; Fy,sy , Yield strength and stress of floc; sy0 ,r, Scaling and empirical parameters forsy ; Fc,p, Cohesive force of primary particle; B1,B2, Empirical parameter; l0 , Kolmogorov micro scale; K0A ,K0B , Empirical dimensionless coefficient. * Corresponding author. Tel.: þ1 352 871 6572; fax: þ1 302 831 3640. E-mail address: [email protected] (M. Son). 0043-1354/$ – see front matter ª 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.watres.2009.05.016

water research 43 (2009) 3582–3592

properties of water, properties of primary particle and so on (Lick et al., 1992). Thus, accurate prediction of cohesive sediment transport may require detailed water column models that resolve time-dependent flow velocity, turbulence and sediment concentration (Winterwerp, 2002; Hsu et al., 2007). In additional to floc size, the density of floc aggregates, which is of great importance to further estimate of settling velocity, has a tendency to decrease or increase as the floc size changes (Dyer, 1989; Mehta, 1987; Kranenburg, 1994). Hence, flocculation process should be appropriately investigated when studying cohesive sediment transport. The purpose of this study is to improve the existing flocculation models to better predict temporal evolution of floc size by incorporating more realistic parameterization on floc yield strength. A brief literature review is given in Section 1.2. Based on the assumption of fractal structure (Section 2.1), the number of primary particles in ruptured plane of a floc is calculated (Section 2.2). The heuristic equation determining the yield strength is theoretically derived based on the fractal theory (Section 2.3). This new formulation of floc strength, along with that proposed empirically by Sonntag and Russel (1987), are incorporated into the flocculation model developed by Son and Hsu (2008) (Section 3). The new flocculation model is validated with experimental data and compared with the previous flocculation models of Winterwerp (1998) and Son and Hsu (2008) based on constant yield strength (Section 4). It is concluded that utilizing variable yield strength and variable fractal dimension is critical to the prediction of the temporal evolution of floc size.

1.2.

Previous studies

It is well known that the flocculation of cohesive sediment depends on collisions resulted from Brownian motion, differential settling, and turbulent shear (Dyer, 1989; Dyer and Manning, 1999; Lick et al., 1993). Hunt (1982) has investigated the effects of Brownian motion, differential settling, and turbulent shear on flocculation and concludes that turbulent shear is dominant unless for weak turbulence condition (see also Stolzenbach and Elimelich (1994), O’Melia (1980), McCave (1984), and van Leussen (1994)). To better control the intensity of turbulent shear, many laboratory experiments have been conducted. Mixing tank experiments are carried out to obtain the temporal evolution of floc size using activated sludge (Biggs and Lant, 2000), a synthetic mineral (Bouyer et al., 2004), and synthetic resin (Spicer and Pratsinis, 1996; Spicer et al., 1998). To investigate the effect of turbulent shear and critical mechanisms causing collision for various particle sizes, Tsai et al. (1987) and Tsai and Hwang (1995) use Couette viscometers composed of two concentric cylinders to generate a velocity gradient. From these experiments, temporal variations of median floc sizes and steady-state floc size distributions are obtained. To quantitatively predict changes of floc properties, such as density and size, many types of flocculation models (FM) have been developed. McAnally and Mehta (2000) developed a dynamic model for aggregation rate of cohesive sediment. This model considers both binary and multi-body collisions. Parker et al. (1972) consider the change of number concentration as a function of turbulent shear quantified by the dissipation

