Model for drop coalescence in a locally isotropic ... - Semantic Scholar
Journal of Colloid and Interface Science 272 (2004) 197–209 www.elsevier.com/locate/jcis
Model for drop coalescence in a locally isotropic turbulent flow field Ganesan Narsimhan Biochemical and Food Process Engineering, Department of Agricultural and Biological Engineering, Purdue University, West Lafayette, IN 47907, USA Received 12 June 2003; accepted 24 November 2003
Abstract The proposed model views drop coalescence in a turbulent flow field as a two-step process consisting of formation of a doublet due to drop collisions followed by coalescence of the individual droplets in a doublet due to the drainage of the intervening film of continuous phase under the action of colloidal (van der Waals and electrostatic) and random turbulent forces. The turbulent flow field was assumed to be locally isotropic. A first-passage-time analysis was employed for the random process of intervening continuous-phase film thickness between the two drops of a doublet in order to evaluate the first two moments of coalescence-time distribution of the doublet. The average drop coalescence time of the doublet was dependent on the barrier for coalescence due to the net repulsive force (net effect of colloidal repulsive and turbulent attractive forces). The predicted average drop coalescence time was found to be smaller for larger turbulent energy dissipation rates, smaller surface potentials, larger drop sizes, larger ionic strengths, and larger drop size ratios of unequal-sized drop pairs. The predicted average drop coalescence time was found to decrease whenever the ratio of average turbulent force to repulsive force barrier became larger. The calculated coalescence-time distribution was broader, with a higher standard deviation, at lower energy dissipation rates, higher surface potentials, smaller drop sizes, and smaller size ratios of unequal drop pairs. The model predictions of average coalescencerate constants for tetradecane-in-water emulsions stabilized by sodium dodecyl sulfate (SDS) in a high-pressure homogenizer agreed fairly well with the inferred experimental values as reported by Narsimhan and Goel (J. Colloid Interface Sci. 238 (2001) 420–432) at different homogenizer pressures and SDS concentrations. 2003 Elsevier Inc. All rights reserved. Keywords: Drop coalescence; Coalescence time distribution; Turbulent flow field; Interdroplet forces; Locally isotropic turbulent flow field; First passage time analysis; Doublet formation; Emulsion formation; High-pressure homogenizer
1. Introduction Emulsions are formed by applying mechanical energy to mixtures of oil and water so that one liquid (oil or water) disperses into the other in the form of fine droplets. The mechanical energy is applied in a variety of emulsion forming equipments such as homogenizers, colloid mills, and mixers. In all this equipment, the oil-and-water mixture is subjected to intense turbulent and shear-flow fields. Turbulence is the predominant mechanism for emulsification in such equipments [1] even though laminar shear and cavitation may also play a role in emulsification. Turbulence leads to break-up of the dispersed phase into small droplets. The relative motion between the drops results in their collision, leading to their coalescence. There usually is a dynamic equilibrium between breakage and coalescence. Consequently, the drop size distribution of the emulsion will be influenced by the E-mail address: [email protected]. 0021-9797/$ – see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2003.11.057
rates of breakage and coalescence. Surfactants and macromolecules are added to help produce small drops. Surfactants decrease the interfacial tension between the oil and water phases through adsorption at the interface. This results in less energy for drop breakage, thus leading to smaller droplets. In addition, surfactant and macromolecules inhibit the coalescence of colliding drops through repulsive electrostatic and steric interaction forces. These repulsive interactions minimize coalescence by slowing down the drainage of the intervening continuous film during a drop-pair encounter. The magnitude of the interaction forces depends on the amount and the nature of the surfactant adsorbed at the oil–water interface, the pH, the ionic strength, and the temperature. Also, the rupture of the intervening continuousphase film, necessary for drop coalescence, is made more difficult when the adsorbed interfacial layer possesses sufficiently high interfacial mechanical properties. The drop-size distribution of the emulsion can be controlled by controlling the rates of drop breakage and coalescence during emulsion
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formation. This requires knowledge of the effect of operating parameters and formulation of the emulsion on drop breakage and coalescence in typical emulsion-forming equipment. A number of models have been reported in the literature for the coalescence efficiency of drops in stirred vessels. These models are based on different ways of visualizing coalescence phenomena. Howarth [2] assumed coalescence to occur immediately if the velocity of approach of two colliding drops along their center line exceeds a critical value. His expression for coalescence efficiency was independent of fluid properties, as the model neglected drainage and rupture of continuous-phase film intervening between a colliding drop pair. Coulaloglou and Tavlarides [3] viewed the influence of the turbulent environment on the two colliding drops as equivalent to the thinning of the intervening film under the application of a constant squeezing force for a certain contact time. They assumed coalescence to occur if the contact time was greater than the time for the draining film to reach a critical thickness for film rupture. Tsouris and Tavlarides [4] accounted for the partial mobility of the drop interface in the evaluation of drop coalescence efficiency in a turbulent flow field. Muralidhar and Ramkrishna [5], Das et al. [6], and Muralidhar et al. [7] visualized a drop pair as being continuously impinged upon by small eddies. They assumed drops to approach each other with a mean force due to asymmetric distribution of pressure fluctuation around the drop and a fluctuating component of the force superimposed on it, arising out of the random nature of these fluctuations. If the time scale of the force fluctuation was much smaller than the time scale of film thinning, the force fluctuation was represented as white noise (uncorrelated). If the two time scales were comparable, the force fluctuation was viewed as correlated. A drop-pair encounter either resulted in coalescence (the thickness of the intervening film reached a critical thickness) or separation (film thickness larger than an arbitrary value). Average probability of coalescence was reported as coalescence efficiency. Das et al. [6] and Muralidhar et al. [7] assumed the drops to be rigid spheres, whereas Muralidhar and Ramkrishna [5] considered the drops to be deformable. These models do not account for the interaction forces (van der Waals, electrostatic, and steric) between the drops due to the use of surfactants in such systems. Tobin and Ramkrishna [8] proposed a model for drop coalescence efficiency in a turbulent flow field based on stochastic analysis of thin film drainage between two colliding drops. They accounted for interdroplet electrostatic forces and considered both rigid and deformable drops. Such an approach is suitable for larger droplets and smaller turbulent intensities when the time scale of drop collision is much larger than the timescale of coalescence. In this paper, we propose a mechanistic model for drop coalescence in a turbulent flow field that accounts for the interdroplet turbulent and colloidal forces. The proposed model is applicable to cases (smaller droplets and high turbulence intensity) where the timescales of collision and coalescence are comparable. Drop coalescence is viewed as
a two-step process, namely, formation of a doublet due to drop collision followed by coalescence of the doublet. Coalescence of a doublet due to random force fluctuations is modeled as a stochastic process and the first two moments of coalescence time distribution are evaluated. The formulation of the model is presented in the next section. Subsequent sections present model simulations and comparisons of the model with drop coalescence rates inferred from the experiments.
2. Model of drop coalescence in a turbulent flow field In a turbulent dispersion, drops are randomly moving about and continually colliding with each other. A colliding drop pair is subjected to interdroplet turbulent and colloidal squeezing forces due to which the intervening continuousphase liquid drains, leading to coalescence of the pair. For relatively large drop sizes (10–100 µm) and relatively lowintensity turbulent flow fields, the timescale of drop collisions (the inverse of the rate of collisions) is much larger than the timescale of coalescence of the drop pair. Consequently, the rate of coalescence can be expressed as rate of coalescence = rate of collisions × coalescence efficiency. Tobin and Ramkrishna [8] have described a model for the evaluation of coalescence efficiency in terms of interdroplet forces and turbulence. However, for sufficiently small drop sizes (0.1 to a few µm) and high-intensity turbulent flow fields, the timescales of collision and coalescence are comparable. Therefore, the rate of coalescence cannot be expressed by the above expression. The model presented below refers to the latter case. Different steps involved in drop coalescence for this case are shown schematically below. In view of the fact that distinct droplets are separated by the continuous phase it is necessary to clarify what is meant by “collision” between droplets. We propose to view collision between two droplets as their approach to one another within some specified proximity. This proximity is assumed to be h0 , the distance beyond which there is negligible colloidal force between droplets. The coalescence process has been modeled as a two-step process. In the first step, the formation of a doublet, which can either coalesce or separate into the original single droplets under the influence of fluctuating turbulent force, occurs. This is followed by the second step, the drainage of the thin liquid film between the liquid drops to a critical thickness at which film rupture occurs, followed by almost instantaneous formation of a larger drop. In this step, the colloidal forces come into the picture. Hence the drop pair experiences both turbulent and colloidal forces under the influence of which the film thickness or the distance of separation reduces from h0 to hcrit . The time it takes for the distance of separation to reduce from h0 to hcrit
G. Narsimhan / Journal of Colloid and Interface Science 272 (2004) 197–209
is called the first passage time or the coalescence time. The coalescence process is shown schematically as follows:
199
its identity upon collision. n2 refers to the bulk number concentration of droplets of radius R2 . Integrating Eq. (5) with the given boundary conditions (7) and (8), we get 7αε1/3 n2 (R1 + R2 )7/3 . (9) 3 The flux of particles across the sphere of influence of radius r = R1 + R2 is equal to [9] dn j = Dturb (10) . dr r=R1 +R2 C1 =
The evolution of number concentration is given by dn1 = −k1 n21 + kd nd , (1) dt dnd = k1 n21 − (kd + k2 )nd , (2) dt dnc = k2 nd , (3) dt where n1 , nd , and nc are the number concentration per unit volume of the monomer, the doublet, and the coalesced droplet, respectively, k1 is the rate constant for the formation of the doublet, kd is the rate of dissociation of the doublet, and k2 is the rate of coalescence of the doublet. These have to be solved with the initial condition
The rate of collisions νc between drops of radius R1 and R2 is then given by [9,10] νc = 4πr 2 j n1 ,
(11)
where n1 is the number concentration of drops of radius R1 . Therefore from Eqs. (9) and (10), the rate of collisions νc is given by
(4)
28π 1/3 (12) αε (R1 + R2 )7/3n1 n2 , 3 √ where α = 2 is a constant. For equal-sized drops of radius R, Eq. (12) reduces to
Consider the collision of drops due to turbulent diffusion onto a reference drop of radius R1 . The rate of collision of drops of radius R2 can be obtained from the evaluation of turbulent diffusive flux of drops onto a sphere of influence with radius (R1 + R2 ). This flux can be calculated from the solution of the diffusion equation for the quasi-steady state, as given by
28π 1/3 (13) αε (2R)7/3 n20 , 3 where n0 is the number concentration of drops. When the drop size d < λ, the relative motion of drops is influenced by eddies that lie in the viscous subrange of the Kolmogorov universal energy spectrum. In this case, the rate of collisions will depend on the viscosity, since these eddies dissipate energy. The turbulent diffusivity in this case is given by [9,10]
t = 0,
n1 = n0 ,
nd = 0,
nc = 0.
2.1. Collision of two drops
∂n = constant = C1 , (5) ∂r where Dturb is the turbulent diffusivity, n is the number concentration of drops of radius R2 , and r is the radial distance from the center of the reference drop of radius R1 . We consider drop coalescence in a turbulent flow field of energy dissipation rates sufficiently high so that the drop sizes lie within the range of local isotropy. For the inertial subrange, the turbulent diffusion coefficient of the second drop Dturb at a distance r from the reference drop is given by [9] Dturb r 2
Dturb ∝ ε
1/3 4/3
r
.
