Chemical Engineering Science 68 (2012) 432–442

Contents lists available at SciVerse ScienceDirect

Chemical Engineering Science

0009-25

doi:10.1

n Corr

Researc

Road, A

E-m

mukher

journal homepage: www.elsevier.com/locate/ces

Relationship between size of oil droplet generated during chemicaldispersion of crude oil and energy dissipation rate: Dimensionless,scaling, and experimental analysis

Biplab Mukherjee a,n, Brian A. Wrenn b, Palghat Ramachandran a

a Department of Energy, Environmental, and Chemical Engineering, Washington University in Saint Louis, One Brookings Drive, St. Louis, MO 63130, USAb Department of Civil and Environmental Engineering, Temple University, 1947 N. 12th Street, Philadelphia, PA 19122, USA

a r t i c l e i n f o

Article history:

Received 28 December 2010

Received in revised form

22 September 2011

Accepted 1 October 2011Available online 12 October 2011

Keywords:

Scaling laws

Dimensionless analysis

Characteristic length

Velocity scale

Energy dissipation rate

Chemical dispersion

09/$ - see front matter & 2011 Elsevier Ltd. A

016/j.ces.2011.10.001

espondence to: National Research Council A

h Laboratory, US Environmental Protection

thens, GA 30605-2700, USA. Tel.: þ1 706 355

ail addresses: mukherjee.biplab@epa.gov,

jeebiplab@rediffmail.com (B. Mukherjee).

a b s t r a c t

Droplet formation mechanisms during the chemical dispersion of crude oil were investigated using

both theoretical and experimental approaches. Dimensionless and force balance analysis identified four

distinct regimes of droplet formations. For d4Z, d scales either with (e�2/5) or (e�1/4) or and for doZ, d

scales either with (e�1/2) or (e�1/4) depending on whether the main restoring force against droplet

breakage is provided by surface tension or oil viscosity. The symbols d, Z, and e represent the droplet

diameter, the Kolmogorov length scale, and energy dissipation rate, respectively. For d4Z and oZ, the

external force, which tries to deform and break the droplet is provided by the pressure difference across

the droplet diameter and viscous shear, respectively. Identification of the relationship d�(e�1/4) for

doZ is a new contribution of this present study. The validity of this relationship was also proven by our

experimental observations over a range of physical properties (dynamic viscosity 0.015–8.6 Pa s; oil–

water interfacial tension 0.0001–0.015 N/m) and mixing energies (0.00075–0.16 W/kg), similar to those

in real environmental settings (e.g., estuary, surface layer of oceans). All these above findings and

observations are vital from the stand point of appropriately scaling droplet formation process, during

chemical dispersion of crude oil, and in the development of reliable predictive models.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Environmental exposure to marine oil spills can have long-lasting negative impacts on our eco-system and the economy.Worldwide, accidental releases of petroleum into marine envir-onments have either increased for small spills (less than10,000 gal) or have relatively remained constant for larger spills(Rourke and Connolly, 2003). On average, more than 10 million -gal of oil get spilled on the marine waters annually worldwide(Rourke and Connolly, 2003)—including 3 million gallons in theUS waters alone (NRC, 2005). However, this figure has spiraledupwards considerably owing to the recent massive oil spill at theGulf of Mexico. In the US, the primary preferred option to mitigatemarine oil-spills involves the use of mechanical recovery systemssuch as booms and skimmers (NRC, 2005). But, for cases, wheremechanical recovery is expected to be ineffective or too slow toprotect sensitive resources, chemical dispersion can provide an

ll rights reserved.

ssociate, National Exposure

Agency, 960 College Station

8135; fax: þ1 706 355 8160.

effectual response alternative (Lessard and Demarco, 2000). Inthis approach, breakup of the oil slick as droplets is promoted bythe application of dispersants over the spilled oil (Clayton et al.,1993; Lessard and Demarco, 2000). The active ingredients of adispersant are the surfactants, which are amphiphilic compoundscontaining both the hydrophobic and hydrophilic componentswithin the same molecule. The amphiphilic nature of the surfac-tants allows them to accumulate and favorably orient along theoil–water interface-phase. This reduces the oil–water interfacialtension and increases the likelihood of formation of oil dropletsand its dispersion in the water column under moderate levels ofmixing energy (NRC, 2005).

Chemical dispersants have been used to treat spilled oil sincethe Torrey Canyon spill in 1967 (Fingas, 1991). A major use ofdispersants was made in the recently Gulf of Mexico oil spill(Schnoor, 2010). Although promising, the use of dispersants hasbeen limited owing to several reasons including low public andregulatory acceptance (NRC, 1985) and our lack of understandingof the complex mechanisms of dispersion (NRC, 2005). The formerreason stems primarily due to concerns of unknown conse-quences from the use of dispersants on the marine ecosystem;however, over the years considerable efforts have been takenand are underway to reduce some of the genuine concerns

B. Mukherjee et al. / Chemical Engineering Science 68 (2012) 432–442 433

(Aurand, 1995; NRC, 2005). The later limitation has deterred ourprogress towards the development of reliable models (Mackayet al., 1978; Vandermeulen, 1982; Daling et al., 1990; NRC, 2005)to predict dispersant performance for specific crude oils indifferent environmental conditions. Model developments arenecessary as they can provide guidance to the regulators andresponders in drawing efficient remediation strategies and inminimizing adversaries arising due to oil spills. Advancementalong the lines of model development would require identifica-tion of the roles and relating the effects of different potentialphysicochemical factors on chemical dispersion.

The crux of the chemical dispersion process is the dropletformation step. Generation of smaller over larger droplets ispreferred; as smaller droplets tend to remain suspended longerin the water column, can easily be transported away from thespilled site, and expose a higher oil surface area per unit volumeof oil dispersed. All these qualities are an added advantage forefficient mitigation of oil spill by chemical dispersants. On theother hand, larger droplets due to higher buoyancy can quicklyrise back to the surface and reform a floating slick. Formation ofdroplets is known to be affected by several factors including theoil–water interfacial tension, mixing energy, viscosity, and den-sity of both the oil and the aqueous phase (Hinze, 1955). Therelative importance of these different factors is unique and variesdepending upon the nature of mechanism that controls dropletformation (Li and Garrett, 1998). Therefore, understanding theunderlying principles and the relative importance of differentfactors on the dynamics of droplet formation is a crucial steptowards the successful implementation of chemical dispersiontechnique and hence, was the focus of our study.