3583

pffiffiffiffiffiffiffi parameter (or shear rate), G ¼ e=n. Herein, e is the turbulent dissipation rate and n is the kinematic viscosity of the fluid. Winterwerp (1998) developed a semi-empirical flocculation model which describes the rate of change of averaged floc size in turbulent flow. This model is based on: the collision frequency derived by Levich (1962); dimensional analysis; and assuming that flocs are of a fractal structure with a constant fractal dimension (van Leussen, 1994). Aggregates of cohesive sediments (flocs) have been considered as fractal entities (Tambo and Watanabe, 1979; Huang, 1994; Logan and Kilps, 1995). However, the assumption of constant fractal dimension for floc aggregate may be too restricted for modeling general cohesive sediment transport that has a wide range of flow condition and sediment concentration. Khelifa and Hill (2006) suggest a model for floc composed of mono-sized primary particles based on variable fractal dimension using a power law (see Eq. (2)). Maggi et al. (2007) also adopt a variable fractal dimension to develop a size-classes flocculation model and conclude that the use of a variable fractal dimension results in better predictions of flocculation process. More recently, Son and Hsu (2008) further extend the floc dynamic equations of Winterwerp (1998) for variable fractal dimension suggested by Khelifa and Hill (2006). Son and Hsu (2008) show that none of the two flocculation models of Winterwerp (1998) and Son and Hsu (2008) is in satisfactory agreement with experimental results for the temporal evolution of floc size in mixing tanks (see Son and Hsu (2008) or Fig. 1 (c) for more details). They conjecture that the constant yield strength adapted by these flocculation models may be the main reason causing such deficiency. The yield strength of a floc is a very important parameter in flocculation process because it has a direct relationship with breakup process of flocculation. Many types of cohesive sediments and techniques have been employed to determine the yield strength of flocs (e.g. Leentvaar and Rebhun, 1983; Francois, 1987; Bache and Rasool, 2001; Wu et al., 2003; Gregory and Dupont, 2001; Wen and Lee, 1998; Yeung and Pelton, 1996; Zhang et al., 1999). For example, Wen and Lee (1998) apply a controllable ultrasonic field to a floc suspension and observe floc erosion. Zhang et al. (1999) squeeze a single floc in suspension between a glass slide and fiber optic using a force transducer. The values of the floc yield strength estimated in these studies are in very wide range between the orders of 2 and 3 (N/m2) (see Jarvis et al. (2005) for more details). McAnally (1999) proposes an equation for the yield stress of floc. His derivation starts with the assumption that the floc yield strength is constant as Kranenburg (1994) suggests. Whereas, Tambo and Hozumi (1979) postulate that the yield strength is related to the net solids area at the plane of rupture. From previous studies, it is known that the fractal dimension and yield strength play an important role in modeling physics of cohesive sediment, flocculation.

2.

Study on floc structure and yield strength

2.1.

Fractal dimension

Under the assumption that the structure of flocs follows the self-similarity, flocs can be considered as fractal entities (e.g.

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water research 43 (2009) 3582–3592

a

b

Result of FM A

300

250

Floc size (micronmeter)

Floc size (micronmeter)

250

Results of FM B

300

200

150

100

200

150

100 Measured result FM B, r=0.7 FM B, r=1.0 FM B, r=1.3

50

50 Measured result FM A 0

0

10

20

30

0

0

Time (min)

c

10

20

30

Time (min)

Results of FM C and FM D

300

Floc size (micronmeter)

250

200

150

100

Measured result FM C FM D

50

0

0

10

20

30

Time (min) Fig. 1 – Experimental result of Spicer et al. (1998) and simulation results of flocculation models. FM A and FM B use the yield strength theoretically derived in this study and the empirical equation for the yield stress of Sonntag and Russel (1987). FM C and FM D of Son and Hsu (2008) and Winterwerp (1998) assume the constant yield strength.

Kranenburg, 1994) and the structure of floc can be described by the fractal dimension. The effective density of floc, rf , is described by the size of a floc, the size and density of a primary particle and the fractal dimension (Kranenburg, 1994):

cohesive sediment in estuary and coastal environments (e.g. Dyer, 1989; Winterwerp, 1998). Khelifa and Hill (2006) propose a power law for fractal dimension that explicitly depends on the diameters of the floc and the primary particle:

 F3 D rf  rw ¼ ðrs  rw Þ d

 b D F¼a d

(1)

where F is the three-dimensional fractal dimension of floc, rs is the density of primary particle, rw is the water density, D is floc diameter, and d is primary particle diameter. In many previous studies on floc dynamics, the fractal dimension is assumed to be a constant and often specified around 2.0 for

(2)

The empirical coefficients in Eq. (2) are specified as a ¼ 3 and b ¼ logðFc =3Þ=logðDfc =dÞ where Fc is a characteristic fractal dimension and Dfc is a characteristic size of flocs. The typical values of Fc and Dfc are suggested to be Fc ¼ 2.0 and Dfc ¼ 2000 mm by Khelifa and Hill (2006). According to Eq. (2),

water research 43 (2009) 3582–3592

the fractal dimension, F, is 3.0 when floc diameter, D, is set to be equal to primary particle diameter, d. This is a reasonable physical constraint in that the fractal dimension of primary particle should be 3.0 by the definition of fractal dimension. On the other hand, the calculated fractal dimension becomes less than 2.0 when the floc size is larger than the characteristic floc size, Dfc. This range of fractal dimension is appropriate for practical application of cohesive sediment that the fractal dimension of very large floc, such as marine snow is less than 2.0 (Kilps et al., 1994). Maggi (2007) also proposes the equation for fractal dimension, defined as the three-dimensional capacity dimension in his study. Essentially, equations of Khelifa and Hill (2006) and Maggi (2007) are very similar. Maggi (2007) suggests b (defined as x in his study) to be 0.1 (see Eq. (2)), based on their calibrations with experimental results (see Maggi (2007) for more details) which is of similar magnitude to that proposed by Khelifa and Hill (2006).

