Effect of Dispersion Properties on the Separation of Batch Liquid

liquid dispersions is experimentally investigated in terms of the variation in the heights of the sedimenting and coalescing interfaces with time. A m...
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Ind. Eng. Chem. Res. 1998, 37, 547-554

547

SEPARATIONS Effect of Dispersion Properties on the Separation of Batch Liquid-Liquid Dispersions S. A. K. Jeelani and S. Hartland* Department of Chemical Engineering and Industrial Chemistry, Swiss Federal Institute of Technology, Universita¨ tstrasse 6, 8092 Zurich, Switzerland

The effect of dispersed phase holdup and dispersion height on the separation of batch liquidliquid dispersions is experimentally investigated in terms of the variation in the heights of the sedimenting and coalescing interfaces with time. A model is presented which predicts these separation profiles, knowing the initial drop diameter and the interfacial coalescence time for a dispersion of given height, dispersed phase holdup, and physical properties. This is verified using the present and published experimental data. 1. Introduction The separation of liquid-liquid dispersions is encountered in petroleum, petrochemical, hydrometallurgical, nuclear fuel processing, and chemical industries. Dispersions are formed in agitated tanks and columns in a solvent extraction process. These have to be separated for the recovery of the expensive solvent. Water in crude oil emulsions are formed during the pressure reduction on offshore platforms. These are stabilized by the natural surfactants present in the crude oil. Chemical demulsifiers are usually added to enhance the separation of the emulsions. The design of the continuous flow gravity separators is based on expensive pilot plant experiments. Attempts have been made to design them using information obtained from small scale batch settling tests (Hartland and Jeelani, 1994). The behavior of these separators can be understood by studying the fundamental processes such as sedimentation and coalescence in batch separators, which are less expensive. In spite of the considerable available literature (Hartland and Jeelani, 1994; Nadiv and Semiat, 1995), simple models do not exist which describe the behavior of batch dispersions. Nadiv and Semiat (1995) experimentally investigated the effect of mixing conditions and dispersion height on the separation of batch dispersions using two settler diameters. They did not systematically vary the dispersion holdup. They presented a model for sedimentation based on the analysis of Aris and Amundson (1973) for batch precipitation of a suspension of solids. An empirical equation was used to describe the coalescence profile. Their model contained four unknown parameters which must be determined using the experimental sedimentation and coalescence profiles. The present paper describes a model which predicts the variation in the heights of the sedimenting and coalescing interfaces with time in a batch settler know* To whom correspondence should be addressed. Telephone: +41-1-632 3060. Fax: +41-1-6321265. E-mail: hartland@tech. chem.ethz.ch.

ing the initial drop diameter and the coalescence time of a single drop at an interface for a dispersion of given height, dispersed phase holdup and physical properties. Experiments are carried out to investigate the effect of dispersed phase holdup and dispersion height on the separation of batch dispersions in terms of the sedimentation and coalescence profiles. The present experimental data and that of Nadiv and Semiat (1995) are used to verify the model. 2. Theory Consider an oil in water dispersion of height H0 with oil holdup fraction 0 in which the average drop diameter is φ0. When the oil is less denser than the water, the oil drops sediment to form a dense-packed layer of holdup fraction p at the top of the dispersion as shown in Figure 1a. The thickness of this layer increases with time as shown in Figure 1b. Interfacial coalescence occurs so the height of the coalescing interface hc (all the heights are measured from bottom) decreases with time t. As the drops sediment so the height of the sedimenting interface (which is also the height of the clear layer of water) at the bottom of the dispersion increases with time. The holdup in the dispersion between the sedimenting interface and the bottom of the dense-packed layer increases with time. The height of the sedimenting dispersion decreases with time becoming zero when the bottom of the dense-packed layer meets the sedimenting interface. As interfacial coalescence proceeds, the height of the dense-packed layer then decreases disappearing at a time tf when complete separation occurs. The height ∆h of the densepacked layer reaches a maximum value ∆hi at time ti at the end of sedimentation. If binary coalescence occurs, the sedimentation rate is increased. Binary coalescence may also occur in the dense-packed zone but is not essential to the separation process. For a dispersion of water in oil, the water drops sediment downward when water is denser than the oil to form a dense-packed layer at the bottom of the dispersion. The profiles are interchanged, the coalesc-