This report examines all the possible mechanisms that controlgeneration of oil droplets during chemical dispersion of crude oilvia theoretical and experimental approaches. In the next section,we use dimensionless analysis followed by scaling analysis toreview research on the mechanisms of droplet formation. Ourtheoretical analysis showed the existence of a new regime of oildroplet sizes in addition to those that are known from literature.The validity of the outcome was confirmed by comparisonwith our experimental observations. We also examined theresults obtained by Delvigne and Sweeney (1988), who investi-gated the breakdown of oil droplets in grid generated turbulentflows and also in breaking waves generated in laboratory tanks.Dispersion experiments, for our study, were conducted in baffled-flask mixing system, which was developed as an alternative to theswirling-flask test to study chemical dispersion of crude oiland to evaluate dispersant performance for inclusion on theNational Contingency Plan (NCP) Product Schedule (Sorial et al.,2004a, 2004b). The hydraulics and mixing characteristics forthe baffled-flask system were studied in detail by Kaku et al.(2006a, 2006b). In this system, mixing is induced by the sloshingmovement of the contents in the flask brought about by therotational motion of the orbital shaker on which the baffle flasksits. The absence of any impeller and paddle in the baffled flask,unlike in other commonly used mixing systems, precludes theoccurrence of extremely high energy dissipation rates, which arenot likely under real environmental scenarios. For our experi-ments, mixing energies of 0.00075 W/kg, 0.016 W/kg, and 0.16 W/kg were used, which varied over a range that encompasseddifferent marine environments (e.g., estuary, open ocean surface)(NRC, 2005). Three weathered crude oils – Arabian Light,Mars, and Lloyd – combined with a commercially availabledispersant, Corexit 9500, at two different concentrations wereused for the experiments. The selection of the above oils anddispersant concentrations provided the flexibility to study dropletscaling over a wide range of physical properties of the dispersedphase.

2. Droplet formation mechanisms

2.1. Dimensionless analysis

Dimensionless analysis provides the basic tool to correlate andinterpret data in terms of key dimensionless groups. In thissection we identify the key dimensionless groups and suggestsuitable forms of correlations for droplet diameter as a function ofphysical properties of dispersed and continuous phase and energydissipation rate. The proposed trends of this analysis will later becompared with scaling analysis, presented in the next section,which is based on estimates of different external and internalforces acting on a droplet.

The analysis was done in accordance with the Buckingham Pitheorem. The details of this approach can be found by Middleman(1998) and in other transport phenomena books. Briefly, allimportant variables that might affect droplet formation, duringdispersion, were identified. This was followed by the selection ofcore or recurring variables. The core variables were later used toexpress the other non-recurring variables in terms of uniquedimensionless groups. Provided below are the relevant para-meters, which were incorporated during our analysis: dropletdiameter – d [L] (the parameter to be correlated), referencevelocity– ur [LT�1] (ur is defined according to whether doZ ord4Z, where Z is the Kolmogorov length scale of the flow),reference length scale – Lr [L] (Lr is defined according to whetherdoZ or d4Z), interfacial tension – s [MT�2], dynamic viscosityof the continuous phase – mc [ML�1T�1], dynamic viscosity of thedispersed phase – md [ML�1T�1], density of the continuous phase– rc [ML�3], density of the dispersed phase – rd [ML�3], andgravity – g [LT�2] (gravity is important in oceans where a floatingoil slick encounters breaking waves, but is not so important inlaboratory measurements). From the above list of nine para-meters, three (equal to the number of fundamental dimensions,length [L], mass [M], and time [T]) basic or core parameters of ur,md, and rd were chosen to express other parameters in dimen-sionless forms as shown below:

Dimensionless diameter¼ dn¼

d

Lrð1Þ

Dimensionless viscosity¼ mn ¼mc

md

ð2Þ

Dimensionless density¼ rn ¼rc

rd

ð3Þ

Reynolds number¼inertial force

viscous force¼ Red ¼

Lrurrd

md

ð4Þ

Weber number¼inertial force

surface tension force¼Wed ¼

Lru2r rd

sð5Þ

Froude number¼inertial force

gravitational force¼ Frd ¼

u2r

gLrð6Þ

Note, selection of different core variables will give seeminglydifferent dimensionless groups; however, they are interrelatedand hence, are not independent dimensionless groups. For exam-ple, the ratio Wed=Red is often referred to as the capillary number(Cad) and is used in many papers on droplet formation analysis.This term, Cad, represents the ratio of the viscous to the surfacetension force (Acrivos, 1983; Bentley and Leal, 1986; Tjaberingaet al., 1993; Groeneweg et al., 1994; Stone, 1994; Sadhal et al., 1996).Similarly,

ffiffiffiffiffiffiffiffiffiffiWed

p=Red is often referred to as the Ohnesorge number

(Ohd), which relates viscous force to inertial and surface tensionforce. The inverse square of the Ohnesorge number ([1/Ohd]2) hasa form similar to Reynolds group with a characteristic velocity of

B. Mukherjee et al. / Chemical Engineering Science 68 (2012) 432–442434

(s/md) (i.e., the capillary velocity) (McKinley, 2005). The character-istic capillary velocity can be assigned a physical meaning as the oneat which a viscous thread of the fluid (i.e., crude oil in our case)would deform, thin and will eventually break (McKinley, 2005).

The effect of Frd will not be included in further discussions;since our study was conducted under laboratory conditions, it isunlikely that the effect of Frd would be important. Also note thatr* is in the order of 1 (between 0.8–1.0) and hence, it may bedifficult to discern its effect. With these restrictions, a suitablecorrelation for the droplet diameter would be of the form

dn¼

d

Lr¼ f mn;rn;Red;Wed

� �ð7Þ

As the breakage and formation of smaller droplet are enhancedat higher Red and/or Wed, an inverse dependency of dropletdiameter on Red and Wed can be considered. This form ofrelationship is also shown to be true as per the scaling analysisin the next section. Moreover as the forces acting on a droplet areadditive in nature and does not bear a power-form correlation(Hinze, 1955; Konno et al., 1977; Calabrese et al., 1986; Sadhalet al., 1996; Shreekumar and Gandhi, 1996), a suggested form ofEq. (7) may presented as

dn¼

d

Lr¼

A1

Redþ

A2

Wedð8Þ

where A1 and A2 are the fitting constants and are functions of m*

and r*.The choice of Lr and ur would depend on whether droplet size

is larger or smaller than the Kolmogorov scale of the flow (Z, [L]).The Kolmogorov scale is defined as the size of the smallest-scaleeddy at which the turbulent energy of the flow is dissipated dueto viscous shear and is given by (Tennekes and Lumley, 1972)

Z¼ ðmc=rcÞ3

e

" #1=4

ð9Þ

where e is the energy dissipation rate (W/kg). The literatureusually addresses cases where d4Z, which is a common scenarioin agitated vessels where impellers and turbines are usuallyemployed for mixing. Mixing scenarios where doZ is not verycommon and had not been extensively studied. However, suchcases can also be present prevalently under certain mixingconditions and in systems as was observed in our experiments.Provided below is the analysis of both the former and the latercases of droplet formation; however, an elaborate analysis of thelater case, for doZ, is one of the new features that has beenpresented in this paper.

2.1.1. Case I: d4ZThe larger of d or Z is taken as the appropriate length scale (Lr);

since, energy dissipation at scales smaller than d will notcontribute to the droplet break-up processes. Hence, when d4Zthe appropriate length scale to be used is the droplet diameter.The velocity scale, ur, corresponded to the energy dissipation rate(e) at the length scale d and hence, is equal to (ed)1/3 (Konno et al.,1977; Shreekumar and Gandhi, 1996). The corresponding Red

and Wed is given as durrd=md ¼ d4=3e1=3rd=md and du2rrd=s¼

d5=3e2=3rd=s, respectively.