2.2. The number of primary particles in ruptured plane of a floc As presented in Eq. (1), the fractal dimension is an indicator to describe how dense a floc is for a given size and density of the primary particles. In other words, the number of primary particles within a floc and in the ruptured plane of a floc is also a function of the fractal dimension. By the definition, the floc density can be calculated as: rf ¼ fs rs þ ð1  fs Þrw

(3)

where fs is the solid volume fraction within a floc. Rearranging Eq. (3), fs is expressed as: Dr fs ¼ f Drs

(4)

where Drf ð¼ rf  rw Þ is the immersed density of floc and Drs ð¼ rs  rw Þ is the immersed density of primary particle. By substituting Eq. (1) to Eq. (4), the equation for fs is rewritten as fs ¼

 F3 D d

(5)

Under the assumption that the floc and the primary particles are spherical for the sake of simplicity, the number of primary particles within a floc, N, is derived from the definition, fs ¼ ðNd3 Þ=D3 , and Eq. (5): F

N ¼ ðD=dÞ

Nrup

 2F=3 pp2=3 D ¼ 4 6 d

In this study, the plane crossing the center of a floc is assumed to be the ruptured plane of the floc due to the action of turbulent shear. This is discussed in more detail in the next section.

2.3.

Yield strength of floc

Existing flocculation models of Winterwerp (1998) and Son and Hsu (2008) adopt a constant yield strength of floc, Fy, under the assumption that the number of bonds (or primary particles) in a ruptured plane is independent of the size of the floc (e.g. Kranenburg, 1994). However, Boadway (1978) and Tsai and Hwang (1995) have observed floc breakup process and concluded that a floc often disaggregates into two roughly equal-sized flocs. Hence, it is assumed here that during floc breakup, a floc is divided by the plane which contains the center of floc as the two daughter flocs have the same size after breakup. The number of primary particles in this plane should be a function of floc size and its fractal dimension. The yield strength of floc depends on the strength of inter-particle bonds between the primary particles and the number of these bonds within a floc (Parker et al., 1972; Boller and Blaser, 1998). Thus, the yield strength is also a function of floc size and its structure (e.g. Yeung and Pelton, 1996). Sonntag and Russel (1987) suggest an empirical equation for the yield stress (units, Pa), which is the yield strength divided by the cross-sectional area of floc ðpD2 Þ=4, based on a power law:   rðF3Þ sy ¼ sy0 frs ¼ sy0 ðD=dÞ

(7)

(8)

where sy0 is a scaling parameter and r is an empirical coefficient. According to a later study by Bache (2004), the value of r is usually in the range between 0.5 and 1.5. Following the definition of the solids volume fraction within a floc, fs (see Eq. (5)), it can be concluded that the yield strength is not constant but a function of fractal dimension and floc size. Our discussion here is consistent with earlier study by Tambo and Hozumi (1979) who postulate that the yield strength of floc is proportional to the net solid area in the ruptured plane. In addition, it is clear that the yield strength of floc has a direct relationship with the cohesive force of each primary particle. Thus, the magnitude of the yield strength is considered as the sum of cohesive force of all primary particles in the ruptured plane and in this study a new equation for floc yield strength is proposed based on Eq. (7):

(6)

In addition, the number of primary particles within a floc is assumed to be sufficient to adopt mensuration by parts. Using mensuration by parts, the average distance between two neighboring primary particles within a floc is determined. From this, one can further determine the averaged area occupied by one primary particle in the ruptured plane. Consequently, an equation for the number of primary particles Nrup in the plane crossing the center of a floc, whose size is D, can be derived as:

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Fy ¼ B2

 2F=3 D d

(9)

where B2 ¼ ðp=4Þðp=6Þ2=3 Fc;p , and Fc,p is the cohesive force of primary particles. Fc,p is considered as an empirical parameter because it depends on the properties of sediment and chemical– biological effects. Further dividing Eq. (9) by the area of ruptured plane, an equation for the yield stress of a floc, sy, is obtained: sy ¼ 

 2F=3 Fy D  ¼ B1 D2 d p=4 D2

(10)

where B1 ¼ ðp=6Þ2=3  Fc;p . It can be seen that in the present formulation, the only parameter that is difficult to obtain and needs to be empirically determined is Fc,p.