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548 Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998

Figure 1. (a) Heights of the sedimenting and coalescing interfaces hs and hc and the dense-packed zone formed at the top of a dispersion due to the upward sedimentation of lighter oil drops in heavier water. (b) Schematic variation in the heights of sedimenting and coalescing interfaces hs and hc with time t for the oil in water dispersion shown in part a. The variation in the height of the boundary between the sedimentation and dense-packed zones hp is also shown.

ing interface being at the bottom of the dispersion and the sedimenting interface at the top of the dispersion. All the equations presented below still apply if the heights hc and hs are replaced respectively by H0 - hc and H0 - hs in all the equations below. When the oil is denser than the water, a layer of oil appears at the bottom and a layer of water at the top. 2.1. Coalescence Profile. The volume rate of interfacial coalescence per unit area ψ ) -dhc/dt is given (Hartland and Vohra, 1978) by

ψ)

2pφ 3τ

(1)

in which τ is the coalescence time for drops of diameter φ at the coalescing interface. We assume the coalescence time τ of a drop at an interface is inversely proportional to the force pressing on the draining film above the drop. For a dense-packed dispersion this force is proportional to the height ∆h and for a monolayer of drops at the interface we assume the force is proportional to the drop diameter φ. The interfacial coalescence time is then given by

( )( )

τ ) τ0

φ φ0 φ0 ∆h

(2)

in which τ0 is the interfacial coalescence time of a single layer of drops of diameter φ0. The interfacial coalescence rate given by eq 1 is then independent of the drop diameter φ and directly proportional to the dense-packed height ∆h

ψ)in which

dhc ψ0 ) ∆h dt φ0

(3a)

ψ0 )

2pφ0 3τ0

(3b)

is the initial interfacial coalescence rate. Since the thickness ∆h of the dense-packed layer increases with time during sedimentation so does the interfacial coalescence rate ψ, reaching a maximum value ψi at time ti. Since the dense-packed height decreases with time between ti and tf so does the interfacial coalescence rate. The maximum interfacial coalescence rate ψi corresponds to an inflection point on the coalescence profile, which is the variation in the height hc of the coalescing interface with time. Similarly the sedimentation profile is the variation in the height hs of the sedimenting interface with time. When t ) ti, ∆h ) ∆hi and ψ ) ψi and eq 3a becomes

ψ)-

dhc ∆h ) ψi dt ∆hi

(4a)

in which

ψi )

ψ0∆hi φ0

(4b)

is the maximum interfacial coalescence rate occurring at the inflection point ti. Combining eqs 3b and 4b gives ψi in terms of the interfacial coalescence time τ0 as

ψi )

2p ∆h 3τ0 i

(4c)

If hp is the height of the position of the boundary between the sedimentation and dense-packed zones, then the height of the dense-packed zone ∆h is given by

∆h ) hc - hp

(5a)

Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998 549

When t ) ti, hp ) hsi, hc ) hci, and ∆h ) ∆hi so that

∆hi ) hci - hsi

(5b)

in which hsi is the height of the sedimenting interface when sedimentation is complete and the whole of the dispersion is dense-packed. Substituting for ∆h from eq 5a into eq 4a gives the first-order linear differential (Euler’s equation) equation

(6)

which can be solved when the variation in hp with time is known before and after the inflection point. Before the Inflection Point. Following Kynch (1952), the height hp of the position of the boundary between the sedimentation and dense-packed zones during the sedimentation (0 < t < ti) can be assumed to decrease linearly with time t by

hp ) H0 - Vt

(7a)

in which the constant rate V ) -dhp/dt of decrease in hp is given by

H0 - hsi V) ti

(7b)

Substituting this in eq 6 and integrating with the initial condition hc ) H0 when t ) 0 gives the variation with time t in the height of the coalescing interface hc during the initial period 0 < t < ti (ti is the time when sedimentation is complete corresponding to the inflection point):

V∆hi (1 - e-ψit/∆hi) ψi

p∆h ) p(hc - hs) ) hc - H0(1 - 0)

0 < t < ti (8)

This equation can be more simply derived in terms of the height of the dense-packed zone ∆h. Differentiating eq 5a gives the variable separable equation

d∆h ∆h ) V - ψi dt ∆hi

(9)