2.1.1.1. I-(a) surface tension control. When RedbWed (i.e.,ðurmd=sÞ51 or when the surface tension force dominates) the1st term on the RHS of Eq. (8) can be neglected and substitution ofWed in Eq. (8) and its rearrangement results in the followingrelationship:

d¼ A3srd

� �3=5

e�2=5 ¼ A4src

� �3=5

e�2=5 ð10Þ

where A3 and A4 ðA4 ¼ A3½rn�3=5Þ are the fitting constant. Here, d

scales as e�2/5, s3/5, and r�3=5c .

2.1.1.2. I-(b) viscous control. When Red5Wed (i.e., ðurmd=sÞb1 orwhen the effect of viscosity dominates), the 2nd term on the RHSof Eq. (8) can be neglected and substitution of Red in Eq. (8) andits rearrangement results in the following relationship:

d¼ A5md

rd

� �3=4

e�1=4 ¼ A6ðrcrdÞ�3=8ðmdÞ

3=4ðeÞ�1=4

ð11Þ

where A5 and A6 ðA6 ¼ A5½rn�3=8Þ are the fitting constant. Here, d

scales as (e)�1/4, (md)3/4, and (rcrd)�3/8.It is clear from the above analysis that when d4Z, d scales

either as relationship shown by Eq. (10) or (11) depending onwhether the surface tension or the viscous forces are important.

2.1.2. Case II: doZThe appropriate length scale, Lr, for this case is Z, which is

equal ðm3c =r3

c eÞ1=4 (Eq. (9)). The appropriate velocity scale, ur is

given as ðemc=rcÞ1=4k , which is known as the Kolmogorov micro-

scale velocity. The subscript ‘‘k’’ in the expression of ur indicatesthat the scale is that of the Kolmogorov eddies. The correspondingRed and Wed is given as Lrurrd=md ¼ ðmn=rnÞ and

Lr u2r rds ¼

mcrc

� �5=4 e1=4rds

� �¼

mn

rn

� �2 m2d

srd

� �e1=4

ðmc=rc Þ3=4

� �, respectively. Note

that substitution of Lr and ur in the expression of Rec

i:e:, Lr urrc

mc

� �, which is the Reynolds number for the continuous

phase, turns out to be 1. Rec of 1 indicates that the flow field inwhich the oil droplet is suspended is laminar in nature, which isexpected when doZ.

2.1.2.1. II-(a) surface tension control. When RedbWed (i.e., whenthe surface tension force dominates) the 1st term on the RHS ofEq. (8) can be neglected and substitution of Wed in Eq. (8) and itsrearrangement results in the following relationship:

d¼ A7rc

mc

� �1=2 se�1=2

rd

¼ A81

mcrc

� �1=2

se�1=2 ð12Þ

where, A7 and A8 (A8¼A7[r * ]) are the fitting constant. Here, d

scales as e�1/2, s, and (mcrc)�1/2.

2.1.2.2. II-(b) viscous control. When Red5Wed (i.e., when theeffect of viscosity dominates) the 2nd term on the RHS of Eq.(8) can be neglected and substitution of Red in Eq. (8) and itsrearrangement results in the following relationship:

d¼ A9md

rd

� �rc

mc

� �1=4

e�1=4 ¼ A10ðmdÞðmcrcÞ�1=4r�1=2

d ðeÞ�1=4ð13Þ

where A9 and A10 ðA10 ¼ A9½rn�1=2Þ are the fitting constant. Here, d

scales as with (e)�1/4, (md), (rd)�1/2, and (mcrc)�1/4.

2.2. Scaling analysis

Scaling analysis involves an estimation of the magnitude ofvarious forces, which causes and prevents droplet breakup andthen balancing these forces in an approximate manner. A dis-persed oil droplet within a water column is subjected to variousexternal and internal forces. Inertial force, which acts on thedroplet surface from outside (Fo [MLT�2]) tries to deform andbreak the droplet. On the other hand, viscous (Vd [MLT�2]) forceacting within the droplet and the surface tension (Sd [MLT�2])force acting on the surface of the droplet tend to resist deforma-tion. Hence, a balance of these opposing forces may be written as(Hinze, 1955; Konno et al., 1977; Calabrese et al., 1986; Sadhal

B. Mukherjee et al. / Chemical Engineering Science 68 (2012) 432–442 435

et al., 1996)

Fo ¼ VdþSd ð14Þ

The inertial force, Fo, is expressed as the surface area of thedroplet, pd2, times the inertial stress, TI, acting on the dropletsurface. An estimate of the inertial stress is based on the

stagnation pressure and is given as rcu2ðdÞ (Sadhal et al., 1996);

where, u2ðdÞ is the average squared velocity difference across twodiametrically opposite points on the surface of the droplet

[L2T�2]. Therefore, Fo is equal to pd2TI ¼ pd2rcu2ðdÞ.

The viscous force, Vd, can be related to the shear stress actingwithin the droplet and is given as (pd2)((md/d)vs), where, vs is thevelocity at the surface of the droplet. According to Hinze (1955),the continuous phase turbulent fluctuations will cause flow

velocities within the droplet in the order offfiffiffiffiffiffiffiffiffiffiffiffiTI=rd

pand therefore,

Vd will be in the order of C1ðpd2Þmd

d

ffiffiffiffiTI

rd

q¼ C1ðpd2

Þmd

d

ffiffiffiffiffiffiffiffiffiffiffiffircu2ðdÞrd

r,

where C1 is the fitting constant.Finally the surface tension force, Sd, is given as pds. Substitu-

tion for Fo, Vd, and Sd in Eq. (14) results in the following relation-ship:

pd2rcu2ðdÞ ¼ C2ðpd2Þmd

d

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffircu2ðdÞ

rd

sþC3pds ð15Þ

where, C2 and C3 are the fitting constants.The relative length scale of the droplet diameter, d, with

respect to the Kolmogorov scale of the flow, Z, dictates themagnitude of the fluctuating velocity at droplet surface. Hence,for the analysis of mechanisms of droplet formation it is neces-sary to consider two cases: I – when d4Z and II – when doZ.

2.2.1. Case I: d4ZFor, d4Z, u2ðdÞ is proportional to �(ed)2/3 (Konno et al., 1977;

Shreekumar and Gandhi, 1996). Substituting the values for u2ðdÞ

in Eq. (15), results in the following relationship:

rcðedÞ2=3¼ C2

md

d

� � rc

rd

� �1=2

ðedÞ1=3þC3

sd

� �ð16Þ

2.2.1.1. I-(a) surface tension control. A common scenario duringmixing of immiscible liquids (e.g., oil and water) is when theexternal inertial force is counteracted by the surface tension force,while the effect due to the viscosity of the dispersed phase isnegligible. Using this assumption, the 1st term on the RHS of Eq.(16) can be neglected resulting in the following relationship:

d

ðs=rcÞ3=5ðeÞ�2=5

¼ A4 ð17Þ

where A4 is the fitting constant. Note that Eq. Eq. (17) is same asEq. (10), obtained using dimensionless analysis. The LHS of Eq.(17) represents the ratio of inertial force to the surface tensionforce and is known as the Weber number (We) (Eq. (5)). A droplet,in the fluid flow field, will disrupt and break when We exceeds acertain critical value, Wecrit (Hinze, 1955).