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3.

water research 43 (2009) 3582–3592

dD cec ped F3þb Fþ4b 1 Gd D ¼ 2rs fs dt blnðD=dÞ þ 1

Development of the flocculation model

It is assumed in this study that for cohesive sediment transport in a natural environment where flow turbulence is usually dominant, the effects of Brownian motion and differential settling are negligible compared to turbulent shear (Hunt, 1982; McCave, 1984; van Leussen, 1994). In order to emphasize the role of yield strength in flocculation models for averaged floc size (e.g. Winterwerp, 1998) and demonstrate the incorporation of variable yield strength into the breakup process, the derivation of the model for floc dynamics of Son and Hsu (2008) based on variable fractal dimension is briefly summarized in this section.

3.1.

By assuming particle diameters are much smaller than the Kolmogorov length scale (l0) and based on the theory of Smoluchowski (1917), Levich (1962) suggests the rate of collision of particles due to flow turbulence can be determined by integrating the diffusion equation over a finite volume: (11)

where n is the number of particles (or flocs in this study) per unit fluid volume and ed is an efficiency parameter for turbulent diffusion. Winterwerp (1998) further assumes that only a certain portion of the collisions causes flocculation and proposes the equation for the rate of aggregation between the flocs in a turbulent fluid: dn 3 ¼  ec ped GD3 n2 dt 2

(12)

where ec is a constant efficiency parameter accounting for the fact that not all collisions result in coagulation. Although ec can be a variable as floc size and floc density change, it is assumed to be constant for simplicity because ec cannot be determined at present on the basis of theoretical arguments (Lick et al., 1992). The volumetric concentration of cohesive sediment floc, f, has the following relationship with the mass concentration and the number of flocs per unit fluid volume, n (Winterwerp, 1998): f¼

! rs  rw c ¼ fs nD3 rf  rw rs

(13)

From Eqs. (1) and (13), the equations for n and dn=dD are derived: c F3 F d D n¼ rs fs

The effect of the turbulent stress of the carrier fluid on flocculation is assumed to be the only mechanism of floc breakup because the flocs are assumed to be small enough to ignore the effect of inter-particle collisions. Winterwerp (1998) suggests that the floc breakup rate due to turbulent shear is a function of the dissipation parameter, G, and the floc yield strength, Fy, and proposes the equation for breakup rate based on dimensional considerations (see Winterwerp (1998) for more details):

 q dD eb Ga mG 1 p dbpþð2q=3ÞF D1bþ2qð2q=3ÞF ðD  dÞ ¼ dt 3 B1 blnðD=dÞ þ 1 (18a) The above equation adopts the yield stress equation theoretically derived in this study (Eq. (10)). Similarly, utilizing the yield stress equation suggested empirically by Sonntag and Russel (1987) (see Eq. (8)), we obtain the following equation for floc breakup:  q dD eb Ga mG 1 p dbprqð3FÞ D1bþrqð3FÞ ðD  dÞ ¼ dt 3 sy0 blnðD=dÞ þ 1 (18b)

Table 1 – A summary of flocculation models used in this study. Model name A

B

(14)

C

(15)

Using chain rule, Eqs. (12) and (15), the equation for flocculation due to collision is obtained (see Son and Hsu (2008) for more details):

(17)

where m is the dynamic viscosity of the fluid, eb is an efficiency parameter for floc breakup and, a, p, and q are the coefficients to be discussed later. Utilizing chain rule, Eqs. (15), (17) and (10) for floc yield stress (see Section 2.3), the equation for floc breakup due to turbulent shear is obtained:

and   dn 3c F3b F1þb D bln þ 1 d D ¼ dD rs fs d

Flocculation due to turbulent shear

 p  q dn Dd mG ¼ neb Ga dt d sy

Flocculation due to collisions of particles

3 Nturb ¼ ped GD3 n2 2

3.2.

(16)

D

Characteristic Variable fractal dimension Variable yield strength theoretically derived Fy in this study Variable fractal dimension Empirical variable yield stress of Sonntag and Russel (1987) Variable fractal dimension Constant Fy Constant fractal dimension Constant Fy

Reference Eq. (19a)

Eq. (19b)

Eq. (11) of Son and Hsu (2008) Eq. (24) of Winterwerp (1998)

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water research 43 (2009) 3582–3592

3.3.