ψi dhc ) (h - H0 + 0H0) dt p∆hi c

V∆hi (1 - e-ψit/∆hi) ψi

hc ) (1 - 0)H0 + p∆hie-ψi(t-ti)/p∆hi

2.2. Sedimentation Profile. Before the Inflection Point. The height hs of the sedimenting interface for the period 0 < t < ti can be represented by the equation

t2 hs ) v0t - (v0 - vi) 2ti

(10)

0 < t < ti

(11)

Thus hc ) H0 and the coalescence rate ψ is equal to zero initially when t ) 0. When t ) ti, hp ) hsi, hc ) hci, and ∆h ) ∆hi so that

hci ) H0 - Vti + ∆hi

(12a)

since

ψi ) V(1 - e

(16)

since (i) hs ) 0 and dhs/dt ) v0 is the sedimentation velocity of drops when t ) 0 and (ii) dhs/dt ) vi is the sedimentation velocity of drops at the inflection point ti. The relationship between the sedimentation velocity vi and the maximum interfacial coalescence rate ψi can be obtained as follows. Equation 13 for the volume balance for the dispersed phase after the completion of sedimentation (t > ti) can be differentiated to give

v)

(1 - p) ψ p

(17a)

since -dhc/dt ) ψ and dhs/dt ) v and p, 0, and H0 are constant. At the inflection point when t ) ti, this becomes

vi )

[

hs ) v0t - v0 -

dhc ) V(1 - e-ψit/∆hi) dt

-ψiti/∆hi

t > ti (15)

(1 - p) ψi p

(17b)

Substituting this in eq 16 gives

0 < t < ti

which reduces to eq 8 by setting ∆h ) hc - hp. Differentiation of eq 8 gives

ψ)-

(14)

that hc ) hci when t ) ti to give the variation with time t in the height of the coalescing interface hc during the final period t > ti when sedimentation is complete after the inflection point:

since -dhc/dt ) ψ ) ψi∆h/∆hi and -dhp/dt ) V. This can be directly integrated to give

∆h )

(13)

so substituting for hp ) hs in eq 6 thus gives the variable separable equation This integrates with the condition

-

ψi ψi dhc + hc ) h dt ∆hi ∆hi p

hc ) H0 - Vt +

After the Inflection Point. After the completion of sedimentation (t > ti), hp ) hs and a volume balance for the dispersed phase then gives

)

(12b)

]

(1 - p) t2 ψi p 2ti

When t ) ti, hs ) hsi so that

[

hsi ) v0 +

0 < t < ti (18)

]

(1 - p) ti ψi p 2

(19)

After the Inflection Point. The variation with time in the height hs of the sedimenting interface after the inflection point can be obtained from the volume balance equation for the dispersed phase (eq 13) in which hc is given by eq 15 so that

hs ) (1 - 0)H0 - (1 - p)∆hie-ψi(t-ti)/p∆hi

t > ti (20)

550 Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998

2.3. Determination of Parameters. Equations 8, 15, 18, and 20 for the coalescence and sedimentation profiles contain the intermediate parameters V, ∆hi, ψi, ti, v0. These can be determined knowing the fundamental parameters, namely the initial drop diameter φ0 and initial interfacial coalescence time τ0 as follows. The relationship between these parameters is first derived. Substituting for hsi from eq 19 in eq 7b gives

V)

H0 v0 (1 - p)ψi - ti 2 2p

(21)

When t ) ti, hc ) hci and hs ) hsi, then eq 13 for the volume balance for the dispersed phase at the inflection point becomes

hci )

(1 - 0)

H0 -

(1 - p)

p

hsi

(1 - p)

(22a)

Substituting the value of hsi, given by eq 19, then gives

hci )

(1 - 0)

H0 -

(1 - p)

pv0ti 2(1 - p)

-

ψiti 2

into eq 24. If this equation is not satisfied, then a new value of ti must be assumed and the iteration procedure repeated until eq 24 is satisfied. The coalescence and sedimentation profiles can then be predicted using eqs 8, 15, 18, and 20 before and after the inflection point. When the initial drop diameter φ0 and the interfacial coalescence time τ0 are not known, they can be determined from the experimental sedimentation and coalescence profiles. The value of φ0 can be obtained from the measured initial sedimentation velocity v0 (the initial slope of the sedimentation profile) using eq 25. The coalescence time τ0 can then be obtained from the measured value of the interfacial coalescence rate ψi (the maximum slope of the coalescence profile) using eq 23b (the time ti being obtained from eq 24 as described above). The coalescence time τ0 can thus be shown to be given by