(6)
The diffusion equation (5) can be solved with the boundary conditions n = 0, n = n2 ,
r = R1 + R2 , r →∞
(7) (8)
to obtain the number concentration of the droplets of radius R2 in the vicinity of the reference drop. The boundary condition (7) is the absorbing boundary condition at the sphere of influence, which implies that the droplet of radius R2 loses
νc =
νc =
Dturb = α(rε)1/3 r ε 2 r =β ν
for r λ,
(14a)
for r < λ.
(14b)
The diffusion equation can again be solved with the boundary conditions (7) and (8) and νc can then be evaluated to give νc =
4πn1 n2
1 1/2 −1/2 1 ε 3ν (R1 +R2 )3
−
1 λ3
+
3 −1/3 −7/3 λ 7α ε
,
(15)
√ where α = 2 and β = 1 are constants. For equal-sized drops of radius R, Eq. (15) reduces to νc =
1 1/2 −1/2 ε 3ν
4πn20 1 − λ13 + (2R)3
3 −1/3 −7/3 λ 7α ε
.
(16)
The time scale of drop collision, τcoll , for equal-sized drops can be defined as τcoll =
n0 , νc
(17)
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where νc is given by Eq. (13) or (16) and n0 , the number of droplets per unit volume, is given by 3φ , (18) 4πR 3 where φ is the dispersed phase fraction and R is the mean droplet radius. The rate constant k1 for the formation of doublets can be taken as the rate constant for the rate of collisions as given by νc k1 = 2 , (19) n0 n0 =
dissociation of the doublet when the initial force fluctuation is F , we get [11]
∞ T (F ) =
G(F , t) dt,
(24)
0
where G(F, t) = Prob T (F ) t =
∞
Using Eqs. (24) and (25), Eq. (23) can be transformed to ∂T ∗ 1 ∂ 2 T ∗ + = −1 ∂F ∗ 2 ∂F ∗ 2 with the boundary conditions
2.2. The rate of dissociation of doublet
F ∗ = 1,
since the colloidal interaction force at the time of doublet formation is negligible. In the above equation, F¯ is the mean turbulent force, δ is the standard deviation, Tf is the timescale of force fluctuation, and ζ (t) is white noise. A negative sign is introduced on the right-hand side of the equation, consistent with the sign convention that the attractive force is negative. It is to be noted that the average turbulent squeezing force is attractive. In order for the doublet to separate, the fluctuating force should overcome the mean force F¯ . The fluctuating force acting on the doublet is modeled as a Poisson process, i.e., a force of magnitude δ (equal to F¯ ), acts on the doublet at random times with a decay timescale Tf (see Eq. (55)). Consequently, the variation of the fluctuating force acting on the doublet is given by the stochastic differential equation [11] ¯ F (t) F −1/2 dt + F¯ Tf − dη(t), dF (t) = (21) Tf Tf
dη(t) = 0, dη(t)2 = dt.
(1 − F ∗ )
(22b)
The evolution of force fluctuation experienced by the doublet is given by the Fokker–Planck equation [12] ¯ F ∂p(F , t|F, 0) F ∂p(F , t|F, 0) = − ∂t Tf Tf ∂F 2 2
¯ 1 F ∂ p(F , t|F, 0) + (23) , 2 Tf ∂F 2 where p(F , t|F, 0) is the conditional probability that the force fluctuation at time t is F given that the force fluctuation at time zero is F . Defining T (F ) as the average time of
T ∗ (1) = 0, ∂T ∗ = 0, ∂F ∗
F ∗ = 0,
(26)
(27a) (27b)
where T ∗ = T /Tf and F ∗ = F /F¯ . The first absorbing boundary condition denotes the dissociation of the doublet whenever the fluctuating force equals the mean turbulent force. The second reflecting boundary condition states that the doublet experiences a fluctuating force between 0 and F¯ . The mean dissociation time of the doublet is T ∗ (0) which is given by [11]
0
∗
T (0) = 2
dy ψ(y)
0 ψ(z) dz,
(28)
y
1
where x ψ(x) = exp
2(1 − x) dx .
(29)
1
Solution of Eq. (28) yields T ∗ (0) = 0.37. Therefore, the rate of dissociation of a doublet kd is given by kd =
(22a)
(25)
0
where νc , the rate of collisions per unit volume, is given by Eq. (13) and Eq. (16) for inertial and viscous subranges, respectively.
Once a doublet is formed, it is subjected to random turbulent force fluctuation. The net turbulent force acting on the doublet at the time of collision is given by [5] 1/2 F = − F¯ − δTf ζ (t) (20)
p(F , t|F, 0) dF .
1 . 0.37Tf
(30)
2.3. Evaluation of the mean coalescence time Once a doublet is formed, the surface-to-surface distance between the two individual droplets within a doublet should be less than hcrit for coalescence to occur. Since the doublet experiences a fluctuating turbulent force, the surface to surface distance h(t) varies randomly with time and is, therefore, a stochastic process. The turbulent mean and fluctuating forces should overcome the colloidal repulsive force for coalescence to occur. The time of coalescence, i.e., the time at which h(t) = hcrit , is random. The evaluation of moments of coalescence time distribution is outlined below.
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2.4. Drainage of the continuous phase film For nondeformable spherical particles, the drainage of continuous-phase liquid between two colliding particles of sizes d and d is given by Taylor’s equation [12], dh 2hF 1 1 2 (31) = + , dt 3πµ d d where h is the surface-to-surface distance between the drops and F is the interaction force between the two emulsion droplets. By convention, the interaction force F is positive if repulsive and negative if attractive. For drops of equal size d, Eq. (31) becomes 8hF dh = . (32) dt 3πµd 2 In a turbulent flow field, the droplet pair is subjected to a random fluctuating turbulent force with mean force F¯ , which will try to squeeze the colliding drop pair toward each other, thus promoting coalescence. The van der Waals attractive force between the two drops would also promote coalescence. On the other hand, the electrostatic repulsive force between the two drops would tend to slow down the film drainage. The expressions for the turbulent and colloidal forces are given in the following sections. 2.5. Turbulent force The mean turbulent force, F¯ , is given by πd 2 ρ 2 u (d), F¯ = 4
(33)
where u2 (d) is the mean squared turbulent velocity fluctuation between the centers of the colliding droplet pair separated by a distance d. For local isotropy, when d λ (inertial subrange), the mean squared velocity fluctuation is given by [9] u2 (d) = 2ε2/3d 2/3 , where ε is the energy dissipation rate per unit mass.
(34)
201
The mean turbulent force F¯ is therefore given by π F¯ = ρε2/3 d 8/3. (35) 2 For local isotropy, when d λ (viscous subrange), the mean squared velocity fluctuation is given by [9] u2 (d) =
εd 2 ρ . µ
(36)
The mean turbulent force F¯ is therefore π ρ 2d 4ε F¯ = . 4 µ
(37)
2.6. Colloidal force A drop pair are deemed to have collided if the interaction force between the two drops is above a certain threshold. In emulsions, surfactants are usually employed to stabilize the emulsion. When the surfactant is ionic, adsorption of surfactants at the emulsion drop interface will impart a charge to the droplet, thus creating an electrical double layer in its vicinity. The colliding droplet pair would experience an electrostatic repulsive force because of the overlap of double layers. In addition, the emulsion droplets also experience a van der Waals attractive force. Typical variation of interdroplet colloidal force with the surface-to-surface distance of separation between the drops is shown in Fig. 1. Van der Waals and electrostatic interactions are included in the evaluation of the colloidal force. Expressions for van der Waals and electrostatic force are given later in the following sections. As expected, at large distances of separation, the interparticle force is negligible. As the drops approach each other, they experience a repulsive (positive) force because of the predominant effect of double-layer repulsion. At very small h, however, the net force is attractive (negative) because of the predominance of van der Waals attraction. Consequently, there exists a critical distance of separation hcrit at which the interparticle force changes from repulsive to attractive (Fig. 1). Once the distance of separation becomes
Fig. 1. Typical variation of colloidal force with the surface-to-surface distance of separation between two drops of diameter 209 nm with ψ0 = 50 mV and I = 0.1 M.
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less than hcrit , the drop pair experiences a strong van der Waals attractive force leading to coalescence. Once this happens, drop coalescence can be considered to be almost instantaneous, due to the strong van der Waals attraction. 2.7. Model equations The drainage of the continuous-phase liquid between two nondeformable drops of equal size is given by Taylor’s equations [12], 8hF dh = . (38) dt 3πµd 2 The net force of interaction experienced by the droplet pair is the sum of the turbulent and colloidal forces. Because of the random nature of the turbulent force, the surface-tosurface distance h(t) can be considered to be a stochastic process. The net interaction force F is given by [6] 1/2 F = − F¯ − Fc − δTf ζ (t) , (39) where F¯ is the mean turbulent force given by Eqs. (35) and (38) and Fc is the colloidal interaction between the droplets due to van der Waals and electrostatic forces, Fc = FVW + FDL ,
(40)
where FVW and FDL refer to the van der Waals and doublelayer interactions, respectively. The last term in Eq. (39) refers to the turbulent fluctuating force, which is explained later in this section. It is to be noted that the hydrodynamic interaction between the two colliding drops is neglected in this analysis. Such an assumption is indeed reasonable, as shown in Appendix A. 2.7.1. Van der Waals attraction The van der Waals interaction is given by [13]
1 1 (1 + λr )2 θ (1 − φ) + + FVW = − (R 2 − θ )2 (R 2 − θ φ)2 2λr (R1 + R2 )2 AH R1 R2 × 2 (41) , (R − θ )(R 2 − θ φ) where R1 , R2 are the droplet radii (R1 = R2 for equal sized drops) and AH , the effective Hamaker constant, is given by the expression 1/2
1/2 − A o )2 , AH = (Aw
(42)
Aw and Ao being the Hamaker constants of the aqueous and oil phases, respectively. φ, θ , and λr are given by 1−λr 2 , φ= (43) 1 + λr θ = (R1 + R2 )2 ,
(44)
λr = R1 /R2 .
(45)
R is the center-to-center distance of separation between the two drops, or R = R1 + R2 + h, where h is the surface-tosurface distance of separation.
For surface-to-surface distances of separation h R1 + R2 , Eq. (41) reduces to FVW = −
AH R1 R2 . 6h2 (R1 + R2 )
(41a)
In writing Eq. (41), the retardation effect is neglected. This is justified since the average droplet size is on the order of a few hundred nanometers. 2.7.2. Electrostatic double-layer repulsion The electrostatic double-layer repulsion is calculated based on a numerical algorithm [14], which calculates the distance of separation between two charged plates at given midpoint potentials by solving the differential equation
2 −1/2 2 Q dX = + cosh Ym − 1 (46) , dQ 2 where X = κx is the scaled distance measured from the midplane. κ, the Debye–Hückel parameter, is given by κ2 =
8πne2 , εε0 kT
(47)
where k is Boltzmann’s constant, T is the absolute temperature, n is the number concentration of the electrolyte, e is the elementary charge, ε is the dielectric constant of the medium, and ε0 is the permittivity constant. We define Q by 1/2 , Q = 2(cosh Y − cosh Ym ) (48) where Y = zeψ/kT is the scaled potential at a distance X from the midplane between the two charged plates in a z : z electrolyte solution of number concentration n, z is the valence number of ions, ψ is the electrostatic potential at a distance x from the midplane, and Ym is the scaled midpoint potential. The midplane in terms of these variables is the point (Q = 0, X = 0). Knowing the value of Q (= Q0 ) corresponding to the surface potential ψ0 , we can solve Eq. (46) from Q = 0 to Q = Q0 by numerical integration. Thus we determine X0 = kh/2, the scaled distance from the midplane to the surface. The electrostatic force of interaction FFP per unit area between two plates separated by a distance h is then given by FFP = 2kT n0 [cosh Ym − 1].