Flocculation model

baffled, stirred tank using a Rushton impeller. The volumetric concentration of primary particle is set to be 1.4  105. From the volumetric concentration and the density of primary particle, the mass concentration is calculated to be 0.0147 kg/ m3. The average dissipation parameter, G, in the tank is 50 s1. Sampling the floc is a crucial step to accurately characterize flocculation. Three kinds of sampling techniques are used: (1) withdrawal of a sample into the sample cell of light scattering instrument using a hand pipette; (2) withdrawal of a sample into the flow-through sample cell using a syringe pump; (3) continuous recycle of the suspension through the sample cell using a peristaltic pump which is a class of mechanical pumps with relatively simple structure and is suitable for miniaturization (Xie et al., 2004). Among three techniques, the results obtained using peristaltic pump show the largest number of samples and most stable shape of evolution curve in the experiment of Spicer et al. (1998). Thus, the result with peristaltic pump is selected and used to validate flocculation models in this study. Fig. 1 presents the experimental result of Spicer et al. (1998) and model results of different flocculation models. The initial floc diameter is set to be 10 mm for all the model simulations presented in Fig. 1. For numerical stability, the size of primary particle is assumed to be 1 mm instead of 0.87 mm. The values of empirical coefficients used to generate these model results are summarized in Table 2. Following prior studies (Winterwerp, 1998; Son and Hsu, 2008), the criteria of specifying these empirical coefficients are to match the equilibrium floc size (except r in FM B, which is used to evaluate temporal evolution of floc size). FM A (Eq. (19a)) using a variable yield strength derived theoretically in this study and variable fractal dimension in modeling flocculation processes shows good agreement with the experimental results (Fig. 1(a)). Fig. 1(b) presents the results for FM B using the empirical yield strength equation of Sonntag and Russel (1987) and variable fractal dimension in flocculation processes. Three values of r have been tested: r ¼ 0.7; r ¼ 1.0; r ¼ 1.3. The model results are sensitive to the choice of r. Specifically, the case of r ¼ 0.7 gives the best result among three values of r. FM B shows numerical instability around r ¼ 0.66. The results given by the flocculation models of Son and Hsu (2008) (FM C) and Winterwerp (1998) (FM D) based on the constant yield strength are shown in Fig. 1(c). These two models use the constant yield strength, which is set to be 1010 N (van Leussen, 1994; Matsuo and Unno, 1981). As mentioned in the previous section, the model of Son and Hsu (2008) uses a variable fractal dimension whereas Winterwerp’s (1998) is based on the constant fractal dimension

The complete flocculation models are obtained by linearly combining flocculation processes due to collisions and turbulent shear (Winterwerp, 1998). For flocculation model that utilize theoretical derivation of floc yield stress developed in this study (Eq. 10), it is denoted as Model ‘‘FM A’’ and is written as (see also Table 1) 0 dD Gdb kA c F3 Fþ4b d D ¼ blnðD=dÞ þ 1 3 rs dt 

 q k0 mG p  B dpþð2q=3ÞF D1bþð2q=3Þð3FÞ ðD  dÞ 3 B1

(19a)

The flocculation model utilizes floc yield stress of Sonntag and Russel (1987) and is denoted as ‘‘FM B’’ in Table 1, is given by: 0 dD Gdb kA c F3 Fþ4b d D ¼ blnðD=dÞ þ 1 3 rs dt 

 q k0 mG p  B dprqð3FÞ D1bþrqð3FÞ ðD  dÞ 3 sy0

(19b)

In both equations shown above, k0A ¼ ð3ec ped Þ=2fs and k0B ¼ aeb are dimensionless empirical coefficients. Essentially, p and q are empirical coefficients. Winterwerp (1998) assumes the values of p ¼ 1 and q ¼ 0.5 based on several additional constrains (see Winterwerp (1998) for more detailed determination of p and q), and these values are also used in this study.

4.