(22b)

τ0 )

[2H0p(1 - 0) - {v0p + ψi(1 - p)}ti] 3ψi(1 - p)

Alternatively, the initial interfacial coalescence time τ0 for drops of diameter φ0 can be directly estimated from the equation derived by Jeelani and Hartland (1994)

When t ) ti, ∆h ) ∆hi, ) hci - hsi, then combining eqs 19 and 22b gives

(1 - 0)

ψiti H0 ∆hi ) (1 - p) 2(1 -p) 2p v0ti

τ0 ) (23a)

The value of ψi is then given in terms of τ0 from eq 4c as

ψi )

p[2H0(1 - 0) - v0ti] (1 - p)(3τ0 + ti)

(23b)

The value of hci given by eq 22b must be equal to that given by eq 12a so that

(1 - 0)

H0 -

(1 - p)

pv0ti ψiti ) 2 2(1 - p) H0 - Vti - ψiti/ln(1 - ψi/V) (24)

in which V and ∆hi are given by eqs 21 and 12b. The experimental coalescence and sedimentation profiles can be predicted when the initial drop diameter φ0, interfacial coalescence time τ0 and the physical properties ∆F, Fc, and µc are known for a given dispersion of initial height H0 and holdup 0. The initial sedimentation velocity v0 can be estimated from equations available in the literature. For instance the equation of Kumar and Hartland (1985) applicable for viscous, intermediate, and turbulent flow regimes is given by

[ x

12µc -1 + v0 ) 0.53Fcφ0

1+

0.53Fc∆Fgφ03(1 - 0)

]

108µc2(1 + 4.5600.73) (25)

in which g is the acceleration due to gravity. A value of ti can then be assumed for a given value of τ0, and the value of ψi can be calculated using eq 23b. This can be used to calculate the value of V using eq 21. The values of v0, ti, ψi, and V can then be substituted

(26a)

3πµcr4 4(1 + 2m)fδr2

(26b)

in which f is the force pressing on the draining continuous phase film of radius r and δr is the critical film thickness at rupture. The surface mobility m is the sum of the mobilities due to induced circulation in the adjacent phases and interfacial tension gradient. For immobile surfaces, the value of m ) 0. For a drop of diameter φ0

π f ) φ03∆Fg 6

(27)

and the radius of the draining film r can be estimated from Derjaguin and Kussakov’s (1939) equation modified for a drop at a deformable interface

r ) φ02

x∆Fg 12σ

(28)

in which σ is the interfacial tension. For small drops the radius of the film r for a drop in a monolayer at an interface is the same as that of a single drop. The critical film thickness at rupture δr, can be estimated from the equation of Vrij and Overbeek (1968)

(

)

πr4Am2 δr ) 0.267 6σf

1/7

(29)

in which Am is the Hamaker constant. Thus the value of τ0 can be estimated from eq 26b for immobile surfaces with m ) 0 using the values of f, r, and δr obtained from eqs 27, 28, and 29. Since f, r, and δr depend only on the initial drop diameter φ0 for a given liquid-liquid system, the coalescence time τ0 given by eq 26b also depends only on φ0. In this case the model for predicting the sedimentation and coalescence profiles

Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998 551

only depends on a single parameter, namely the initial drop diameter φ0. 3. Experimental Section 3.1. Liquid-Liquid System. The continuous phase was demineralized water having a density of 996 kg/ m3 and viscosity of 1 mPa‚s. The dispersed phase was 25% (by volume) decane in paraffin oil having a density of 837.3 kg/m3 and viscosity of 1.26 mPa‚s at 20 °C. The interfacial tension was 52.4 mN/m. The viscosities and densities were measured by a Ubbelhode viscometer and a Paar (Type DMA 46) density meter. The interfacial tension was measured by the drop volume method. 3.2. Equipment. A 89 mm internal diameter and 164 mm tall QVF glass beaker served as the mixing vessel. The mixing vessel was provided with four 7 mm wide and 100 mm tall baffles and a 45 mm diameter six (14 mm wide and 12 mm tall) flat bladed turbine impeller driven by a 0.5 kW variable speed electric motor. The baffles and the impeller were made of stainless steel. The settler was a 25 mm internal diameter and 1000 mm tall QVF glass tube blinded at the bottom by a steel flange. 3.3. Procedure. The mixing vessel was filled with known volumes of demineralized water and the organic phase corresponding to the desired dispersed organic phase holdup fraction 0. The liquids were agitated for 2 h to ensure mutual saturation of the phases. This was confirmed by the fact that the separation time was constant in two different batch settling experiments. A typical experiment consisted of filling the mixing vessel with known volumes of the mutually saturated aqueous and organic phases to give the desired holdup fraction 0, locating the impeller at about 20 mm below the interface between the phases, switching on the motor, and mixing the liquids for 60 min at a constant impeller speed of 10.5 s-1. The agitation was then stopped, the dispersion was quickly poured into the settler, and the positions of the sedimenting and coalescing interfaces as a function of time (referred to as sedimentation and coalescence profiles) were recorded until two clear phases were formed. The experiments were repeated by varying the initial height of the dispersion in the settler at different values of the holdup. The impeller speed was found to be much higher than the minimum required for complete mixing. The mixing time was found to be more than sufficient from preliminary experiments resulting in constant separation time. These values of the impeller speed and mixing time are kept constant in all the experiments. The temperature was constant at about 21 °C.

Figure 2. Experimental (symbols) variation in the heights of sedimenting and coalescing interfaces hs and hc with time t for the present data when the initial holdup 0 ) 0.3. Open circles and filled diamonds, squares, triangles, and circles, respectively, correspond to the initial dispersion heights H0 ) 183, 320, 457, 686, and 915 mm. The full lines correspond to the variation predicted by the model using the values of the parameters listed in Table 1. Table 1. Effect of Initial Dispersion Height H0 on Model Parameters for the Present Data When the Initial Dispersed Phase Holdup E0 ) 0.3, Ep ) 0.65, v0 ) 9.2 mm/s, O0 ) 0.847 mm, τ0 ) 24 s, and ψ0 ) 0.0153 mm/s parameter

H0 ) 183 mm

H0 ) 320 mm

H0 ) 457 mm

H0 ) 686 mm

H0 ) 915 mm

ψi, mm/s vi, mm/s V, mm/s hci, mm hsi, mm ∆hi, mm ti, s tf, s

1.16 0.63 3.60 169.9 105.6 64.3 21.5 47.0

1.68 0.90 3.39 284.4 191.5 93.0 37.9 75.0

2.03 1.09 3.24 393.0 280.5 112.5 54.5 100.0

2.36 1.27 3.03 565.1 434.5 130.6 83.0 131.3

2.51 1.35 2.88 730.8 591.9 138.9 112.2 173.8

4. Comparison with Experimental Data The experimental (symbols) variation in the positions of the coalescing and sedimenting interfaces hc and hs with time t compared with that predicted (full lines) by eqs 8 and 15 for 0 < t < ti and eqs 18 and 20 for τi < t < tf is shown in Figure 2 at different initial dispersion heights H0 ) 183, 320, 457, 686, and 915 mm when the initial holdup 0 ) 0.3 for the present data using the values of the model parameters listed in Table 1. Good agreement is obtained which is also true for the data when the initial holdup 0 ) 0.4, 0.5, and 0.6. The agreement is good in spite of the fact that a single constant value of the interfacial coalescence time τ0 ) 24 s is used for all the data at different values of 0.

Figure 3. Experimental (symbols) variation in the heights of sedimenting and coalescing interfaces hs and hc with time t for the present data when the initial dispersion height H0 ) 915 mm. Filled circles, triangles, squares, and diamonds, respectively, correspond to the initial dispersed phase holdups 0 ) 0.3, 0.4, 0.5, and 0.6. The full lines correspond to the variation predicted by the model using the values of the parameters listed in Tables 1-4.