(49)
Using the Derjaguin approximation, the interaction force FDL (h) between two droplets of radius R separated by a surface-to-surface distance h can be obtained by integration to give
∞ FDL (h) = πR
FFP (x) dx. h
(50)
G. Narsimhan / Journal of Colloid and Interface Science 272 (2004) 197–209
2.8. Turbulent fluctuation
and
The last term in Eq. (39) refers to the fluctuating part of the turbulent force, which is modeled as a band-limited white noise ζ (t) with ζ (t) = 0, (51)
2 |t − t | , ζ(t)ζ(t ) = (52) exp − Tf Tf where . denotes ensemble average. At any time, ζ (t) has a Gaussian distribution with mean zero and standard deviation δ, which is taken to be equal to F¯ [5]. Tf , the time scale of force fluctuation, can be defined as [5] Tf =
d [u2 (d)]1/2
,
(53)
where d is the drop size and u2 (d) is the mean squared velocity fluctuation at a distance of separation d. For d λ, from Eq. (34), we get Tf = 2−1/2 ε−1/3 d 2/3 .
(54)
For d λ, from Eq. (37), we get 1/2 ν µ . Tf = = ερ ε
(55)
It is assumed that the random squeezing force, which determined the coalescence probability of the doublet, is spatially uniform and temporally a white noise process. From Eqs. (38) and (39), 1/2 2h{F¯ − Fcoll − δTf ζ (t)} dh =− , dt 3πµR 2
dh = −
2h{F¯ − Fcoll } dt + 3πµR 2
1/2 2δhTf 3πµR 2
(56) dW (t),
(57)
where dW (t) is a Wiener process. Equation (57) is the stochastic differential equation describing the evolution of h(t). This equation is written in the Stratanovich sense [11], dh = f2 (h) + g2 (h) dW (t),
(58)
where f2 (h) = −
2(F¯ − Fcoll )h 3πµR 2
(59)
and 1/2
g2 (h) =
2δhτf
3πµR 2
203
.
(60)
Converting the above equation to its equivalent in Ito form [11] gives dh = f1 (h) + g1 (h)ξ(t),
(61)
g1 (h) = g2 (h). This gives f1 (h) = −
2δ 2 τf h 2(F¯ − Fcoll )h + 3πµR 2 9π 2 µ2 R 4 1/2
g1 (h) =
2δhτf
3πµR 2
.
(65)
Since h(t) is a stochastic process, the evolution of surface-to-surface distance between the two droplets of a doublet after its formation (i.e., h(0) = h0 ) is random and is usually referred to as a sample path of the process. For an ensemble of a large number of doublets subjected to colloidal and turbulent forces, the evolution of their surface-to-surface distance of separation gives an ensemble of sample paths. From this ensemble one can then define the probability that a doublet will have a surface-to-surface distance of x (say) at time t. One can then define p(x , t|x, 0) as the conditional probability that h = x at time t with the initial condition x. The existence of the doublet implies that hcrit < h < h0 . When h = hcrit , the coalescence of the doublet occurs since colloidal force becomes attractive for h < hcrit . Therefore, p(hcrit , t|x, 0) = 0.
(66)
Since the colliding drop pair is subjected to an attractive turbulent force and the dissociation of the doublet at the time of collision is accounted for through kd , we impose a reflecting boundary at h = h0 ; i.e., the flux at h0 is zero: dp(h0 , t|x, 0) = 0. (67) dx As pointed out earlier, we are interested in the average coalescence time. Consequently, the final point in the sample path is hcrit . Therefore, we are interested in the evolution of this conditional probability backward in time. This is given by the backward Fokker–Planck equation [11], ∂p(x , t|x, 0) ∂p(x , t|x, 0) = f1 (x) ∂t ∂x ∂ 2 p(x , t|x, 0) 1 . + g12 (x) 2 ∂x 2 Defining
G(x, t) = dx p(x , t|x, 0) = Prob(T t),
(68)
(69)
we have from Eq. (68) ∂G(x, t) ∂G(x, t) 1 2 ∂2 = f1 (x) + g1 (x) 2 G(x, t). ∂t ∂x 2 ∂x Defining the mean first passage time,
∞
(62)
(64)
and
where 1 f1 (h) = f2 (h) + g2 (h)g2 (h) 2
(63)
T (x) = T = − 0
∂G(x, t) dt = t ∂t
(70)
∞ G(x, t) dt, 0
(71)
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where T (x) is the average coalescence time of the doublet when the initial separation of drops is x. We can derive a simple ordinary differential equation for T (x) by using Eq. (71) and integrating Eq. (70) over (0, ∞). Noting that
∞
∞ Tn (x) = −
tn
∂G(x, t) dt ∂t
(82)
0
∂G(x, t) dt = G(x, ∞) − G(x, 0) = −1 ∂t
(72)
0
dT (x) 1 2 d 2 T (x) + g1 (x) = −1 dx 2 dx 2 with the boundary conditions
f1 (x)
(73)
dT = 0, dx T = 0.
x = h0 , x = hcrit ,
(74a)
∂Tn (x) 1 2 ∂ 2 Tn (x) + g1 (x) = −nTn−1 (x), (83) ∂x 2 ∂x 2 which can be solved recursively to obtain the moments of the coalescence time distribution. In terms of dimensionless quantities, we have
x T (x) = 2
dy ψ(y)
where
h0
ψ(z) dz, g12 (z)
y
hcrit
x
ψ(x) = exp
dx
(75)
2f12 (x )/g1 (x )
.
(76)
hcrit
Equation (75) at x = h0 becomes
h0
x
exp −
T (h0 ) = 2 hcrit
h0 × y
hcrit
1 exp g12
2f1 dx g12
x
hcrit
2f1
dx dx dy. g12
T (x) , Tf
x∗ =
(x − hcrit ) , (h0 − hcrit )
(77)
(78) 1/2
g1 (x)Tf f1 (x)Tf , g1∗ (x ∗ ) = , (h0 − hcrit ) (h0 − hcrit ) x ∗ 2f1∗ (x )
∗ ∗ ψ (x ) = exp dx , g1∗ 2 (x ) f1∗ (x ∗ ) =
(79)
(80)
0
we have
x ∗
T (x ) = 2 0
dy ψ ∗ (y)
1 y
ψ ∗ (z) g1∗ 2 (z)
dz.
x ∗
dy ψ ∗ (y)
0
1 y
∗ (z)ψ ∗ (z) nTn−1
g1∗ 2 (z)
dz,
(84)
where Tn∗ (x ∗ ) =
Tn (x) . Tfn
(85)
The rate of formation of the coalesced droplet, k2 (x), in Eq. (3) is given by k2 =
1 . T (h0 )
(86)
The first and the second moments of the distribution of coalescence time were evaluated from Eqs. (81) and (84), respectively. Even though it is possible to evaluate higher moments, these calculations were found to be computationally intensive and therefore were not carried out. The moments were then employed to fit the beta distribution, 1 x α−1 f (x) = (87) x exp − . Γ (α)β α β
3. Simulation results
In terms of dimensionless quantities, T ∗ (x ∗ ) =
Tn∗ (x ∗ ) = 2
(74b)
Note that (74a) and (74b) are the corresponding reflecting and absorbing boundary conditions for the mean coalescence time. The solution for Eq. (73) with boundary conditions (74) is given by [10]
∗
from Eq. (70), we get f1 (x)
we get
∗
Defining the nth moment of the coalescence time distribution Tn (x) as
(81)
Typical variation of the average dimensionless coalescence time T ∗ (x ∗ ) with the dimensionless initial position x ∗ is shown in Fig. 2. As expected, the average coalescence time is zero for the initial position x ∗ = 0 (x = hcrit ) since drop coalescence is deemed to occur immediately when the surface-to-surface distance between two colliding drops is hcrit . The average coalescence time increases dramatically as the surface-to-surface distance of separation increases (Fig. 2) because of the presence of a high force barrier (see Fig. 1) for intermediate distances of separation. At greater distances of separation, however, the average coalescence time tends to level off since most of the resistance to coalescence lies near the repulsive force barrier. As a result, the average coalescence time is insensitive to the actual distance of separation at the time of drop collision. This is important for the evaluation of coalescence rates since drop collision (or, equivalently, the initial distance of separation) is assumed
G. Narsimhan / Journal of Colloid and Interface Science 272 (2004) 197–209
205
Fig. 2. Variation of dimensionless coalescence time T ∗ with initial dimensionless surface-to-surface distance x ∗ .
Fig. 4. Effect of energy dissipation rate on drop coalescence time distribution for equal-sized droplets of R = 2.5×10−7 m, I = 0.05, ψ0 = 58.8 mV. The three distributions are for three different energy dissipation rates in W/kg as indicated.
Fig. 3. Effect of energy dissipation rate on drop coalescence time and Ftur /Fcoll for equal-sized droplets of R = 2.5 × 10−7 m, I = 0.05, ψ0 = 58.8 mV. F, Average drop coalescence time; 2, Ftur /Fcoll .
to occur when the interdroplet force is much smaller than the barrier force (1% of barrier force in these calculations). Consequently, any small variation in the initial distance of separation at the time of drop collision will not significantly affect the calculated average drop coalescence time. In the subsequent calculations, therefore, T ∗ (1) (which, in dimensional average coalescence time, is T (h0 )) is reported as the average dimensionless drop coalescence time. The variation of average coalescence time with energy dissipation rate is shown in Fig. 3. The average coalescence time decreases exponentially from about 4 × 108 to 2.96 × 10−6 s as the energy dissipation rate increases from 106 to 6 ×106 W/kg. The ratio of the average turbulent force to the barrier force versus the energy dissipation rate is also
shown in Fig. 3. It is interesting to note that the coalescence time decreases dramatically as this ratio increases. The coalescence time distribution as given by Eq. (88) was evaluated and is shown for different energy dissipation rates in Fig. 4. The coalescence time distribution becomes broader with a larger standard deviation at lower energy dissipation rates. Average coalescence time versus surface potential is shown in Fig. 5. The coalescence time is found to increase dramatically with the surface potential. For example, the average coalescence time increases from ∼10−5 to ∼103 s as the surface potential increases from 35 to 55 mV. The increase in the force barrier (or the corresponding decrease in FT /Fcoll , where FT and Fcoll refer to the average turbulent force and force barrier, respectively, since the average turbulent force remains constant) with surface potential can be seen from Fig. 5. The dimensionless coalescence time distribution becomes broader at higher surface potentials (Fig. 6). The variation of average coalescence time with drop size for coalescence of equal-sized droplets is shown in Fig. 7.
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Fig. 5. Effect of surface potential on drop coalescence time and Ftur /Fcoll for equal-sized droplets of R = 2.5 × 10−7 m, I = 0.05, ε = 106 W/kg. F, Average drop coalescence time; 2, Ftur /Fcoll . Fig. 8. Effect of drop size on drop coalescence time distribution for equal-sized droplets of I = 0.05, ε = 4.8 × 107 W/kg, ψ0 = 58.8 mV. F, Average drop coalescence time; 2, Ftur /Fcoll .