2.2.1.2. I-(b) viscous control. An alternative scenario, which is notoften considered, is the one in which the external inertial force iscounteracted by the viscosity of the dispersed phase and theeffect of surface tension force is negligible. Using this assumption,the 2nd term on the RHS of Eq. (16) can be neglected resulting inthe following relationship:

d

ðrcrdÞ�3=8ðmdÞ

3=4ðeÞ�1=4

¼ A6 ð18Þ

where A6 is the fitting constant. Note that Eq. (18) is same asEq. (11), obtained using dimensionless analysis. The LHS ofEq. (18) represents the ratio of inertial force to the effect dueto the viscosity of the dispersed phase; a droplet, in the flowfield, will disrupt and break when this ratio exceeds a certaincritical value. It may be also noted here that Karbstein and

Schubert (1995) used Ohd ¼mdffiffiffiffiffiffiffiffiffirdsLr

p ¼

ffiffiffiffiffiffiffiWed

p

Red

� �to describe,

which of the mechanisms, represented by Eqs. (17) and (18),will control droplet breakage. Eq. (17) applies in cases of low Oh,whereas Eq. (18) applies in cases of high Oh (Karbstein andSchubert, 1995). Although exact cut off values for Oh was notprovided, Eqs. (17) and (18) was found to be true whenmdo0.01 Pa s and b 0.01 Pa s, respectively (Karbstein andSchubert, 1995).

2.2.2. Case II: doZFor, doZ, uðdÞ is proportional to the Kolmogorov microscale

velocity, which is equal to e1=4ðmc=rcÞ1=4. Energy dissipation in the

viscous sub-range is the major factor in the droplet breakage atthese scales. On the other hand, energy scales, which are largerthan Z simply transport the droplet in the flow field and do notcontribute towards inertial stress. Substituting the values for uðdÞ

in Eq. (15), results in the following relationship:

rce1=2 mc

rc

� �1=2

¼ C4md

d

� � 1

rd

� �1=2

rce1=2 mc

rc

� �1=2 !1=2

þC5sd

� �ð19Þ

2.2.2.1. II-(a) surface tension control. When the main restoringforce is the surface tension force and the effect due to theviscosity of the dispersed phase is negligible the 1st termon the RHS of Eq. (19) can be neglected resulting in thefollowing relationship (Shinnar, 1961; Sprow, 1967; Li andGarrett, 1998):

d

sr�1=2c m�1=2

c e�1=2¼ A8 ð20Þ

where A8 is the fitting constant. Note that Eq. (20) is same as Eq.(12), obtained using dimensionless analysis. The LHS of Eq. (20)represents the ratio of viscous to the surface tension force and isknown as the capillary number (Ca). A droplet, in the flow field,will break when ratio of the viscous to the surface tension forceacting on the droplet exceeds a certain value known as the criticalcapillary number (Cac) (Acrivos, 1983; Bentley and Leal, 1986;Tjaberinga et al., 1993; Groeneweg et al., 1994; Stone, 1994;Sadhal et al., 1996).

2.2.2.2. II-(b) viscous control. An alternative scenario, whichis also possible, is when the main restoring force is due to theeffect of dispersed phase viscosity and the effect of surfacetension force is negligible. Under this condition, the 2nd termon the RHS of Eq. (19) can be neglected resulting in the followingrelationship:

d

mdm�1=4c r�1=2

d r�1=4c e�1=4

¼ A10 ð21Þ

where A10 is the fitting constant. Note that Eq. (21) is same asEq. (13), obtained using dimensionless analysis. The LHS ofEq. (21) represents the ratio of the external viscous force to theeffect due to the viscosity of the dispersed phase; a droplet, in theflow field, will disrupt and break when this ratio exceeds a certaincritical value.

Table 1Dimensionless and scaling analysis for liquid-liquid droplet formation.

Droplet

formation

scenarios

Appropriate

length scale

(Lr)

Appropriate

velocity scale

(ur)

Dominant

deforming

force

Dominant restoring

force

Ratio of the

dominating

deforming to

restoring force

Criteria Scaling relationships for

droplet diameter

d4Z d (ed)1/3 Inertia Surface-tension ðrcu2r Þ

ðs=dÞRedbWed

� src

� �3=5e�2=5

Viscous effect of

the dispersed phase

ðrcu2r Þ

ðmdvs=dÞRed5Wed � ðrcrdÞ

�3=8ðmdÞ

3=4ðeÞ�1=4

doZZ¼ m3

c

r3c e

� �1=4 emc

rc

� �1=4

k

Viscous shear Surface-tension ðmcur=LrÞ

ðs=dÞRedbWed

� 1mcrc

� �1=2se�1=2

Viscous effect of

the dispersed phase

ðmcur=LrÞ

ðmdvs=dÞRed5Wed � ðmdÞðmcrcÞ

�1=4r�1=2dðeÞ�1=4

Table 2Physical properties of mixtures of evaporatively weathered crude oils with

Corexit 9500.

B. Mukherjee et al. / Chemical Engineering Science 68 (2012) 432–442436

The four droplet formation scenarios, obtained using dimen-sionless and scaling analysis, and their respective scaling para-meters are summarized in Table 1.

Oil type Weathering

extent (%)

DORa density

(rd)

(kg/m3)

Viscosity (md)

(Pa s)b

Interfacial

tension (s)

(N/m)�103

Arabian

light

23 1:25 895 0.0182 0.18

1:100 880 0.0162 0.21

Mars 21 1:25 928 0.153 0.45

1:100 930 0.163 1.35

Lloyd 17 1:25 960 6.410 7.01

1:100 960 8.580 14.73

a DOR is the dispersant-to-oil ratio.b Pa s¼1000 cP.

2.2.3. Case III: other cases

There can be several other scenarios of droplet breakagesdepending on the nature of the external deforming force, Fo.Gravitational force may play a significant role in oil dropletgeneration processes, especially if the floating oil slick encountersbreaking waves. Under this scenario, TI can be approximatedusing the quantity rcgd (Sadhal et al., 1996); where g is theacceleration due to gravity [LT�2]. Substitution and simplificationof Eq. (14) results in the following relationships:

d

ðs=rcgÞ1=2¼ A11 ð22Þ

d

ðmdÞ2=3ð1=rcrdgÞ1=3

¼ A12 ð23Þ

where A11 and A12 are the fitting constants. Eqs. (22) and (23) areapplicable when the main restoring force against droplet defor-mation is provided by the surface tension and the viscosity of thedispersed phase, respectively. The LHS of Eq. (22) represents therelative importance of the gravitational force compared tothe surface tension force and is known as the Eotvos number(Eo). The LHS of Eq. (23) represents the relative importance thegravitational force compared to the effect of dispersed phaseviscosity. These mechanisms, as represented by Eqs. (22) and (23),are applicable for d4Z. When Froude number (Fr), which repre-sents the relative importance of the inertial to the gravitationalforce, is high, mechanisms as expressed by Eqs. (17) and (18)controls droplet formations. When Fr is low, the dominantmechanisms of droplet formation are as represented by Eqs. (22)and (23).