Model results

In this section, the new flocculation models based on variable yield strength, i.e., Eq. (19a) (FM A) and Eq. (19b) (FM B), are validated with several existing experimental data sets. Comparisons with the previous flocculation models developed by Son and Hsu (2008) (FM C) and Winterwerp (1998) (FM D) based on the constant yield strength are also carried out in order to demonstrate the effect of a variable yield strength. A summary on the flocculation models tested in this study is given in Table 1. Numerical solutions of flocculation models are obtained using an explicit Runge–Kutta method (ODE45 function of MATLAB is used in this study). Spicer et al. (1998) carried out an experiment on flocculation and measured the temporal evolution of floc size in a mixing tank. In this experiment, polystyrene particles, whose primary particle diameter and density are 0.87 mm and 1050 kg/m3 (Spicer and Pratsinis, 1996), are mixed in a 2.8 L,

Table 2 – Empirical parameters of the flocculation models used for experiment of Spicer et al. (1998). FM FM FM FM FM FM FM

A B B B C D

k0A

k0B

B1

r

sy0

Fy

6.74 5.99 3.74 2.99 2.50 0.44

4.59  106 4.08  106 2.55  106 2.04  106 1.72  106 1.10  106

2.63  1014 N.A. N.A. N.A. N.A. N.A.

N.A. 0.7 1.0 1.3 N.A. N.A.

N.A. 3.02  102 1.07  101 3.81  101 N.A. N.A.

N.A. N.A. N.A. N.A. 1010 1010

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water research 43 (2009) 3582–3592

Table 3 – Empirical parameters of the flocculation models used for experiment of Biggs and Lant (2000). k0A

FM FM FM FM FM FM FM

A B B B C D

k0B

B1 5

0.02 0.02 0.02 0.01 0.02 0.004

2.88  10 2.89  105 2.41  105 1.92  105 2.40  105 2.23  105

4.20  10 N.A. N.A. N.A. N.A. N.A.

(F ¼ 2.0). However, two models predict similar results on temporal evolutions of floc size and they both do not agree with measured data. When comparing the new model to the existing approaches using the constant yield strength, it is notable that results obtained with a variable yield strength (FM A and B) are clearly better than those of a constant yield

b

Result of FM A

160

140

120

120

100 80 60 40

0

20

40

60

80

N.A. 0.7 1.0 1.3 N.A. N.A.

N.A. 2.82  102 5.24  102 9.65  102 N.A. N.A.

N.A. N.A. N.A. N.A. 1010 1010

80 60 40

Measured result FM B, r=0.7 FM B, r=1.0 FM B, r=1.3

20 0

100

0

20

Time (min)

c

Fy

100

Measured result FM A 0

sy0

Results of FM B

160

140

20

r

strength (Between 0 and 11 min, the values of root mean square error of FM A, B, C, and D are 17.4, 24.7, 60.3, and 58.7 mm). Overall, a variable yield strength has significant effect on the temporal evolution of floc size. FM A and B adopting variable yield strength improve the prediction of time-dependent behavior of flocculation.

Floc size (micronmeter)

Floc size (micronmeter)

a

13

40

60

80

100

Time (min) Results of FM C and FM D

160

Floc size (micronmeter)

140 120 100 80 60 40 Measured result FM C FM D

20 0

0

20

40

60

80

100

Time (min) Fig. 2 – Experimental result of Biggs and Lant (2000) and model results of flocculation models. FM A and FM B use the yield strength theoretically derived in this study and the empirical equation for the yield stress of Sonntag and Russel (1987). FM C and FM D of Son and Hsu (2008) and Winterwerp (1998) assume the constant yield strength.

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water research 43 (2009) 3582–3592

Table 4 – Experimental conditions of Burban et al. (1989). Case

B12 B4

G (s1)

Mass conc. (kg/m3)

Equilibrium time (min)

Equilibrium floc size (mm)

200 200

0.05 0.80

45 10

80–89 25

It is also emphasized here that FM B adopting Sonntag and Russel (1987) with r ¼ 0.7 show very similar results with FM A. This is not surprising if we further compare Eqs. (8)–(10). The powers of D in Eqs. (8) and (9) are r(F  3) and 2F/3  2, respectively. It can be seen that when r ¼ 2/3 ¼ 0.67, these two powers become identical. In other words, our new formulation for variable yield strength provides a theoretical approach to determine empirical coefficient r required in the formation of Sonntag and Russel (1987). Using FM A with yield strength derived theoretically in this study, the flocculation model has one less empirical parameter. Biggs and Lant (2000) report the temporal evolution of floc size of activated sludge under the conditions of G ¼ 19.4 s1 and with sludge volumetric concentration of 0.05. For this experiment, a baffled batch vessel and a flat six blade impeller are used. Because the total diluted mass concentration is not reported, the mass concentration is estimated from the volumetric concentration under the assumption that the density of sludge is 1300 kg/m3 and the density of primary particle is 2650 kg/m3. Hence, the calculated mass concentration is 24.19 kg/m3. In addition, the size of primary particle is also assumed to be 4 mm. Empirical coefficients used to model this experiment are shown in Table 3. The experimental result of Biggs and Lant (2000) and the model results are presented in Fig. 2. Similar to that shown in Fig. 1, FM A and FM B using variable fractal dimension show more smooth S-curves and are in better agreement with the experimental data. Overall, results predicted by FM A and B are quite similar. Comparing the model performance with the case of Spicer et al. (1998) (Fig. 1), it can be noted that the predicted temporal evolution of floc size in this case agrees less favorably with experimental data. However, the adoption of variable yield