Even better agreement is obtained if different values of τ0 corresponding to different values of 0 are used. The experimental (symbols) variation in hs and hc with time t compared with that predicted (full lines) is shown in Figure 3 at different initial holdups 0 ) 0.3, 0.4, 0.5,

552 Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998 Table 2. Effect of Initial Dispersion Height H0 on Model Parameters for the Present Data When the Initial Dispersed Phase Holdup E0 ) 0.4, Ep ) 0.65, v0 ) 4.25 mm/s, O0 ) 0.68 mm, τ0 ) 24 s, and ψ0 ) 0.0123 mm/s parameter

H0 ) 183 mm

H0 ) 320 mm

H0 ) 457 mm

H0 ) 686 mm

H0 ) 915 mm

ψi, mm/s vi, mm/s V, mm/s hci, mm hsi, mm ∆hi, mm ti, s tf, s

1.33 0.72 2.93 157.8 84.0 73.8 33.8 80

1.79 0.96 2.69 256.3 157.4 98.9 60.4 120

2.00 1.08 2.51 346.2 235.4 110.8 88.4 150

2.10 1.13 2.29 487.1 370.9 116.2 137.9 209

2.08 1.12 2.14 623.7 508.8 114.9 189.6 250

Table 3. Effect of Initial Dispersion Height H0 on Model Parameters for the Present Data When the Initial Dispersed Phase Holdup E0 ) 0.5, Ep ) 0.65, v0 ) 2.86 mm/s, O0 ) 0.71 mm, τ0 ) 24 s, and ψ0 ) 0.0127 mm/s parameter

H0 ) 183 mm

H0 ) 320 mm

H0 ) 457 mm

H0 ) 686 mm

H0 ) 915 mm

ψi, mm/s vi, mm/s V, mm/s hci, mm hsi, mm ∆hi, mm ti, s tf, s

1.74 0.94 4.12 154.2 57.7 96.5 30.4 90

2.35 1.27 3.70 244.6 114.5 130.1 55.5 120

2.61 1.41 3.35 322.5 177.9 144.6 83.4 165

2.64 1.42 2.89 437.8 292.0 145.8 136.5 210

2.57 1.38 2.64 549.8 407.8 142.0 192.3 290

Figure 4. Experimental (symbols) variation in the heights of sedimenting and coalescing interfaces hs and hc with time t for the data of Nadiv and Semiat (1995) for 40% n-heptane in paraffin oil dispersed in water obtained using a 23 mm diameter settler. Open triangles and circles and filled diamonds, squares, triangles, and circles respectively correspond to the initial dispersion heights H0 ) 400, 452, 578, 744, 888, and 1000 mm. The full lines correspond to the variation predicted by the model using the values of the parameters listed in Table 5.

Table 4. Effect of Initial Dispersion Height H0 on Model Parameters for the Present Data When the Initial Dispersed Phase Holdup E0 ) 0.6, Ep ) 0.7, v0 ) 1.8 mm/s, O0 ) 0.73 mm, τ0 ) 24 s, and ψ0 ) 0.014 mm/s parameter

H0 ) 183 mm

H0 ) 320 mm

H0 ) 457 mm

H0 ) 686 mm

H0 ) 915 mm

ψi, mm/s vi, mm/s V, mm/s hci, mm hsi, mm ∆hi, mm ti, s tf, s

2.10 0.90 4.71 148.9 40.8 108.1 30.2 100

2.72 1.17 4.03 226.0 86.0 140.0 58.0 150

2.93 1.26 3.55 288.4 137.6 150.8 90.0 175

2.77 1.19 2.93 374.2 231.6 142.6 155.0 250

2.60 1.12 2.64 459.7 325.8 133.9 223.5 310

and 0.6 when H0 ) 915 mm using the values of the model parameters listed in Tables 1-4. For each value of the dispersed phase holdup 0, the drop diameter φ0 is obtained from the measured initial sedimentation velocity v0 using eq 25. This value of φ0 is only a function of the dispersion holdup 0 and mixing conditions. In Tables 1-4 a constant value of φ0 is used for all the dispersion heights. Only a single value of τ0