Similarly, external tangential forces can also play an active rolein droplet formation. These cases usually occur when dropletdeformation is driven by gradients in the surface tension, whichmay be caused by variations in temperature or surfactant con-centrations (Stone, 1994; Sadhal et al., 1996). The two majordimensionless numbers that must be considered for scaling ofdroplet deformation under this scenario are the Peclet number(Pe) and the Marangoni number (Ma). Details for this case arebeyond the scope of this work and may be found in Stone (1994)and Sadhal et al. (1996).

3. Experimental materials and methods

3.1. Oil–dispersant mixtures

Three crude oils of Arabian Light (initial density – 830 kg/m3),Mars (initial density – 876 kg/m3), and Lloyd (initial density –921 kg/m3) were used in our experiments. Each of the oils wasevaporatively weathered under a stream of air for 24 h at roomtemperature (20–22 1C) in a fume hood. The extent of mass lossvaried among oils due to differences in the relative concentrationsof volatile components. The mass of Arabian Light was reduced by23%, Mars was reduced by 21%, and Lloyd was reduced by 17%.The oils were evaporatively weathered to prevent compositionalchanges from occurring during preparation, storage, or testing ofthe oil samples, which is known to effect dispersion (Mukherjeeet al., 2011a). Changes in composition during sample preparationwould have introduced an uncontrolled confounding factor intothe experimental design.

Dispersant used was the commercially available Corexit 9500(Nalco Energy Services, Sugar Land, TX, USA), which is made up of48% nonionic and 35% anionic surfactants in a solvent consistingof a mixture of food-grade aliphatic hydrocarbons (NRC, 2005).The nonionic surfactants include ethoxylated sorbitan mono- andtri-oleates and sorbitan monooleate. Sodium dioctyl sulfosucci-nate is the major anionic surfactant in the dispersant (Pollino andHoldway, 2002).

Two sets of oil–dispersant mixtures were prepared with eachof the weathered oil and Corexit 9500. In the first set thedispersant-to-oil ratio was kept at 1:25 (v/v), while in the secondset the ratio was at 1:100 (v/v). To ensure homogeneous distribu-tion of dispersant in the oil, the oil–dispersant mixtures were

B. Mukherjee et al. / Chemical Engineering Science 68 (2012) 432–442 437

mixed for 24 h at 200 rpm on a gyratory shaker. The physicalproperties of the oil–dispersant mixtures are provided in Table 2.

3.2. Continuous phase

Artificial sea water was used as the continuous phase, whichwas prepared by dissolving 35 g of artificial sea salt (Sigma–Aldrich) in 1 L of ultrapure deionized water. The sea water wasfiltered through a 0.2 mm membrane filter (Millipore, Billerica,MA, USA) before use. Filtration removed any suspended materialsthat might interfere with droplet size measurements.

3.3. Mixing system and energy

Dispersion experiments were conducted using baffled flasks,which were constructed by modifying 150-mL trypsinizing flaskswith four glass baffles (Wheaton Science Products; Millville, NJ,USA). The flasks were modified by attaching a drain port near thebottom to facilitate sample collection (Sorial et al., 2004a, 2004b).The contents in the baffled flasks were mixed by shaking on a Lab-Line Orbit Environ Shaker (Lab-Line Instruments, Inc., MelrosePark, IL, USA) with an orbital diameter of 1.9 cm. Three rotationalspeeds-125 rpm, 150 rpm, and 200 rpm were used. The bulkaverage energy dissipation rates corresponding to these speedswere 0.00075, 0.016, and 0.16 W/kg water, respectively (Table 3).These average energy dissipation rates were determined by Kakuet al. (2006a, 2006b) using a mixing system with the samecharacteristics as described above. For one rotational speed(125 rpm), the average energy dissipation rate was estimated byexponential extrapolation of the reported data for 50 and100 rpm. In the baffled-flask system, over-and-under type ofmixing occurs due to three dimensional fluid flow induced bythe baffles and can be considered to be isotropic (Kaku et al.(2006a, 2006b)). The spatial distributions of fluid velocity andmixing energy are relatively uniform in the baffled-flask systemand become more uniform at higher rotational speeds (Kaku et al.,2006b).

3.4. Dispersion experiments

One-hundred and twenty milliliters of filtered artificial seawater was added to each baffled flask, and 0.1-mL of an oil–dispersant mixture was added on the water surface using aRepeater Plus Pipette (Eppendorf, Westbury, NY, USA). The con-tents of the flasks were mixed for 45 min at a predeterminedrotational speed. Preliminary results showed that dispersion of allsix oil–dispersant combinations used in this research reachedsteady state within 45 min of mixing. After mixing, approximately1-mL of the dispersed phase was drained through the stopcock atthe bottom of the baffled flask and discarded and a 5-mL samplewas collected for further analysis. Three independent replicateswere conducted for each experimental condition for a total of 54independent tests. The order of the tests was randomized to

Table 3Bulk average mixing energies and their corresponding rotational speeds in baffled-

flask mixing system.

Rotational speed (rpm) Kolmogorov

length scale

Bulk average

mixing

energy (e) (W/kg)

Baffled flask Z (mm)

125 192 7.5�10�4

150 90 1.6�10�2

175 68 4.8�10�2

200 50 1.6�10�1

preclude confounding of systematic (e.g., temporal) effects withtreatment effects.

3.5. Analytical methods

The size distribution and number concentration of the dis-persed oil droplets was measured using an optical particlecounter (OPC) (Particle Measuring Systems Inc., Boulder, CO,USA). The OPC was equipped with an LS-200 sample moduleand a Liquilaz E20P detector with a built-in 12-mW laser diode(l¼785 nm). The OPC reported droplet sizes in 15 adjustablechannels from 2 to 125 mm. The number concentration of theparticles must be o104 mL�1 to prevent coincidence counting.Therefore, samples were diluted with deionized water by a factorof 100–5000 before measurement. The pipette tips used totransfer these samples were cut to enlarge the opening to at least2 mm to minimize the effects of sample transfer on the droplet-size distribution (APHA 1999). Our previous research (Mukherjeeand Wrenn, 2009b) showed that the size of the oil dropletsreported by the OPC (diluted sample) matched well with thatobtained using microscopy (undiluted sample).

The interfacial tension between the oil–dispersant mixturesand the filtered artificial sea water was measured using a digitaltensiometer (Model: KSV Sigma 703 D) using the Du Nouy ringmethod. The ring diameter was 1.909 cm and was made ofplatinum-iridium wire with diameter of 0.037 cm. The ring washung from a balance hook and was slowly lowered into thefiltered artificial sea water. A thick layer of oil–dispersant mixturewas then carefully added on to the water surface. The Du Nouyring was then slowly raised through the oil–water interface andthe force needed to break through the interface was recorded toestimate the oil–water interfacial tension. The lower and uppermeasuring range for the instrument was 0.001 and 1000 dyn/cm,respectively, at room temperature (20–22 1C).