Fig. 6. Effect of surface potential on drop coalescence time distribution for equal-sized droplets of R = 2.5 × 10−7 m, I = 0.05, ε = 107 W/kg.
Fig. 9. Effect of size ratio on average drop coalescence time and Ftur /Fcoll for R = 1.5 × 10−7 m, I = 0.05, ε = 4.8 × 107 W/kg, ψ0 = 58.8 mV. F, Average drop coalescence time; 2, Ftur /Fcoll .
Fig. 7. Effect of drop size on drop coalescence time and Ftur /Fcoll for equal-sized droplets of I = 0.05, ε = 5 × 106 W/kg, ψ0 = 58.8 mV. F, Average drop coalescence time; 2, Ftur /Fcoll .
The coalescence time is found to decrease from 6.3 × 109 to 1.9 × 10−6 s as the drop radius increases from 1.5 × 10−7 to 3 × 10−7 m. Fig. 7 also shows the variation of FT /Fcoll versus drop size. Even though the force barrier increases with drop size, the turbulent force increases much more rapidly than the barrier force, thus resulting in a smaller average coalescence time. The coalescence time distributions for two different drop sizes, as shown in Fig. 8, indicate narrowing of the distribution for larger drop size. The results for unequal drop sizes (Figs. 9 and 10) indicate a decrease in
Fig. 10. Effect of size ratio on drop coalescence time distribution for R = 1.5 × 10−7 m, I = 0.05, ε = 4.8 × 107 W/kg, ψ0 = 58.8 mV.
average drop coalescence time with a narrower coalescence time distribution for larger size ratios because of the increase in turbulent squeezing force compared to the interdroplet colloidal force. The variation of average coalescence time with ionic strength for constant surface potential is shown in Fig. 11.
G. Narsimhan / Journal of Colloid and Interface Science 272 (2004) 197–209
Fig. 11. Effect of ionic strength on average drop coalescence time for R = 2.5 × 10−7 m, ε = 4.8 × 107 W/kg, ψ0 = 58.8 mV.
The coalescence time decreases from 1.66 × 10−7 to 1.95 × 10−8 s as the ionic strength increases from 0.01 to 0.5 M. Such behavior is understandable since an increase in the ionic strength compresses the electrical double layer, thus allowing the colliding drop pair to come nearer before they experience repulsion. Since the model imposes a reflecting boundary condition at h = h0 in the evaluation of the first passage time, it does not allow the dissociation of the doublet during the random drainage process. Such an assumption is reasonable whenever the average turbulent squeezing force is sufficiently high. Since the surface-to-surface separation of colliding droplets is only about 10 times larger than the microscale of turbulence, local isotropy may not accurately reflect the turbulent flow field.
4. Comparison of model predictions with experimental data Average drop coalescence rates and average coalescence rate constants were inferred from the experimental measurement of the evolution of drop number concentrations of tetradecane-in-water emulsions stabilized by sodium dodecyl sulfate for a negative stepchange in homogenizer pressure [15]. The experimental data and the methodology for the inference of average drop-coalescence rate constants are given elsewhere [15]. Here we compare the experimental average drop coalescence rate constants with model predictions. In order to account for the drop size distribution, the drop sizes were divided into six size intervals and the distribution was represented by a discretized beta distribution. Equations similar to Eqs. (1)–(3) were written for the coalescence of drops of sizes i and j . The effect of hydrodynamic interaction is accounted for in the evaluation of drop collision rates, as described in Appendix A. The drop collision rate constant ki,j for each drop size pair was evaluated using Eqs. (19), (A.1), and (A.2). It is to be noted that the effect of hydrodynamic interaction on drop coalescence time is not significant, as explained in Appendix A. The rate constants kd and k2 were evaluated from Eqs. (30) and (86), respectively, for each drop size pair. The average drop coalescence
207
Fig. 12. Comparison of predicted and experimental drop coalescence rate constants in a high-pressure homogenizer at different homogenizer pressures. The model predictions are made for R¯ = 1.951 × 10−7 m, σ = 3.03 × 10−8 m, I = 0.05, ε = 7.526 × 107 W/kg, ψ0 = 63.9 mV. F, Experimental; 2, model prediction.
rate constant kav was then calculated using kav =
nc,ij i
j
θ n20
,
(88)
where nc,ij is the number concentration of coalesced droplets due to collision of droplets of sizes i and j , which was evaluated from the solution of Eqs. (1)–(3) subject to the initial condition (4). Calculation of the energy dissipation rate ε and the average residence time θ through the homogenizer valve is explained elsewhere [15]. The experimental conditions and the parameters (interfacial tension, zeta potential, mean drop size, and standard deviation of drop size distribution) are given in Narsimhan and Goel [15]. A comparison of model predictions with inferred average coalescence-rate constants from experiments at different homogenizer pressures is shown in Fig. 12. As expected, the coalescence-rate constant increases with homogenizer pressure, though the dependence is less pronounced for the model predictions. The model predictions agree fairly well with the values inferred from the data considering that only the coalescence-rate constants averaged over drop sizes are compared. Comparison of size-specific coalescence-rate constants could not be made since experimental size-specific coalescence-rate constants were not obtained. A comparison of model predictions with inferred average coalescencerate constants from experiments at different SDS concentrations is shown in Fig. 13. The model predictions follow the same trend as the experiments and agree fairly well with the inferred values. Of course, the predicted average dropcoalescence-rate constants were found to be higher than the experimental values, especially at higher SDS concentrations. This is believed to be due to the simplifying assumption of a reflecting boundary condition at h0 for the evaluation of the coalescence time of a doublet. In other words, the model allows dissociation of the doublet only at the time of its formation (in the absence of colloidal forces) and assumes that once the doublet is formed, it will eventually coalesce into a single droplet with the coalescence time depending on the relative magnitudes of turbulent and colloidal forces.
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G. Narsimhan / Journal of Colloid and Interface Science 272 (2004) 197–209
model is believed to be due to the simplifying assumption of a reflecting boundary condition at h = h0 for the evaluation of coalescence time of a doublet.
Acknowledgment This research was supported in part by USDA National Research Initiative Grant #9602252.
Fig. 13. Comparison of predicted and experimental drop coalescence rate constants in a high-pressure homogenizer at different SDS concentrations. The model predictions are made for R¯ = 2.32 × 10−7 m, σ = 3.203 × 10−8 m, I = 0.1, ε = 4.82 × 107 W/kg. F, Experimental; 2, model prediction.
5. Conclusions A model for collision and coalescence of a drop pair in a turbulent flow field that accounts for interdroplet colloidal as well as fluctuating turbulent forces is developed. Colloidal forces include van der Waals and double-layer electrostatic forces. The model views drop coalescence as a two-step process consisting of formation of a doublet due to drop collisions followed by coalescence of the individual droplets in a doublet as a result of drainage of the intervening film of continuous phase. The fluctuating turbulent force was modeled as band-limited white noise. The average as well as higher moments of the coalescence-time distribution were evaluated from the analysis of the random process of intervening continuous-phase film thickness and were found to be dependent on the force barrier to drop coalescence. Even though the average drop-coalescence time was dependent on the initial separation between the two colliding drop pairs, it was found to be fairly insensitive at sufficiently large separations. The average drop-coalescence time decreased dramatically with an increase in energy dissipation rate and the coalescence-time distribution became narrower at higher energy dissipation rates. The average coalescence time was longer and the coalescence time distribution broader at higher surface potentials as well as lower ionic strength because of higher force barriers to drop coalescence. Even though the turbulent as well as colloidal forces increased for larger drop sizes and size ratios of the unequal drop pair, the turbulent force increased faster than the colloidal force, thus resulting in a smaller average drop coalescence time and a narrower distribution. The model predictions of average drop-coalescence-rate constants that account for the drop-size distributions compared fairly well with the values inferred from the experimental evolution of drop number concentration to a negative step change in homogenizer pressure for tetradecane-inwater emulsions stabilized by SDS at different homogenizer pressures and SDS concentrations. The overprediction of the
Appendix A. Effect of hydrodynamic interaction on drop coalescence A.1. Collision rate Following the analysis of Honig et al. [16], Eq. (5) can be modified to account for van der Waals interactions and hydrodynamic interactions, giving the following expression h for the collision rate constant ki,j between particles of radii Ri and Rj , λ ∗ −1 h ki,j
= 4π 0
vw exp(Ui,j /kT )βi,j (y)
dy
(ε/ν)1/2 (Ri + Rj )4 (y + 1)4
∞
vw /kT )β (y) exp(Ui,j i,j
+
αε1/3 (Ri + Rj )10/3(y + 1)10/3
λ∗ −1
for (Ri + Rj ) < λ, λ∗
dy (A.1)
where = λ/(Ri + Rj ) and is the van der Waals interaction potential. The hydrodynamic interaction correction factor βi,j (y) is evaluated using the iterative equations given elsewhere [17]. For (Ri + Rj ) > λ, vw Ui,j
∞ h ki,j
= 4π 0
vw /kT )β (y) exp(Ui,j i,j
αε1/3 (Ri + Rj )10/3 (y + 1)10/3
dy.
(A.2)
A.2. Drop coalescence time The Taylor equation for the drainage of a continuous film between two colliding nondeformable drops of equal size can be written to account for the hydrodynamic resistance, 8h dh = (FT + Fcoll + Fhyd ), dt 3πµd 2
(A.3)
where Fhyd is the hydrodynamic resistance. Expressing the hydrodynamic resistance in terms of friction coefficient, we get 8h dh 8h dh = , (A.4) (FT + Fcoll ) − f dt dt 3πµd 2 3πµd 2 where f is the friction coefficient. Defining the dimension∗ =F less variables h∗ = κh, FT∗ = FT /κkT , Fcoll coll /κkT ,
G. Narsimhan / Journal of Colloid and Interface Science 272 (2004) 197–209
t ∗ = t/Tf and recognizing that the friction coefficient is given by f ∼ = 6πµR(0.645) [18], the above equation yields dh∗ = dt ∗
2κkT Tf ∗ ∗ ∗ ) h (FT + Fcoll 3πµR 2 1+
8(0.645) κd
.
(A.5)
For typical values of ionic strength and drop size, κd 1. For example, for an ionic strength of 0.01 and drop diameter of 100 nm, kd ∼ = 30, and for an ionic strength of 0.1 and drop diameter of 100 nm, kd ∼ = 100. Consequently, the denominator on the right hand side is close to unity. As a result, Eqs. (38) and (39) constitute a good approximation of the above equation. Therefore, hydrodynamic interaction is not important in the evaluation of drop coalescence time.
References [1] P. Walstra, in: P. Becher (Ed.), Encyclopedia of Emulsion Technology, vol. 1, Dekker, New York, 1983. [2] W.J. Howarth, Chem. Eng. Sci. 19 (1967) 33.