(24 s) is used to represent the data in Tables 1-4. This imposes a severe test upon the model; even better agreement could be obtained if the individual values of φ0 and τ0 are used for each experiment. Because the drops in the dense-packed zone are moving toward the coalescing interface, the magnitude of the holdup p is somewhat less than the value of 0.74 corresponding to close-packed spheres. Figure 4 shows the experimental (symbols) variation in hs and hc with time t for the data of Nadiv and Semiat (1995) for 40% n-heptane in paraffin oil dispersed in water obtained using a 23 mm diameter settler at different initial dispersion heights H0 ) 400, 452, 578, 744, 888, and 1000 mm. The full lines correspond to the variation predicted by the present model using the values of the parameters listed in Table 5. A similar agreement is obtained for their larger (65 mm) diameter settler using the values of the parameters listed in Table 6. The agreement is good in spite of the fact that a single constant value of the interfacial coalescence time τ0 ) 35 s is used for both the settlers at different initial holdups and dispersion heights. Better agreement is obtained using the actual values of τ0 corresponding to the different holdups. The continuous water phase

Table 5. Effect of Initial Dispersion Height H0 on Model Parameters for the Data of Nadiv and Semiat (1995)a parameter

H0 ) 400 mm

H0 ) 452 mm

H0 ) 578 mm

H0 ) 744 mm

H0 ) 888 mm

H0 ) 1000 mm

0 v0, mm/s φ0, mm τ0, s ψ0, mm/s ψi, mm/s vi, mm/s V, mm/s hci, mm hsi, mm ∆hi, mm ti, s tf, s

0.408 4.4 0.65 35 0.0080 1.76 0.95 3.04 329.5 187.2 142.3 70.0 140

0.416 4.3 0.66 35 0.0081 1.91 1.03 3.06 364.4 210.4 154.0 79.0 150

0.434 4.1 0.67 35 0.0083 2.25 1.21 3.17 444.8 263.6 181.3 99.3 200

0.440 4.0 0.67 35 0.0083 2.42 1.30 3.00 543.8 348.8 195.0 131.6 240

0.466 3.3 0.64 35 0.0079 2.43 1.31 2.74 601.6 405.5 196.1 176.0 300

0.422 4.2 0.66 35 0.0081 2.35 1.27 2.62 701.3 511.2 190.1 187.0 325

a 40% n-heptane in paraffin oil (density and viscosity are 799 kg/m3 and 4.15 mPa‚s) dispersed in water (density and viscosity are 998 kg/m3 and 0.98 mPa‚s) obtained using a 23 mm diameter settler. The interfacial tension is 58.9 mN/m. The temperature was about 20 °C. The value of p ) 0.65.

Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998 553 Table 6. Effect of Initial Dispersion Height H0 on Model Parameters for the Data of Nadiv and Semiat (1995)a parameter

H0 ) 320 mm

H0 ) 460 mm

H0 ) 600 mm

H0 ) 760 mm

H0 ) 950 mm

0 v0, mm/s φ0, mm τ0, s ψ0, mm/s ψi, mm/s vi, mm/s V, mm/s hci, mm hsi, mm ∆hi, mm ti, s tf, s

0.419 3.92 0.63 35 0.0089 1.41 0.47 2.16 260.0 161.3 98.7 73.5 140

0.435 3.91 0.65 35 0.0093 1.72 0.58 2.25 350.5 229.8 120.7 102.5 150

0.345 5.1 0.61 35 0.0087 1.52 0.51 1.80 471.6 365.5 106.1 130.4 200

0.380 5.3 0.68 35 0.0097 1.93 0.64 2.19 572.1 437.3 134.8 147.2 240

0.396 5.1 0.69 35 0.0098 2.06 0.69 2.22 682.2 537.9 144.3 185.9 300

a 40% n-heptane in paraffin oil dispersed in water obtained using a 65 mm diameter settler at 20 °C. The value of p ) 0.75.

Table 7. Values of τ0 Obtained Using Eq 26b for Single Drop Coalescence Times for Immobile Surfaces (so m ) 0) at Different Holdups Using the Values f, r, and δr Obtained from Eqs 27-29 (a) Present Data parameter