The viscosities of the oil–dispersant mixtures were measuredusing a constant stress and strain rheometer (TA Instruments AR2000). The rheometer had a cone and plate geometry with adiameter of 60 mm (cone angle 01 59 min, 49 s: truncation27 mm). The torque was between 0.1 and 200 mN m. The lowerlimit of the stress that the equipment can accurately measure was0.0008 Pa. The temperature was maintained at 25 1C duringmeasurement using a Peltier plate. The Peltier plate was cleanedwith DCM and isopropyl alcohol (Sigma–Aldrich) and dried beforeand after all measurements.

The density was determined by measuring the mass of a knownvolume of oil–dispersant mixture using a Mettler AE 200 balance(Mettler-Toledo, Inc., Columbus, OH, USA), which has a readability of0.1 mg and accuracy better than 0.2 mg under typical laboratoryoperating conditions. Using an Eppendorf repeater plus pipette,1-mL of the oil–dispersant sample was transferred to a plasticweigh boat, and the mass was recorded after the value stabilized.Six replicate measurements were made, and the average valueswere used to calculate the oil–dispersant mixture density.

3.6. Sauter mean diameter (d32)

The effects of the physical properties of the dispersed andcontinuous phase and the energy dissipation rate on dropletformation were evaluated using Sauter mean diameter (d32) ofthe droplet size distributions. The droplet-size distributionsvaried with treatments, but in general were tri-modal withdroplet sizes ranging between 2 mm and 125 mm. However,droplets450 mm were rare in most of the treatments. Threerepresentative cases of the normalized-volume distribution ofdroplets generated by dispersion of oil–dispersant mixtures, inthe artificial sea water, are provided in Fig. 1. The Sauter mean

Fig. 2. Effect of average energy dissipation rate (e) on d32 generated by the

dispersion of weathered (A) Arabian Light, (B) Mars, and (C) Lloyd crude oil at a

dispersant-to-oil (DOR) of 1:25 (K) and 1:100 ( ) in baffled flask mixing system.

The solid, dotted, and long dash lines represent the fit when d32 scales with e�1/4,

e�1/2, and C1 [e]�1/4þC2 [e]�1/2, respectively, where C1 and C2 are fitting

constants. The error bar represents one standard deviation of three independent

experiments. Note that in few cases the errors bars are too small and are obscured

Fig. 1. Normalized volume distributions of oil droplets generated by the disper-

sion of weathered Arabian Light (J), Mars ( ), and Lloyd (.) crude oil at

dispersant-to-oil (DOR) of 1:25 and average energy dissipation rate (e) of

0.016 W/kg in baffled-flask mixing system. The Arabian Light, Mars, and Lloyd

were evaporatively weathered by 23%, 21%, and 17%, respectively.

B. Mukherjee et al. / Chemical Engineering Science 68 (2012) 432–442438

diameter represents the ratio of the volume of oil that getsdispersed upon mixing per unit area of the oil–water interfacethat gets exposed and is expressed as

d32 ¼

P15i ¼ 1 Nidi

3

P15i ¼ 1 Nidi

2

0@

1A ð24Þ

where Ni is the number concentration of droplets in channel (orbin) ‘‘i’’ (droplets/mL) and di is the average droplet diameter inchannel i (mm), which is given by the arithmetic average of theupper and lower size limits of the channel. Ntot is the numberconcentration of all droplets measured by the OPC (i.e., alldroplets with diameters greater than 2 mm and less than125 mm). The upper limit of ‘‘i’’ (i.e., 15) in Eq. (24) representsthe maximum number of adjustable size channels in the OPC.

Although some modelers (e.g., Li and Garrett, 1998) havesuggested that the diameter of the largest stable droplet that cansurvive in a flow field without undergoing any further breakage(dmax) as the appropriate parameter for characterizing scaling trends,d32 was preferred in this study. The reason being, that the computa-tion of d32 is relatively easy and is less prone to error as it considersthe entire size-distribution data instead of just one extreme of thedistribution in the case of dmax, which exists in very low numberconcentration (Shaw, 2003). Also, d32 and dmax are proportional toeach other with only a few exceptions (Sprow, 1967; Calabrese et al.,1986; Zhou and Kresta, 1998; Pacek et al., 1999). Therefore, anyinformation on the droplet-scaling behavior based on either of thediameters should lead to the similar conclusions.

by the data points. The Arabian Light, Mars, and Lloyd were evaporatively

weathered by 23%, 21%, and 17%, respectively.

4. Results and discussions

The variation of d32 with the average energy dissipation rate(e) for the weathered Arabian Light, Mars and Lloyd crude oils isshown in Fig. 2. For all the cases, d32 decreased with an increase inthe mixing energy and DOR (dispersant-to-oil ratio). Both of theseobservations are consistent with our a priori expectations.Increase in the mixing energy provides additional energy to breaklarge-sized oil droplets into small-sized daughter droplets(Narsimhan et al., 1980); this result in the generation of a dropletsize distribution that is dominated by small-sized droplets andtherefore a smaller d32. With an increase in DOR from 1:100 to1:25 (i.e., increase in the dispersant concentration), the interfacial

tension of the resulting oil–dispersant mixture decreases(Table 2). A decrease in the oil–water interfacial tension favorseasy breakage and the formation of small-sized droplets bylowering the energy requirement for the creation of new oil–water interfacial areas. Note that at a fixed mixing energy andDOR, d32 for the Arabian Light containing treatments were smallerthan that of the Mars, which were smaller compared to d32

generated by the dispersion of the Lloyd crude oil. For example,compare d32 values of 7.5, 11, and 22.5 mm for the Arabian Light,Mars, and Lloyd crude oil, respectively, at 0.16 W/kg and DOR1:100. This characteristic variation of d32, at a fixed mixingenergy, is due to the combined effect of the oil viscosity and the

Fig. 3. Effect of average energy dissipation rate (e) on d32 generated by the

dispersion of 18.5% weathered Mars crude oil combined with laboratory made

dispersant of HLB 10 (K) and 12 ( ) at a dispersant-to-oil (DOR) of 1:25 in baffled

flask mixing system. The solid, dotted, and long dash lines represent the fit when

d32 scales with e�1/4, e�1/2, and C1 [e]�1/4þC2 [e]�1/2, respectively, where C1 and

C2 are fitting constants. The error bar represents one standard deviation of three

independent experiments. Note that in most of the cases the errors bars are too

small and are obscured by the data points.

B. Mukherjee et al. / Chemical Engineering Science 68 (2012) 432–442 439

oil–water interfacial tension. The Lloyd crude oil is approximately60–40 times more viscous than that of the Mars and is severalhundred times more viscous than the Arabian Light (Table 2).Higher viscosity means larger cohesive force within the oil phase,which resists disruption and breakage of the oil droplets intosmaller ones. Similarly, higher oil–water interfacial tension of theLloyd crude oil (s�10�10�3 N/m), compared to the Mars(s�1�10�3 N/m) and the Arabian Light (s�1.9�10�4 N/m)also prevented the formation of small-sized oil droplets.