209
[3] C.A. Coulaloglou, L.L. Tavlarides, Chem. Eng. Sci. 32 (1977) 1289– 1297. [4] C. Tsouris, L.L. Tavlarides, AIChE J. 40 (1994) 395. [5] R. Muralidhar, D. Ramkrishna, Ind. Eng. Chem. Fundam. 25 (1986) 554. [6] P.K. Das, R. Kumar, D. Ramkrishna, Chem. Eng. Sci. 42 (1987) 213. [7] R. Muralidhar, D. Ramkrishna, P.K. Das, R. Kumar, Chem. Eng. Sci. 43 (1988) 1559. [8] T. Tobin, D. Ramkrishna, Can. J. Chem. Eng. 77 (1999) 1090. [9] V.G. Levich, Physicochemical Hydrodynamics, Prentice–Hall, Englewood Cliffs, NJ, 1962. [10] P. Walstra, Netherlands Milk Dairy J. 29 (1975) 279. [11] C.W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and Natural Sciences, Springer-Verlag, Berlin/New York, 1983. [12] I.B. Ivanov, D.S. Dimitrov, in: I.B. Ivanov (Ed.), Thin Liquid Films, Dekker, New York, 1983. [13] D.H. Melik, H.S. Fogler, J. Colloid Interface Sci. 101 (1984) 72. [14] D.Y.C. Chan, R.M. Pahsley, L.R. White, J. Colloid Interface Sci. 77 (1980) 283. [15] G. Narsimhan, P. Goel, J. Colloid Interface Sci. 238 (2001) 420–432. [16] E.P. Honig, G.J. Roebersen, P.H. Wiersema, J. Colloid Interface Sci. 36 (1971) 97. [17] L.A. Spielmen, J. Colloid Interface Sci. 33 (1970) 562. [18] J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics, Noordhoff, Leyden, 1973.
Model for drop coalescence in a locally isotropic turbulent flow field Ganesan Narsimhan Biochemical and Food Process Engineering, Department of Agricultural and Biological Engineering, Purdue University, West Lafayette, IN 47907, USA Received 12 June 2003; accepted 24 November 2003
Abstract The proposed model views drop coalescence in a turbulent flow field as a two-step process consisting of formation of a doublet due to drop collisions followed by coalescence of the individual droplets in a doublet due to the drainage of the intervening film of continuous phase under the action of colloidal (van der Waals and electrostatic) and random turbulent forces. The turbulent flow field was assumed to be locally isotropic. A first-passage-time analysis was employed for the random process of intervening continuous-phase film thickness between the two drops of a doublet in order to evaluate the first two moments of coalescence-time distribution of the doublet. The average drop coalescence time of the doublet was dependent on the barrier for coalescence due to the net repulsive force (net effect of colloidal repulsive and turbulent attractive forces). The predicted average drop coalescence time was found to be smaller for larger turbulent energy dissipation rates, smaller surface potentials, larger drop sizes, larger ionic strengths, and larger drop size ratios of unequal-sized drop pairs. The predicted average drop coalescence time was found to decrease whenever the ratio of average turbulent force to repulsive force barrier became larger. The calculated coalescence-time distribution was broader, with a higher standard deviation, at lower energy dissipation rates, higher surface potentials, smaller drop sizes, and smaller size ratios of unequal drop pairs. The model predictions of average coalescencerate constants for tetradecane-in-water emulsions stabilized by sodium dodecyl sulfate (SDS) in a high-pressure homogenizer agreed fairly well with the inferred experimental values as reported by Narsimhan and Goel (J. Colloid Interface Sci. 238 (2001) 420–432) at different homogenizer pressures and SDS concentrations. 2003 Elsevier Inc. All rights reserved. Keywords: Drop coalescence; Coalescence time distribution; Turbulent flow field; Interdroplet forces; Locally isotropic turbulent flow field; First passage time analysis; Doublet formation; Emulsion formation; High-pressure homogenizer
1. Introduction Emulsions are formed by applying mechanical energy to mixtures of oil and water so that one liquid (oil or water) disperses into the other in the form of fine droplets. The mechanical energy is applied in a variety of emulsion forming equipments such as homogenizers, colloid mills, and mixers. In all this equipment, the oil-and-water mixture is subjected to intense turbulent and shear-flow fields. Turbulence is the predominant mechanism for emulsification in such equipments [1] even though laminar shear and cavitation may also play a role in emulsification. Turbulence leads to break-up of the dispersed phase into small droplets. The relative motion between the drops results in their collision, leading to their coalescence. There usually is a dynamic equilibrium between breakage and coalescence. Consequently, the drop size distribution of the emulsion will be influenced by the E-mail address: [email protected]. 0021-9797/$ – see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2003.11.057
rates of breakage and coalescence. Surfactants and macromolecules are added to help produce small drops. Surfactants decrease the interfacial tension between the oil and water phases through adsorption at the interface. This results in less energy for drop breakage, thus leading to smaller droplets. In addition, surfactant and macromolecules inhibit the coalescence of colliding drops through repulsive electrostatic and steric interaction forces. These repulsive interactions minimize coalescence by slowing down the drainage of the intervening continuous film during a drop-pair encounter. The magnitude of the interaction forces depends on the amount and the nature of the surfactant adsorbed at the oil–water interface, the pH, the ionic strength, and the temperature. Also, the rupture of the intervening continuousphase film, necessary for drop coalescence, is made more difficult when the adsorbed interfacial layer possesses sufficiently high interfacial mechanical properties. The drop-size distribution of the emulsion can be controlled by controlling the rates of drop breakage and coalescence during emulsion
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G. Narsimhan / Journal of Colloid and Interface Science 272 (2004) 197–209
formation. This requires knowledge of the effect of operating parameters and formulation of the emulsion on drop breakage and coalescence in typical emulsion-forming equipment. A number of models have been reported in the literature for the coalescence efficiency of drops in stirred vessels. These models are based on different ways of visualizing coalescence phenomena. Howarth [2] assumed coalescence to occur immediately if the velocity of approach of two colliding drops along their center line exceeds a critical value. His expression for coalescence efficiency was independent of fluid properties, as the model neglected drainage and rupture of continuous-phase film intervening between a colliding drop pair. Coulaloglou and Tavlarides [3] viewed the influence of the turbulent environment on the two colliding drops as equivalent to the thinning of the intervening film under the application of a constant squeezing force for a certain contact time. They assumed coalescence to occur if the contact time was greater than the time for the draining film to reach a critical thickness for film rupture. Tsouris and Tavlarides [4] accounted for the partial mobility of the drop interface in the evaluation of drop coalescence efficiency in a turbulent flow field. Muralidhar and Ramkrishna [5], Das et al. [6], and Muralidhar et al. [7] visualized a drop pair as being continuously impinged upon by small eddies. They assumed drops to approach each other with a mean force due to asymmetric distribution of pressure fluctuation around the drop and a fluctuating component of the force superimposed on it, arising out of the random nature of these fluctuations. If the time scale of the force fluctuation was much smaller than the time scale of film thinning, the force fluctuation was represented as white noise (uncorrelated). If the two time scales were comparable, the force fluctuation was viewed as correlated. A drop-pair encounter either resulted in coalescence (the thickness of the intervening film reached a critical thickness) or separation (film thickness larger than an arbitrary value). Average probability of coalescence was reported as coalescence efficiency. Das et al. [6] and Muralidhar et al. [7] assumed the drops to be rigid spheres, whereas Muralidhar and Ramkrishna [5] considered the drops to be deformable. These models do not account for the interaction forces (van der Waals, electrostatic, and steric) between the drops due to the use of surfactants in such systems. Tobin and Ramkrishna [8] proposed a model for drop coalescence efficiency in a turbulent flow field based on stochastic analysis of thin film drainage between two colliding drops. They accounted for interdroplet electrostatic forces and considered both rigid and deformable drops. Such an approach is suitable for larger droplets and smaller turbulent intensities when the time scale of drop collision is much larger than the timescale of coalescence. In this paper, we propose a mechanistic model for drop coalescence in a turbulent flow field that accounts for the interdroplet turbulent and colloidal forces. The proposed model is applicable to cases (smaller droplets and high turbulence intensity) where the timescales of collision and coalescence are comparable. Drop coalescence is viewed as
a two-step process, namely, formation of a doublet due to drop collision followed by coalescence of the doublet. Coalescence of a doublet due to random force fluctuations is modeled as a stochastic process and the first two moments of coalescence time distribution are evaluated. The formulation of the model is presented in the next section. Subsequent sections present model simulations and comparisons of the model with drop coalescence rates inferred from the experiments.
2. Model of drop coalescence in a turbulent flow field In a turbulent dispersion, drops are randomly moving about and continually colliding with each other. A colliding drop pair is subjected to interdroplet turbulent and colloidal squeezing forces due to which the intervening continuousphase liquid drains, leading to coalescence of the pair. For relatively large drop sizes (10–100 µm) and relatively lowintensity turbulent flow fields, the timescale of drop collisions (the inverse of the rate of collisions) is much larger than the timescale of coalescence of the drop pair. Consequently, the rate of coalescence can be expressed as rate of coalescence = rate of collisions × coalescence efficiency. Tobin and Ramkrishna [8] have described a model for the evaluation of coalescence efficiency in terms of interdroplet forces and turbulence. However, for sufficiently small drop sizes (0.1 to a few µm) and high-intensity turbulent flow fields, the timescales of collision and coalescence are comparable. Therefore, the rate of coalescence cannot be expressed by the above expression. The model presented below refers to the latter case. Different steps involved in drop coalescence for this case are shown schematically below. In view of the fact that distinct droplets are separated by the continuous phase it is necessary to clarify what is meant by “collision” between droplets. We propose to view collision between two droplets as their approach to one another within some specified proximity. This proximity is assumed to be h0 , the distance beyond which there is negligible colloidal force between droplets. The coalescence process has been modeled as a two-step process. In the first step, the formation of a doublet, which can either coalesce or separate into the original single droplets under the influence of fluctuating turbulent force, occurs. This is followed by the second step, the drainage of the thin liquid film between the liquid drops to a critical thickness at which film rupture occurs, followed by almost instantaneous formation of a larger drop. In this step, the colloidal forces come into the picture. Hence the drop pair experiences both turbulent and colloidal forces under the influence of which the film thickness or the distance of separation reduces from h0 to hcrit . The time it takes for the distance of separation to reduce from h0 to hcrit
G. Narsimhan / Journal of Colloid and Interface Science 272 (2004) 197–209
is called the first passage time or the coalescence time. The coalescence process is shown schematically as follows:
199
its identity upon collision. n2 refers to the bulk number concentration of droplets of radius R2 . Integrating Eq. (5) with the given boundary conditions (7) and (8), we get 7αε1/3 n2 (R1 + R2 )7/3 . (9) 3 The flux of particles across the sphere of influence of radius r = R1 + R2 is equal to [9] dn j = Dturb (10) . dr r=R1 +R2 C1 =
The evolution of number concentration is given by dn1 = −k1 n21 + kd nd , (1) dt dnd = k1 n21 − (kd + k2 )nd , (2) dt dnc = k2 nd , (3) dt where n1 , nd , and nc are the number concentration per unit volume of the monomer, the doublet, and the coalesced droplet, respectively, k1 is the rate constant for the formation of the doublet, kd is the rate of dissociation of the doublet, and k2 is the rate of coalescence of the doublet. These have to be solved with the initial condition
The rate of collisions νc between drops of radius R1 and R2 is then given by [9,10] νc = 4πr 2 j n1 ,
(11)
where n1 is the number concentration of drops of radius R1 . Therefore from Eqs. (9) and (10), the rate of collisions νc is given by
(4)
28π 1/3 (12) αε (R1 + R2 )7/3n1 n2 , 3 √ where α = 2 is a constant. For equal-sized drops of radius R, Eq. (12) reduces to
Consider the collision of drops due to turbulent diffusion onto a reference drop of radius R1 . The rate of collision of drops of radius R2 can be obtained from the evaluation of turbulent diffusive flux of drops onto a sphere of influence with radius (R1 + R2 ). This flux can be calculated from the solution of the diffusion equation for the quasi-steady state, as given by
28π 1/3 (13) αε (2R)7/3 n20 , 3 where n0 is the number concentration of drops. When the drop size d < λ, the relative motion of drops is influenced by eddies that lie in the viscous subrange of the Kolmogorov universal energy spectrum. In this case, the rate of collisions will depend on the viscosity, since these eddies dissipate energy. The turbulent diffusivity in this case is given by [9,10]
t = 0,
n1 = n0 ,
nd = 0,
nc = 0.