strength allows the prediction of floc size that increases more rapidly in the initial stage of flocculation and, after larger aggregates are created, the floc size increases more gradually as it eventually approaches the equilibrium value. Burban et al. (1989) perform experiments with Detroit River sediment in a Couette chamber. The experiments have been reproduced by McAnally (1999) and two cases of temporal evolution of floc size (Case B12 and Case B4) are shown in McAnally (1999). The mass concentrations are 0.05 kg/m3 for Case B12 and 0.80 kg/m3 for Case B4. The dissipation parameter for both cases is set to be G ¼ 200 s1. To simulate this experiment, the size and density of primary particle are assumed to be 4 mm and 2650 kg/m3 due to absence of more information. In addition, McAnally (1999) provides information on the time required for the floc size to reach equilibrium state. Hence, empirical parameters of the flocculation models are calibrated according to equilibrium time determined by McAnally (1999). More details about experiment conditions and model coefficients are shown in Tables 4 and 5. Figs. 3 and 4 present the experimental results of Burban et al. (1989) and model results for Case B12 and Case B4. Consistent previous model–data comparisons presented in Figs. 1 and 2, flocculation models combined with a variable yield strength predict better temporal evolution of floc size than that using the constant yield strength in Case B12 (see Fig. 3). However, there is less significant difference among the model results for Case B4 (see Fig. 4) and in fact all models predict temporal evolution of floc size that agree reasonably well with measured data. Similar to the previous simulations, results predicted by FM A and FM B with r ¼ 0.7 are almost identical and show the best agreement with experimental data. In this study, the equations for variable yield strength are combined with the flocculation model of Son and Hsu (2008) (FM C). As mentioned previously, FM C uses a variable fractal dimension whereas the flocculation model of Winterwerp (1998) (FM D) uses the fixed fractal dimension of 2.0. According to results presented in Figs. 1–4, we have established that it is necessary to utilize flocculation models based on variable fractal dimension and using variable yield strength in order to predict the temporal evolution of floc size. However, it is not yet clear if one can simply implement variable yield strength in a flocculation model based on fixed fractal dimension and

Table 5 – Empirical parameters of the flocculation models used for experiment of Burban et al. (1989). Case

FM

k0A

k0B

B1 6

r

sy0

Fy

13

N.A. 0.7 1.0 1.3 N.A. N.A.

N.A. 4.55  102 7.45  102 1.21  101 N.A. N.A.

N.A. N.A. N.A. N.A. 1010 1010

N.A. 0.7 1.0 1.3 N.A. N.A.

N.A. 2.60  102 3.10  101 3.60  101 N.A. N.A.

N.A. N.A. N.A. N.A. 1010 1010

B12

FM FM FM FM FM FM

A B B B C D

1.05 1.05 0.88 0.77 0.58 0.18

1.38  10 1.38  106 1.16  106 1.02  106 7.56  107 9.60  107

6.92  10 N.A. N.A. N.A. N.A. N.A.

B4

FM FM FM FM FM FM

A B B B C D

0.30 0.30 0.29 0.29 0.18 0.07

2.95  105 9.23  106 2.75  105 2.71  105 1.64  105 2.09  105

4.10  1012 N.A. N.A. N.A. N.A. N.A.