The physical properties of oil and water phases (i.e., viscosity,oil–water interfacial tension, density) and mixing energy do notbear a simple one-to-one correlation with the sizes of the oildroplets formed during dispersion. But rather, as shown by ourscaling and dimensionless analysis and by our previous study(Mukherjee and Wrenn, 2011b), the relationships are interrelatedand complex. For example when d4Z and the main restoringforce is provided by surface tension (Eq. (17)), d scales withðs=rcÞ

3=5e�2=5 and not as an individual function of density, surfacetension, or mixing energy, separately. However, note that therelative importance of different potential factors on dropletformation varies depending upon the mechanisms involved. Forexample when the main restoring force is provided by surfacetension, the droplet diameter scales with s3/5 for d4Z but with sfor doZ (Eqs. (17) and (20)). Among all the potential factors thataffect droplet formations, mixing energy is considered to be themost vital in studies relating to chemical dispersion of crude oil(NRC, 2005). Mixing energy not only plays a significant role in thebreakage and coalescence kinetics of the dispersed oil droplets(Coulaloglou and Tavlarides, 1977; Aravamudan et al., 1981; Liand Garrett, 1998; Ernest et al., 1995; Kaku et al. (2006a, 2006b))but also in the transportation of the generated droplets awayfrom the spilled site. Considering the importance of mixingenergy on droplet formation, we investigated the best-form ofrelationship that describes the variation of d32 with mixingenergy. Our analysis showed that for all the oil–dispersantcombinations, d32 scaled proportionally with �[e]�1/4 (Fig. 2).As shown in the sections on dimensionless and scaling analysis,there are two mixing scenarios under which d scales with �[e]�1/4.Under the first condition d4Z (Eq. (18)), while under the seconddoZ (Eq. (21)) and for the both conditions RedoWed. Refer toTable 3 and Fig. 2, the values of Z at 0.00075, 0.016, 0.16 W/kgwere all greater than d32. The fact that in all treatments d32 scaleswith [e]�1/4 and was less than Z, implied that the same mechan-ism, as expressed by Eq. (21), controlled droplet formations.Under this condition d scales with ðmdÞðmcrcÞ

�1=4r�1=2d ðeÞ�1=4 and

the viscous effect of the oil phase is the main restoring force whilethe effect of surface tension force is negligible. The appropriatelength scale and the velocity scale for this regime of dropletformation is given by Z and ðemc=rcÞ

1=4k , respectively, while Red

and Wed is expressed as (mcrd/mdrc) and ðmc=rcÞ5=4ðe1=4rd=sÞ,

respectively. For the applicability of Eq. (21), Red must be less thanWed, which was confirmed to be true for most of the treatments.However for a few cases, especially at 0.00075 W/kg, Wed/Red wasfound to be r1 and a fit of [e]�1/4 failed to explain the variation ofd32 with e. As shown previously by our dimensionless and scalinganalysis, when doZ and Wed/Redr1, d scales with [e]�1/2 (Eq. (20))and not with [e]�1/4, which is also obvious from the better fitprovided by d32�[e]�1/2 at lower mixing energies (Fig. 2). Note thatfor a fixed mixing energy, oil–dispersant combination with low oil–water interfacial tension favored the formation of smaller sizeddroplets (Fig. 2), which implied that although the effect of surfacetension force was small but was not completely absent in treat-ments even when the viscous effect of the dispersed phase wasdominant. This also explains why a mixer model, which includedboth the effect of the viscous and surface tension forcesðd32 � C1½e��1=4þC2½e��1=2Þ was able to explain, more effectively,

the variation of d32 with e than when either of the forces wasconsidered alone (Fig. 2). Although Wed4Red, for most treatments,perhaps the ratio of A2/A1 is larger than one (Eq. (8)). Therefore the2nd term on the RHS of Eq. (8) and hence, the surface tension forcestill had some minor contribution on droplet formation process.More detailed studies are needed to accurately estimate A1 and A2

and realize their physical significance and relative importance in thedroplet formation process.

Some additional experiments were conducted in the baffled-flaskmixing system to investigate the droplet scaling behavior for oils ofmedium viscosity (e.g., Mars) and low oil–water interfacial tension(e.g., Arabian Light). For this purpose, two combinations of oil–dispersant mixtures were prepared using 18.5% weathered Marscrude oil and laboratory made dispersants with hydrophilic–liphophilic balance (HLB) of 10 and 12 at DOR 1:25. Both dispersantscontained the same surfactants – Tween 80 (ethoxylated sorbitanmonooleate; HLB 15) and Span 80 (sorbitan monooleate; HLB 4.3) –and the same solvent (hexadecane). Hexadecane constituted 30% ofthe mixture by mass, and the surfactants constituted 70% of thedispersant mass. The relative proportions of the surfactants werevaried to produce final HLB of 10 (30% hexadecane, 32.5% Span 80,and 37.5% Tween 80 by mass) and 12 (30% hexadecane, 19.5% Span80, and 50.5% Tween 80 by mass). The viscosities of both the oil–dispersant mixtures were 0.132 Pa s, while the oil–water interfacialtensions were 0.00015 and 0.0001 N/m for the mixture containingdispersant of HLB 12 and 10, respectively. For both the oil–dispersant combinations, d32 scaled proportionally with [e]�1/4

(Fig. 3). The estimated values of Z, corresponding to the energydissipation rates, were found to be less than d32 (Table 3 and Fig. 3).It was also confirmed that in each treatment Red was less Wed,which means that the mechanism expressed by Eq. (21) controlleddroplet formation as was also observed in the earlier cases. Notethat the oil–dispersant mixture with HLB 10 (i.e., lower oil–waterinterfacial tension) compared to HLB 12 (i.e., higher oil–waterinterfacial tension) consistently generated small-sized droplets at afixed mixing energy; however, Eq. (21) suggests that the effect ofsurface tension force should be absent. This difference between thetheoretical analysis and experimental observation indicate thatalthough the effect of the surface tension force was low, for theabove sets of experiments, but was not completely absent. Also note,

B. Mukherjee et al. / Chemical Engineering Science 68 (2012) 432–442440

that the viscous effect of the dispersed phase and surface tensionforce together were able to help better explain the slight non-linearity in the observed d32 versus e plot as opposed to when theviscous effect of the dispersed phase is considered alone (Fig. 3).

Research till date had shown a scaling behavior of the formd�[e]�1/2 for doZ (Eq. (20)) for which the main restoring force isprovided by surface tension while the effect of the dispersedphase viscosity is negligible (Acrivos, 1983; Bentley and Leal,1986; Tjaberinga et al., 1993; Groeneweg et al., 1994; Stone,1994; Sadhal et al., 1996). However, our current research indi-cates that under certain range of physical properties and mixingenergies, for doZ, droplet sizes can scale with [e]�1/4. Under thisscenario, the main restoring force is provided by the dispersedphase viscosity while the effect of surface tension force isnegligible. Note, that when doZ, the continuous phase Reynoldsnumber (Rec) should be very low. This means that the externalforce, which tries to disrupt and break the droplets, is notpressure but is viscous in nature. The estimated values of Rec

were found to be 1, in our case, which is indicative of the presenceof a laminar flow regime for the droplets.