2.1. Collision of two drops
∂n = constant = C1 , (5) ∂r where Dturb is the turbulent diffusivity, n is the number concentration of drops of radius R2 , and r is the radial distance from the center of the reference drop of radius R1 . We consider drop coalescence in a turbulent flow field of energy dissipation rates sufficiently high so that the drop sizes lie within the range of local isotropy. For the inertial subrange, the turbulent diffusion coefficient of the second drop Dturb at a distance r from the reference drop is given by [9] Dturb r 2
Dturb ∝ ε
1/3 4/3
r
.
(6)
The diffusion equation (5) can be solved with the boundary conditions n = 0, n = n2 ,
r = R1 + R2 , r →∞
(7) (8)
to obtain the number concentration of the droplets of radius R2 in the vicinity of the reference drop. The boundary condition (7) is the absorbing boundary condition at the sphere of influence, which implies that the droplet of radius R2 loses
νc =
νc =
Dturb = α(rε)1/3 r ε 2 r =β ν
for r λ,
(14a)
for r < λ.
(14b)
The diffusion equation can again be solved with the boundary conditions (7) and (8) and νc can then be evaluated to give νc =
4πn1 n2
1 1/2 −1/2 1 ε 3ν (R1 +R2 )3
−
1 λ3
+
3 −1/3 −7/3 λ 7α ε
,
(15)
√ where α = 2 and β = 1 are constants. For equal-sized drops of radius R, Eq. (15) reduces to νc =
1 1/2 −1/2 ε 3ν
4πn20 1 − λ13 + (2R)3
3 −1/3 −7/3 λ 7α ε
.
(16)
The time scale of drop collision, τcoll , for equal-sized drops can be defined as τcoll =
n0 , νc
(17)
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where νc is given by Eq. (13) or (16) and n0 , the number of droplets per unit volume, is given by 3φ , (18) 4πR 3 where φ is the dispersed phase fraction and R is the mean droplet radius. The rate constant k1 for the formation of doublets can be taken as the rate constant for the rate of collisions as given by νc k1 = 2 , (19) n0 n0 =
dissociation of the doublet when the initial force fluctuation is F , we get [11]
∞ T (F ) =
G(F , t) dt,
(24)
0
where G(F, t) = Prob T (F ) t =
∞
Using Eqs. (24) and (25), Eq. (23) can be transformed to ∂T ∗ 1 ∂ 2 T ∗ + = −1 ∂F ∗ 2 ∂F ∗ 2 with the boundary conditions
2.2. The rate of dissociation of doublet
F ∗ = 1,
since the colloidal interaction force at the time of doublet formation is negligible. In the above equation, F¯ is the mean turbulent force, δ is the standard deviation, Tf is the timescale of force fluctuation, and ζ (t) is white noise. A negative sign is introduced on the right-hand side of the equation, consistent with the sign convention that the attractive force is negative. It is to be noted that the average turbulent squeezing force is attractive. In order for the doublet to separate, the fluctuating force should overcome the mean force F¯ . The fluctuating force acting on the doublet is modeled as a Poisson process, i.e., a force of magnitude δ (equal to F¯ ), acts on the doublet at random times with a decay timescale Tf (see Eq. (55)). Consequently, the variation of the fluctuating force acting on the doublet is given by the stochastic differential equation [11] ¯ F (t) F −1/2 dt + F¯ Tf − dη(t), dF (t) = (21) Tf Tf
dη(t) = 0, dη(t)2 = dt.
(1 − F ∗ )
(22b)
The evolution of force fluctuation experienced by the doublet is given by the Fokker–Planck equation [12] ¯ F ∂p(F , t|F, 0) F ∂p(F , t|F, 0) = − ∂t Tf Tf ∂F 2 2
¯ 1 F ∂ p(F , t|F, 0) + (23) , 2 Tf ∂F 2 where p(F , t|F, 0) is the conditional probability that the force fluctuation at time t is F given that the force fluctuation at time zero is F . Defining T (F ) as the average time of
T ∗ (1) = 0, ∂T ∗ = 0, ∂F ∗
F ∗ = 0,
(26)
(27a) (27b)
where T ∗ = T /Tf and F ∗ = F /F¯ . The first absorbing boundary condition denotes the dissociation of the doublet whenever the fluctuating force equals the mean turbulent force. The second reflecting boundary condition states that the doublet experiences a fluctuating force between 0 and F¯ . The mean dissociation time of the doublet is T ∗ (0) which is given by [11]
0
∗
T (0) = 2
dy ψ(y)
0 ψ(z) dz,
(28)
y
1
where x ψ(x) = exp
2(1 − x) dx .
(29)
1
Solution of Eq. (28) yields T ∗ (0) = 0.37. Therefore, the rate of dissociation of a doublet kd is given by kd =
(22a)
(25)
0
where νc , the rate of collisions per unit volume, is given by Eq. (13) and Eq. (16) for inertial and viscous subranges, respectively.
Once a doublet is formed, it is subjected to random turbulent force fluctuation. The net turbulent force acting on the doublet at the time of collision is given by [5] 1/2 F = − F¯ − δTf ζ (t) (20)
p(F , t|F, 0) dF .
1 . 0.37Tf
(30)
2.3. Evaluation of the mean coalescence time Once a doublet is formed, the surface-to-surface distance between the two individual droplets within a doublet should be less than hcrit for coalescence to occur. Since the doublet experiences a fluctuating turbulent force, the surface to surface distance h(t) varies randomly with time and is, therefore, a stochastic process. The turbulent mean and fluctuating forces should overcome the colloidal repulsive force for coalescence to occur. The time of coalescence, i.e., the time at which h(t) = hcrit , is random. The evaluation of moments of coalescence time distribution is outlined below.
G. Narsimhan / Journal of Colloid and Interface Science 272 (2004) 197–209
2.4. Drainage of the continuous phase film For nondeformable spherical particles, the drainage of continuous-phase liquid between two colliding particles of sizes d and d is given by Taylor’s equation [12], dh 2hF 1 1 2 (31) = + , dt 3πµ d d where h is the surface-to-surface distance between the drops and F is the interaction force between the two emulsion droplets. By convention, the interaction force F is positive if repulsive and negative if attractive. For drops of equal size d, Eq. (31) becomes 8hF dh = . (32) dt 3πµd 2 In a turbulent flow field, the droplet pair is subjected to a random fluctuating turbulent force with mean force F¯ , which will try to squeeze the colliding drop pair toward each other, thus promoting coalescence. The van der Waals attractive force between the two drops would also promote coalescence. On the other hand, the electrostatic repulsive force between the two drops would tend to slow down the film drainage. The expressions for the turbulent and colloidal forces are given in the following sections. 2.5. Turbulent force The mean turbulent force, F¯ , is given by πd 2 ρ 2 u (d), F¯ = 4
(33)
where u2 (d) is the mean squared turbulent velocity fluctuation between the centers of the colliding droplet pair separated by a distance d. For local isotropy, when d λ (inertial subrange), the mean squared velocity fluctuation is given by [9] u2 (d) = 2ε2/3d 2/3 , where ε is the energy dissipation rate per unit mass.
(34)
201
The mean turbulent force F¯ is therefore given by π F¯ = ρε2/3 d 8/3. (35) 2 For local isotropy, when d λ (viscous subrange), the mean squared velocity fluctuation is given by [9] u2 (d) =
εd 2 ρ . µ
(36)
The mean turbulent force F¯ is therefore π ρ 2d 4ε F¯ = . 4 µ
(37)
2.6. Colloidal force A drop pair are deemed to have collided if the interaction force between the two drops is above a certain threshold. In emulsions, surfactants are usually employed to stabilize the emulsion. When the surfactant is ionic, adsorption of surfactants at the emulsion drop interface will impart a charge to the droplet, thus creating an electrical double layer in its vicinity. The colliding droplet pair would experience an electrostatic repulsive force because of the overlap of double layers. In addition, the emulsion droplets also experience a van der Waals attractive force. Typical variation of interdroplet colloidal force with the surface-to-surface distance of separation between the drops is shown in Fig. 1. Van der Waals and electrostatic interactions are included in the evaluation of the colloidal force. Expressions for van der Waals and electrostatic force are given later in the following sections. As expected, at large distances of separation, the interparticle force is negligible. As the drops approach each other, they experience a repulsive (positive) force because of the predominant effect of double-layer repulsion. At very small h, however, the net force is attractive (negative) because of the predominance of van der Waals attraction. Consequently, there exists a critical distance of separation hcrit at which the interparticle force changes from repulsive to attractive (Fig. 1). Once the distance of separation becomes
Fig. 1. Typical variation of colloidal force with the surface-to-surface distance of separation between two drops of diameter 209 nm with ψ0 = 50 mV and I = 0.1 M.
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less than hcrit , the drop pair experiences a strong van der Waals attractive force leading to coalescence. Once this happens, drop coalescence can be considered to be almost instantaneous, due to the strong van der Waals attraction. 2.7. Model equations The drainage of the continuous-phase liquid between two nondeformable drops of equal size is given by Taylor’s equations [12], 8hF dh = . (38) dt 3πµd 2 The net force of interaction experienced by the droplet pair is the sum of the turbulent and colloidal forces. Because of the random nature of the turbulent force, the surface-tosurface distance h(t) can be considered to be a stochastic process. The net interaction force F is given by [6] 1/2 F = − F¯ − Fc − δTf ζ (t) , (39) where F¯ is the mean turbulent force given by Eqs. (35) and (38) and Fc is the colloidal interaction between the droplets due to van der Waals and electrostatic forces, Fc = FVW + FDL ,
(40)
where FVW and FDL refer to the van der Waals and doublelayer interactions, respectively. The last term in Eq. (39) refers to the turbulent fluctuating force, which is explained later in this section. It is to be noted that the hydrodynamic interaction between the two colliding drops is neglected in this analysis. Such an assumption is indeed reasonable, as shown in Appendix A. 2.7.1. Van der Waals attraction The van der Waals interaction is given by [13]
1 1 (1 + λr )2 θ (1 − φ) + + FVW = − (R 2 − θ )2 (R 2 − θ φ)2 2λr (R1 + R2 )2 AH R1 R2 × 2 (41) , (R − θ )(R 2 − θ φ) where R1 , R2 are the droplet radii (R1 = R2 for equal sized drops) and AH , the effective Hamaker constant, is given by the expression 1/2
1/2 − A o )2 , AH = (Aw
(42)
Aw and Ao being the Hamaker constants of the aqueous and oil phases, respectively. φ, θ , and λr are given by 1−λr 2 , φ= (43) 1 + λr θ = (R1 + R2 )2 ,
(44)
λr = R1 /R2 .
(45)
R is the center-to-center distance of separation between the two drops, or R = R1 + R2 + h, where h is the surface-tosurface distance of separation.