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water research 43 (2009) 3582–3592

a

b

Results of FM A and FM B

120

100

Floc size (micronmeter)

Floc size (micronmeter)

100

80

60

40

20

0

Results of FM A, FM C and FM D

120

20

40

60

60

40 Measured result FM A FM C FM D

20

Measured result FM A FM B, r = 0.7 0

80

0

80

0

20

40

60

80

Time (min)

Time (min)

Fig. 3 – Experimental results of case B12 of Burban et al. (1989) and results of the flocculation models.

a

b

Results of FM A and FM B

35

30

Floc size (micronmeter)

Floc size (micronmeter)

30

25

20

15

10 Measured result FM A FM B, r = 0.7

5

0

Results of FM A, FM C and FM D

35

0

5

10

15

25

20

15

10 Measured result FM A FM C FM D

5

0

20

0

5

10

15

20

Time (min)

Time (min)

Fig. 4 – Experimental results of case B4 of Burban et al. (1989) and results of the flocculation models.

obtain similar model performance. To examine the effect of variable fractal dimension, Eq. (9) is combined with the flocculation model of Winterwerp (1998) with a fixed fractal dimension, F ¼ 2.0. Fig. 5 shows the results of experiment of Spicer et al. (1998), FM A, the flocculation model of Winterwerp (1998) combined with variable yield strength of Eq. (9), and the original model of Winterwerp (1998) with constant yield strength (FM D). Although the model results of Winterwerp (1998) combined with Eq. (9) is slightly better than FM D using constant yield strength, FM A based on variable fractal dimension and variable yield strength remains to be superior. Hence, it can be concluded that flocculation model based on variable fractal dimension is a more physically based

mathematical formulation while variable yield strength is critical process during floc breakup that needs to be carefully parameterized.

5.

Discussion

The empirical yield stress proposed by Sonntag and Russel (1987) (see Eq. (8)) shows the best agreement with measured data when using r ¼ 0.7. It is also demonstrated that when specifying r ¼ 2/3 in Sonntag and Russel (1987), it reduces to theoretical model for yield stress developed in this study. Hence, we suggest that the theoretical model proposed in this study is robust and it

water research 43 (2009) 3582–3592

constant yield strength. Although incorporating solely a variable fractal dimension in the flocculation models may not predict the temporal evolution of floc size well, it gives good agreement with measured data when it is further combined with variable yield stress formulations. However, it shall be also emphasized here that when simply using variable yield strength in a flocculation model based on fixed fractal dimension, the results for temporal evolution of floc size remains unsatisfactory (Fig. 5). Hence, it is recommended in this study that both variable yield stress and variable fractal dimension are critical to predict flocculation processes.

300

Floc size (micronmeter)

250

200

150

100 Measured result FM A Model of Winterwerp with variable Fy

50

FM D

0

0

5

10

15

20

25

30

Time (min) Fig. 5 – When using the flocculation model of Winterwerp (1998) based on constant fractal dimension (F [ 2) combined with a variable yield strength (Eq. (9)), the predicted temporal evolution of floc size (dashed curve) remains unsatisfactory.

may be appropriate to specify the empirical parameter r in Sonntag and Russel (1987) to be around 0.7. We believe there still remain several weak points in the present model of flocculation process. To describe flocculation due to collision, the constant efficiency parameter, ec, has been adopted in this study and it is assumed that collisions cause only aggregation. However, it has been observed that collisions can make both aggregation and breakup (McAnally, 1999). When the collisional stress is larger than the yield stress of floc, the breakup due to collision is expected rather than aggregation. It is not easy to adopt a variable efficiency parameter because it can be highly empirical and explicit formulation has not been proposed at present although it is clear that ec is a function of potential and yield stress. In this study, it has been assumed that a floc is simply disaggregated into two roughly equal-sized flocs (Boadway, 1978; Tsai and Hwang, 1995) due to lack of more detailed evidence. However, it is possible that a floc can fragment into a number of particles having a range of sizes (e.g. Srivastava, 1971). More complicated flocculation model assuming more general types of breakup process is warranted. In addition, more studies are needed to understand parameters p and q (in Eq. (17)) because they are currently highly empirical. Winterwerp (1998) uses the assumption that the equilibrium floc size is independent of primary particle size and fractal dimension is 2.0 (see Eq. (25) of Winterwerp (1998). If p þ nf  3 equals to zero, De is not a function of primary particle size) to determine their values. In the context of variable fractal dimension, more physicalbased criterion shall be incorporated to determine p and q.

6.

3591

Conclusion

It can be concluded that variable yield strength is a more reasonable approach to flocculation modeling than the

Acknowledgements This study is supported by the U.S. Office of Naval Research (N00014-09-1-0134) and National Science Foundation (OCE0913283) to University of Delaware. M. Son is also partially supported by ONR grant (N00014-07-1-0494) to University of Florida. The authors thank Dr. Mehta (U. of Florida) for helpful discussions and suggestions during this study.

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