Fig. 4 shows the effect of the average energy dissipation rateon dmax for the Prudhoe Bay and the Ekosfisk crude oil. These datawere taken from Delvigne and Sweeney (1988), who investigatedthe breakdown of oil droplets in grid generated turbulent flowsfor mixing energies ranging between 0.001 and 5 W/kg. We fittedall the droplet formation models those were discussed in thedimensionless and scaling analysis section, to the data. Not onesingle, but combinations of two models successfully explained thevariation of dmax with the mixing energy. For er and Z0.5 W/kg,dmax was found to be proportional to [e]�1/4 (i.e., either Eq. (18) or

Fig. 4. Variation of dmax with average energy dissipation rate (e) for (A) Prudhoe

Bay and (B) Ekofisk crude oil. The lines represent the fit of dmax scaling with e in

different regimes of droplet formation. These data were taken from the original

work by Delvigne and Sweeney (1988). The numbers corresponding to each data

points, in the graph, represents the Kolmorogov length scale (Z, mm).

(21)) and [e]�2/5 (Eq. (17)), respectively. The relationshipdmax � ½e��2=5, matched closely with the empirical fit ofdmax�[e]–0.570.1 suggested by Delvigne and Sweeney (1988) foreZ0.1 W/kg. The relationship dmax � ½e��1=2 (i.e., Eq. (20)) alsoprovided a reasonable fit for eZ0.5 W/kg; however, was notapplicable as dmax4Z (Fig. 4). Therefore the fit dmax � ½e��2=5

seemed to be the most appropriate for e beyond 0.5 W/kg forboth the oils. The scaling relationship for er0.5 W/kgði:e:, dmax � ½e��1=4Þ, matched as was obtained for our experimentalresults. However in our experiments d was less Z as opposed to dgreater than Z in the case of Delvigne and Sweeney (1988) (Fig. 4).Therefore the relationship as expressed by Eq. (18) explains thedroplet formation mechanism for er0.5 W/kg in Fig. 4. Note, thatthroughout the whole range of mixing energy tested by Delvigne andSweeney (1988), d4Z and the external force, which tried to disruptand break the droplets, was due to the pressure difference across thedroplet diameter. For er0.5 W/kg the main restoring force wasbecause of the viscosity of the oil phase, while for eZ0.5 W/kg thesurface tension force was the dominating restoring force.

5. Conclusions

The mechanisms of droplet formations, during the chemicaldispersion of crude oil, were investigated using theoretical andexperimental approaches. For the theoretical work, dimensionlessas well as force balance analysis were employed. Both of theseanalyses lead to the same correlations for the dependency of thedrop diameter (d) on the energy dissipation rates (e) in thesystem. The outcomes of the analyses are as follows:

(i)

d4Z: Here the appropriate length and velocity scale is d and(ed)1/3, respectively. The droplet size, d, scales either with e�2/

5 or e�1/4 depending on whether the main restoring force isprovided by the surface tension or the oil-phase viscosity. Theexternal deforming force, which tries to disrupt and break thedroplet, is due to pressure difference across droplet diameter.

(ii)

doZ: Here the appropriate length and velocity scale is Z (theKolmogorov length scale) and ðemc=rcÞ

1=4k (the Kolmogorov

microscale velocity), respectively. The droplet size, d, scaleseither with e�1/2 or e�1/4 depending on whether the mainrestoring force is provided by the surface tension or the oil-phase viscosity. The external deforming force, which tries todisrupt and break the droplet, is viscous in nature. While theformer case (i.e., d�e�1/2) is a well established droplet forma-tion mechanism in the literature for doZ, the latter case (i.e.,d�e�1/4) is a new contribution of this present study. This case ismore pertinent to our study while the former case is morepertinent to unit operations such as liquid–liquid extraction orsystems with large energy dissipations.

In the experimental part of the work, studies were conductedin baffled-flask mixing system using different crude oils and withcommercial and lab-made dispersants over a range of mixingenergy similar to those that might be encountered under realenvironmental settings. For all the treatments, d32 scaled withe�1/4 and was found to be less than Z. These observationsconfirmed our theoretical findings that within a certain range ofphysical properties and mixing energy, it is possible for thedroplets less than Z to scale with e�1/4. In addition, our theoreticalmodels were able to successfully explain the variation of dmax as afunction of energy dissipation rates for experiments conducted byDelvigne and Sweeney (1988). For e40.5 W/kg, dmax scaled withe�2/5 and was found to be greater than Z. Our model matchedwell with the empirical fit of dmax�[e]–0.570.1 provided byDelvigne and Sweeney (1988) for this regime of mixing energy.

B. Mukherjee et al. / Chemical Engineering Science 68 (2012) 432–442 441

We were also able to identify the mechanism and provided anappropriate scaling behavior of dmax with e�1/4 for eo0.5 W/kg,which was not done by any prior studies. This scaling behavior ofthe form dmax � e�1=4 is different from the commonly observedbehavior of dmax � e�2=5 for d4Z. An important significance of ourwork is the identification of all the scaling behavior of d with e forboth doand4Z, in addition to the already known ones. Theeffect of mixing energy on droplet formations would grosslyremain over estimated if one uses the relationships d�e�1/2 fordoZ and d�e�2/5 for d4Z, without taking into account d�e�1/4.All these above findings and observations will have a directimpact in the appropriate scaling of droplet sizes as a functionof different physicochemical factors, which is important for thedevelopment of reliable predictive models. Model developmentsare undeniably important as they can increase the possibility topredict at sea-performances based on lab-scale results and canprovide guidance to spill responders in effective remediation ofoil-spills on natural water bodies. To be able to accurately scaleand extrapolate lab results will however, require a thoroughunderstanding and incorporation of the effects of flow-fieldheterogeneities on droplet formation, which will especially havean important effect at larger scales of study (Mukherjee andWrenn, 2009a). Droplet formation is a microscale phenomenon,and is known to be a strong function of local fluid properties(Konno et al., 1983) and the flow characteristic (Acrivos, 1983;Bentley and Leal, 1986; Tjaberinga et al., 1993; Groeneweg et al.,1994; Stone, 1994; Ha and Leal, 2001). In most of the mixingsystems, the space variations of local hydraulic properties areinhomogeneous (Galinat et al., 2007) and therefore, the likelihoodof droplet formation (i.e., breakage probability) depends on theposition in the flow field rather than on the global fluid properties(e.g., bulk average mixing energy) (Galinat et al., 2007).

Author disclosure statement

No competing financial interests exist.

Acknowledgments

The authors would like to thank Larry Heugatter (ConocoPhilips Co.) for providing the crude oils used in the research,and Pratim Biswas (Washington University) for providing accessto the optical particle counter. The Department of Energy,Environmental, and Chemical Engineering provided financialsupport to BM during this research.

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