For surface-to-surface distances of separation h R1 + R2 , Eq. (41) reduces to FVW = −
AH R1 R2 . 6h2 (R1 + R2 )
(41a)
In writing Eq. (41), the retardation effect is neglected. This is justified since the average droplet size is on the order of a few hundred nanometers. 2.7.2. Electrostatic double-layer repulsion The electrostatic double-layer repulsion is calculated based on a numerical algorithm [14], which calculates the distance of separation between two charged plates at given midpoint potentials by solving the differential equation
2 −1/2 2 Q dX = + cosh Ym − 1 (46) , dQ 2 where X = κx is the scaled distance measured from the midplane. κ, the Debye–Hückel parameter, is given by κ2 =
8πne2 , εε0 kT
(47)
where k is Boltzmann’s constant, T is the absolute temperature, n is the number concentration of the electrolyte, e is the elementary charge, ε is the dielectric constant of the medium, and ε0 is the permittivity constant. We define Q by 1/2 , Q = 2(cosh Y − cosh Ym ) (48) where Y = zeψ/kT is the scaled potential at a distance X from the midplane between the two charged plates in a z : z electrolyte solution of number concentration n, z is the valence number of ions, ψ is the electrostatic potential at a distance x from the midplane, and Ym is the scaled midpoint potential. The midplane in terms of these variables is the point (Q = 0, X = 0). Knowing the value of Q (= Q0 ) corresponding to the surface potential ψ0 , we can solve Eq. (46) from Q = 0 to Q = Q0 by numerical integration. Thus we determine X0 = kh/2, the scaled distance from the midplane to the surface. The electrostatic force of interaction FFP per unit area between two plates separated by a distance h is then given by FFP = 2kT n0 [cosh Ym − 1].
(49)
Using the Derjaguin approximation, the interaction force FDL (h) between two droplets of radius R separated by a surface-to-surface distance h can be obtained by integration to give
∞ FDL (h) = πR
FFP (x) dx. h
(50)
G. Narsimhan / Journal of Colloid and Interface Science 272 (2004) 197–209
2.8. Turbulent fluctuation
and
The last term in Eq. (39) refers to the fluctuating part of the turbulent force, which is modeled as a band-limited white noise ζ (t) with ζ (t) = 0, (51)
2 |t − t | , ζ(t)ζ(t ) = (52) exp − Tf Tf where . denotes ensemble average. At any time, ζ (t) has a Gaussian distribution with mean zero and standard deviation δ, which is taken to be equal to F¯ [5]. Tf , the time scale of force fluctuation, can be defined as [5] Tf =
d [u2 (d)]1/2
,
(53)
where d is the drop size and u2 (d) is the mean squared velocity fluctuation at a distance of separation d. For d λ, from Eq. (34), we get Tf = 2−1/2 ε−1/3 d 2/3 .
(54)
For d λ, from Eq. (37), we get 1/2 ν µ . Tf = = ερ ε
(55)
It is assumed that the random squeezing force, which determined the coalescence probability of the doublet, is spatially uniform and temporally a white noise process. From Eqs. (38) and (39), 1/2 2h{F¯ − Fcoll − δTf ζ (t)} dh =− , dt 3πµR 2
dh = −
2h{F¯ − Fcoll } dt + 3πµR 2
1/2 2δhTf 3πµR 2
(56) dW (t),
(57)
where dW (t) is a Wiener process. Equation (57) is the stochastic differential equation describing the evolution of h(t). This equation is written in the Stratanovich sense [11], dh = f2 (h) + g2 (h) dW (t),
(58)
where f2 (h) = −
2(F¯ − Fcoll )h 3πµR 2
(59)
and 1/2
g2 (h) =
2δhτf
3πµR 2
203
.
(60)
Converting the above equation to its equivalent in Ito form [11] gives dh = f1 (h) + g1 (h)ξ(t),
(61)
g1 (h) = g2 (h). This gives f1 (h) = −
2δ 2 τf h 2(F¯ − Fcoll )h + 3πµR 2 9π 2 µ2 R 4 1/2
g1 (h) =
2δhτf
3πµR 2
.
(65)
Since h(t) is a stochastic process, the evolution of surface-to-surface distance between the two droplets of a doublet after its formation (i.e., h(0) = h0 ) is random and is usually referred to as a sample path of the process. For an ensemble of a large number of doublets subjected to colloidal and turbulent forces, the evolution of their surface-to-surface distance of separation gives an ensemble of sample paths. From this ensemble one can then define the probability that a doublet will have a surface-to-surface distance of x (say) at time t. One can then define p(x , t|x, 0) as the conditional probability that h = x at time t with the initial condition x. The existence of the doublet implies that hcrit < h < h0 . When h = hcrit , the coalescence of the doublet occurs since colloidal force becomes attractive for h < hcrit . Therefore, p(hcrit , t|x, 0) = 0.
(66)
Since the colliding drop pair is subjected to an attractive turbulent force and the dissociation of the doublet at the time of collision is accounted for through kd , we impose a reflecting boundary at h = h0 ; i.e., the flux at h0 is zero: dp(h0 , t|x, 0) = 0. (67) dx As pointed out earlier, we are interested in the average coalescence time. Consequently, the final point in the sample path is hcrit . Therefore, we are interested in the evolution of this conditional probability backward in time. This is given by the backward Fokker–Planck equation [11], ∂p(x , t|x, 0) ∂p(x , t|x, 0) = f1 (x) ∂t ∂x ∂ 2 p(x , t|x, 0) 1 . + g12 (x) 2 ∂x 2 Defining
G(x, t) = dx p(x , t|x, 0) = Prob(T t),
(68)
(69)
we have from Eq. (68) ∂G(x, t) ∂G(x, t) 1 2 ∂2 = f1 (x) + g1 (x) 2 G(x, t). ∂t ∂x 2 ∂x Defining the mean first passage time,
∞
(62)
(64)
and
where 1 f1 (h) = f2 (h) + g2 (h)g2 (h) 2
(63)
T (x) = T = − 0
∂G(x, t) dt = t ∂t
(70)
∞ G(x, t) dt, 0
(71)
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where T (x) is the average coalescence time of the doublet when the initial separation of drops is x. We can derive a simple ordinary differential equation for T (x) by using Eq. (71) and integrating Eq. (70) over (0, ∞). Noting that
∞
∞ Tn (x) = −
tn
∂G(x, t) dt ∂t
(82)
0
∂G(x, t) dt = G(x, ∞) − G(x, 0) = −1 ∂t
(72)
0
dT (x) 1 2 d 2 T (x) + g1 (x) = −1 dx 2 dx 2 with the boundary conditions
f1 (x)
(73)
dT = 0, dx T = 0.
x = h0 , x = hcrit ,
(74a)
∂Tn (x) 1 2 ∂ 2 Tn (x) + g1 (x) = −nTn−1 (x), (83) ∂x 2 ∂x 2 which can be solved recursively to obtain the moments of the coalescence time distribution. In terms of dimensionless quantities, we have
x T (x) = 2
dy ψ(y)
where
h0
ψ(z) dz, g12 (z)
y
hcrit
x
ψ(x) = exp
dx
(75)
2f12 (x )/g1 (x )
.
(76)
hcrit
Equation (75) at x = h0 becomes
h0
x
exp −
T (h0 ) = 2 hcrit
h0 × y
hcrit
1 exp g12
2f1 dx g12
x
hcrit
2f1
dx dx dy. g12
T (x) , Tf
x∗ =
(x − hcrit ) , (h0 − hcrit )
(77)
(78) 1/2
g1 (x)Tf f1 (x)Tf , g1∗ (x ∗ ) = , (h0 − hcrit ) (h0 − hcrit ) x ∗ 2f1∗ (x )
∗ ∗ ψ (x ) = exp dx , g1∗ 2 (x ) f1∗ (x ∗ ) =
(79)
(80)
0
we have
x ∗
T (x ) = 2 0
dy ψ ∗ (y)
1 y
ψ ∗ (z) g1∗ 2 (z)
dz.
x ∗
dy ψ ∗ (y)
0
1 y
∗ (z)ψ ∗ (z) nTn−1
g1∗ 2 (z)
dz,
(84)
where Tn∗ (x ∗ ) =
Tn (x) . Tfn
(85)
The rate of formation of the coalesced droplet, k2 (x), in Eq. (3) is given by k2 =
1 . T (h0 )
(86)
The first and the second moments of the distribution of coalescence time were evaluated from Eqs. (81) and (84), respectively. Even though it is possible to evaluate higher moments, these calculations were found to be computationally intensive and therefore were not carried out. The moments were then employed to fit the beta distribution, 1 x α−1 f (x) = (87) x exp − . Γ (α)β α β
3. Simulation results
In terms of dimensionless quantities, T ∗ (x ∗ ) =
Tn∗ (x ∗ ) = 2
(74b)
Note that (74a) and (74b) are the corresponding reflecting and absorbing boundary conditions for the mean coalescence time. The solution for Eq. (73) with boundary conditions (74) is given by [10]
∗
from Eq. (70), we get f1 (x)
we get
∗
Defining the nth moment of the coalescence time distribution Tn (x) as
(81)
Typical variation of the average dimensionless coalescence time T ∗ (x ∗ ) with the dimensionless initial position x ∗ is shown in Fig. 2. As expected, the average coalescence time is zero for the initial position x ∗ = 0 (x = hcrit ) since drop coalescence is deemed to occur immediately when the surface-to-surface distance between two colliding drops is hcrit . The average coalescence time increases dramatically as the surface-to-surface distance of separation increases (Fig. 2) because of the presence of a high force barrier (see Fig. 1) for intermediate distances of separation. At greater distances of separation, however, the average coalescence time tends to level off since most of the resistance to coalescence lies near the repulsive force barrier. As a result, the average coalescence time is insensitive to the actual distance of separation at the time of drop collision. This is important for the evaluation of coalescence rates since drop collision (or, equivalently, the initial distance of separation) is assumed
G. Narsimhan / Journal of Colloid and Interface Science 272 (2004) 197–209
205
Fig. 2. Variation of dimensionless coalescence time T ∗ with initial dimensionless surface-to-surface distance x ∗ .
Fig. 4. Effect of energy dissipation rate on drop coalescence time distribution for equal-sized droplets of R = 2.5×10−7 m, I = 0.05, ψ0 = 58.8 mV. The three distributions are for three different energy dissipation rates in W/kg as indicated.
Fig. 3. Effect of energy dissipation rate on drop coalescence time and Ftur /Fcoll for equal-sized droplets of R = 2.5 × 10−7 m, I = 0.05, ψ0 = 58.8 mV. F, Average drop coalescence time; 2, Ftur /Fcoll .
to occur when the interdroplet force is much smaller than the barrier force (1% of barrier force in these calculations). Consequently, any small variation in the initial distance of separation at the time of drop collision will not significantly affect the calculated average drop coalescence time. In the subsequent calculations, therefore, T ∗ (1) (which, in dimensional average coalescence time, is T (h0 )) is reported as the average dimensionless drop coalescence time. The variation of average coalescence time with energy dissipation rate is shown in Fig. 3. The average coalescence time decreases exponentially from about 4 × 108 to 2.96 × 10−6 s as the energy dissipation rate increases from 106 to 6 ×106 W/kg. The ratio of the average turbulent force to the barrier force versus the energy dissipation rate is also
shown in Fig. 3. It is interesting to note that the coalescence time decreases dramatically as this ratio increases. The coalescence time distribution as given by Eq. (88) was evaluated and is shown for different energy dissipation rates in Fig. 4. The coalescence time distribution becomes broader with a larger standard deviation at lower energy dissipation rates. Average coalescence time versus surface potential is shown in Fig. 5. The coalescence time is found to increase dramatically with the surface potential. For example, the average coalescence time increases from ∼10−5 to ∼103 s as the surface potential increases from 35 to 55 mV. The increase in the force barrier (or the corresponding decrease in FT /Fcoll , where FT and Fcoll refer to the average turbulent force and force barrier, respectively, since the average turbulent force remains constant) with surface potential can be seen from Fig. 5. The dimensionless coalescence time distribution becomes broader at higher surface potentials (Fig. 6). The variation of average coalescence time with drop size for coalescence of equal-sized droplets is shown in Fig. 7.
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Fig. 5. Effect of surface potential on drop coalescence time and Ftur /Fcoll for equal-sized droplets of R = 2.5 × 10−7 m, I = 0.05, ε = 106 W/kg. F, Average drop coalescence time; 2, Ftur /Fcoll . Fig. 8. Effect of drop size on drop coalescence time distribution for equal-sized droplets of I = 0.05, ε = 4.8 × 107 W/kg, ψ0 = 58.8 mV. F, Average drop coalescence time; 2, Ftur /Fcoll .
Fig. 6. Effect of surface potential on drop coalescence time distribution for equal-sized droplets of R = 2.5 × 10−7 m, I = 0.05, ε = 107 W/kg.
Fig. 9. Effect of size ratio on average drop coalescence time and Ftur /Fcoll for R = 1.5 × 10−7 m, I = 0.05, ε = 4.8 × 107 W/kg, ψ0 = 58.8 mV. F, Average drop coalescence time; 2, Ftur /Fcoll .
Fig. 7. Effect of drop size on drop coalescence time and Ftur /Fcoll for equal-sized droplets of I = 0.05, ε = 5 × 106 W/kg, ψ0 = 58.8 mV. F, Average drop coalescence time; 2, Ftur /Fcoll .
The coalescence time is found to decrease from 6.3 × 109 to 1.9 × 10−6 s as the drop radius increases from 1.5 × 10−7 to 3 × 10−7 m. Fig. 7 also shows the variation of FT /Fcoll versus drop size. Even though the force barrier increases with drop size, the turbulent force increases much more rapidly than the barrier force, thus resulting in a smaller average coalescence time. The coalescence time distributions for two different drop sizes, as shown in Fig. 8, indicate narrowing of the distribution for larger drop size. The results for unequal drop sizes (Figs. 9 and 10) indicate a decrease in
Fig. 10. Effect of size ratio on drop coalescence time distribution for R = 1.5 × 10−7 m, I = 0.05, ε = 4.8 × 107 W/kg, ψ0 = 58.8 mV.
average drop coalescence time with a narrower coalescence time distribution for larger size ratios because of the increase in turbulent squeezing force compared to the interdroplet colloidal force. The variation of average coalescence time with ionic strength for constant surface potential is shown in Fig. 11.
G. Narsimhan / Journal of Colloid and Interface Science 272 (2004) 197–209
Fig. 11. Effect of ionic strength on average drop coalescence time for R = 2.5 × 10−7 m, ε = 4.8 × 107 W/kg, ψ0 = 58.8 mV.
The coalescence time decreases from 1.66 × 10−7 to 1.95 × 10−8 s as the ionic strength increases from 0.01 to 0.5 M. Such behavior is understandable since an increase in the ionic strength compresses the electrical double layer, thus allowing the colliding drop pair to come nearer before they experience repulsion. Since the model imposes a reflecting boundary condition at h = h0 in the evaluation of the first passage time, it does not allow the dissociation of the doublet during the random drainage process. Such an assumption is reasonable whenever the average turbulent squeezing force is sufficiently high. Since the surface-to-surface separation of colliding droplets is only about 10 times larger than the microscale of turbulence, local isotropy may not accurately reflect the turbulent flow field.
4. Comparison of model predictions with experimental data Average drop coalescence rates and average coalescence rate constants were inferred from the experimental measurement of the evolution of drop number concentrations of tetradecane-in-water emulsions stabilized by sodium dodecyl sulfate for a negative stepchange in homogenizer pressure [15]. The experimental data and the methodology for the inference of average drop-coalescence rate constants are given elsewhere [15]. Here we compare the experimental average drop coalescence rate constants with model predictions. In order to account for the drop size distribution, the drop sizes were divided into six size intervals and the distribution was represented by a discretized beta distribution. Equations similar to Eqs. (1)–(3) were written for the coalescence of drops of sizes i and j . The effect of hydrodynamic interaction is accounted for in the evaluation of drop collision rates, as described in Appendix A. The drop collision rate constant ki,j for each drop size pair was evaluated using Eqs. (19), (A.1), and (A.2). It is to be noted that the effect of hydrodynamic interaction on drop coalescence time is not significant, as explained in Appendix A. The rate constants kd and k2 were evaluated from Eqs. (30) and (86), respectively, for each drop size pair. The average drop coalescence
207
Fig. 12. Comparison of predicted and experimental drop coalescence rate constants in a high-pressure homogenizer at different homogenizer pressures. The model predictions are made for R¯ = 1.951 × 10−7 m, σ = 3.03 × 10−8 m, I = 0.05, ε = 7.526 × 107 W/kg, ψ0 = 63.9 mV. F, Experimental; 2, model prediction.
rate constant kav was then calculated using kav =
nc,ij i
j
θ n20
,
(88)
where nc,ij is the number concentration of coalesced droplets due to collision of droplets of sizes i and j , which was evaluated from the solution of Eqs. (1)–(3) subject to the initial condition (4). Calculation of the energy dissipation rate ε and the average residence time θ through the homogenizer valve is explained elsewhere [15]. The experimental conditions and the parameters (interfacial tension, zeta potential, mean drop size, and standard deviation of drop size distribution) are given in Narsimhan and Goel [15]. A comparison of model predictions with inferred average coalescence-rate constants from experiments at different homogenizer pressures is shown in Fig. 12. As expected, the coalescence-rate constant increases with homogenizer pressure, though the dependence is less pronounced for the model predictions. The model predictions agree fairly well with the values inferred from the data considering that only the coalescence-rate constants averaged over drop sizes are compared. Comparison of size-specific coalescence-rate constants could not be made since experimental size-specific coalescence-rate constants were not obtained. A comparison of model predictions with inferred average coalescencerate constants from experiments at different SDS concentrations is shown in Fig. 13. The model predictions follow the same trend as the experiments and agree fairly well with the inferred values. Of course, the predicted average dropcoalescence-rate constants were found to be higher than the experimental values, especially at higher SDS concentrations. This is believed to be due to the simplifying assumption of a reflecting boundary condition at h0 for the evaluation of the coalescence time of a doublet. In other words, the model allows dissociation of the doublet only at the time of its formation (in the absence of colloidal forces) and assumes that once the doublet is formed, it will eventually coalesce into a single droplet with the coalescence time depending on the relative magnitudes of turbulent and colloidal forces.
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model is believed to be due to the simplifying assumption of a reflecting boundary condition at h = h0 for the evaluation of coalescence time of a doublet.
Acknowledgment This research was supported in part by USDA National Research Initiative Grant #9602252.
Fig. 13. Comparison of predicted and experimental drop coalescence rate constants in a high-pressure homogenizer at different SDS concentrations. The model predictions are made for R¯ = 2.32 × 10−7 m, σ = 3.203 × 10−8 m, I = 0.1, ε = 4.82 × 107 W/kg. F, Experimental; 2, model prediction.
5. Conclusions A model for collision and coalescence of a drop pair in a turbulent flow field that accounts for interdroplet colloidal as well as fluctuating turbulent forces is developed. Colloidal forces include van der Waals and double-layer electrostatic forces. The model views drop coalescence as a two-step process consisting of formation of a doublet due to drop collisions followed by coalescence of the individual droplets in a doublet as a result of drainage of the intervening film of continuous phase. The fluctuating turbulent force was modeled as band-limited white noise. The average as well as higher moments of the coalescence-time distribution were evaluated from the analysis of the random process of intervening continuous-phase film thickness and were found to be dependent on the force barrier to drop coalescence. Even though the average drop-coalescence time was dependent on the initial separation between the two colliding drop pairs, it was found to be fairly insensitive at sufficiently large separations. The average drop-coalescence time decreased dramatically with an increase in energy dissipation rate and the coalescence-time distribution became narrower at higher energy dissipation rates. The average coalescence time was longer and the coalescence time distribution broader at higher surface potentials as well as lower ionic strength because of higher force barriers to drop coalescence. Even though the turbulent as well as colloidal forces increased for larger drop sizes and size ratios of the unequal drop pair, the turbulent force increased faster than the colloidal force, thus resulting in a smaller average drop coalescence time and a narrower distribution. The model predictions of average drop-coalescence-rate constants that account for the drop-size distributions compared fairly well with the values inferred from the experimental evolution of drop number concentration to a negative step change in homogenizer pressure for tetradecane-inwater emulsions stabilized by SDS at different homogenizer pressures and SDS concentrations. The overprediction of the
Appendix A. Effect of hydrodynamic interaction on drop coalescence A.1. Collision rate Following the analysis of Honig et al. [16], Eq. (5) can be modified to account for van der Waals interactions and hydrodynamic interactions, giving the following expression h for the collision rate constant ki,j between particles of radii Ri and Rj , λ ∗ −1 h ki,j
= 4π 0
vw exp(Ui,j /kT )βi,j (y)
dy
(ε/ν)1/2 (Ri + Rj )4 (y + 1)4
∞
vw /kT )β (y) exp(Ui,j i,j
+
αε1/3 (Ri + Rj )10/3(y + 1)10/3
λ∗ −1
for (Ri + Rj ) < λ, λ∗
dy (A.1)
where = λ/(Ri + Rj ) and is the van der Waals interaction potential. The hydrodynamic interaction correction factor βi,j (y) is evaluated using the iterative equations given elsewhere [17]. For (Ri + Rj ) > λ, vw Ui,j
∞ h ki,j
= 4π 0
vw /kT )β (y) exp(Ui,j i,j
αε1/3 (Ri + Rj )10/3 (y + 1)10/3
dy.
(A.2)
A.2. Drop coalescence time The Taylor equation for the drainage of a continuous film between two colliding nondeformable drops of equal size can be written to account for the hydrodynamic resistance, 8h dh = (FT + Fcoll + Fhyd ), dt 3πµd 2
(A.3)
where Fhyd is the hydrodynamic resistance. Expressing the hydrodynamic resistance in terms of friction coefficient, we get 8h dh 8h dh = , (A.4) (FT + Fcoll ) − f dt dt 3πµd 2 3πµd 2 where f is the friction coefficient. Defining the dimension∗ =F less variables h∗ = κh, FT∗ = FT /κkT , Fcoll coll /κkT ,
G. Narsimhan / Journal of Colloid and Interface Science 272 (2004) 197–209
t ∗ = t/Tf and recognizing that the friction coefficient is given by f ∼ = 6πµR(0.645) [18], the above equation yields dh∗ = dt ∗
2κkT Tf ∗ ∗ ∗ ) h (FT + Fcoll 3πµR 2 1+
8(0.645) κd
.
(A.5)
For typical values of ionic strength and drop size, κd 1. For example, for an ionic strength of 0.01 and drop diameter of 100 nm, kd ∼ = 30, and for an ionic strength of 0.1 and drop diameter of 100 nm, kd ∼ = 100. Consequently, the denominator on the right hand side is close to unity. As a result, Eqs. (38) and (39) constitute a good approximation of the above equation. Therefore, hydrodynamic interaction is not important in the evaluation of drop coalescence time.
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