0 ) 0.3

0 ) 0.4

0 ) 0.5

0 ) 0.6

φ0, mm f, µN r, mm δr, µm τ0, s

0.85 0.54 0.035 0.01 64.4

0.68 0.28 0.023 0.0087 29.4

0.70 0.31 0.024 0.0089 33.2

0.73 0.34 0.026 0.0091 37.5

(b) Data of Nadiv and Semiat (1995) for Their 23 mm Diameter Settler parameter

0 ) 0.408

0 ) 0.416

0 ) 0.434

0 ) 0.44

0 ) 0.466

0 ) 0.422

φ0, mm f, µN r, mm δr, µm τ0, s

0.657 0.290 0.0227 0.0087 28.8

0.662 0.297 0.023 0.0087 29.5

0.674 0.313 0.024 0.0088 31.6

0.675 0.315 0.024 0.0088 31.7

0.648 0.278 0.022 0.0086 27.3

0.663 0.298 0.023 0.0087 29.7

(c) Data of Nadiv and Semiat (1995) for Their 65 mm Diameter Settler parameter

0 ) 0.419

0 ) 0.435

0 ) 0.345

0 ) 0.38

0 ) 0.396

φ0, mm f, µN r, mm δr, µm τ0, s

0.633 0.256 0.021 0.0084 25.3

0.658 0.291 0.0228 0.0086 29.1

0.613 0.235 0.0197 0.0082 93.2

0.683 0.326 0.0245 0.0089 33.3

0.695 0.343 0.0254 0.0090 35.4

released when the oil drops coalesce at the disengaging interface flows downwards against the rising oil drops, thereby reducing the dense-packed holdup p. The values of p shown in Tables 5 and 6 for the 23 and 65 mm diameter settlers are 0.65 and 0.75, respectively. This difference could be due to the fact that the interfacial coalescence rate ψi for the smaller settler is necessarily greater than that for the larger settler. The average values of ψi are 2.2 and 1.7 mm/s, the corresponding dense-packed heights ∆hi being 176.5 and 121 mm. The predicted values of the separation time tf in Figures 2-4 are longer than the experimental values listed in Tables 1-5, since the variations with time in the heights of the coalescing and sedimenting interfaces after the inflection point given by eqs 15 and 20 contain an exponential term, and the final value of hc ) hs ) H0(1 - 0) is approached asymptotically.

The value of τ0 can, in principle, be obtained from independent single drop interfacial coalescence time experiments. Alternatively, it can be estimated from the available theoretical equations for single drop coalescence times. The values of τ0 calculated using eq 26b for single drop coalescence times when the surfaces are immobile (so m ) 0) are listed in Table 7a for the present data and Table 7b,c for the data of Nadiv and Semiat (1995). In this equation the values of f, r, and δr, are obtained using eqs 27, 28, and 29 and the value of the Hamaker constant Am ) 2.15 × 10-21 (Jeelani and Hartland, 1994). These values of τ0 are close to the value obtained from the experimental sedimentation and coalescence profiles for the present data and that obtained from the data of Nadiv and Semiat (1995). All systems inevitably become contaminated with surface active agents. These collect at the coalescing interface, rendering it immobile. Drops at the interface are particularly affected as they have spent a considerable time in the dispersion. Furthermore, any binary coalescence occurring in the dense-packed zone reduces the interfacial area per unit volume, thereby increasing the concentration of the surface active contaminants in both our experiments and those of Nadiv and Semiat (1995). Acknowledgment The experiments were carried out by Dr. Donghao Chen. Nomenclature Am ) Hamaker constant, J f ) force pressing on the film, N g ) acceleration due to gravity, m/s2 H0 ) initial height of the dispersion, m hc ) height of the coalescing interface, m hci ) height of the coalescing interface at the inflection point, m hp ) height of the boundary between the dense-packed and sedimentation zones, m hs ) height of the sedimenting interface, m hsi ) height of the sedimenting interface at the inflection point, m ∆h ) height of the dense-packed layer, m ∆hi ) height of the dense-packed layer at the injection point, m t ) time, s ti ) time taken for the completion of sedimentation, s tf ) time taken for the separation of dispersion, s v ) sedimentation velocity of drops, m/s v0 ) initial sedimentation velocity of drops, m/s vi ) sedimentation velocity of drops at the inflection point, m/s V ) rate of decrease in the height of the position of the boundary between sedimentation and dense-packed zones, m/s Greek Symbols δr ) critical film thickness at rupture, m 0 ) initial dispersed phase holdup fraction p ) dispersed phase holdup fraction in the dense-packed layer µc ) viscosity of the continuous phase, Pa‚s µd ) viscosity of the dispersed phase, Pa‚s Fc ) density of the continuous phase, kg/m3 Fd ) density of the dispersed phase, kg/m3 ∆F ) density difference between phases, kg/m3 σ ) interfacial tension, N/m φ ) drop diameter at the coalescing interface, m

554 Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998 φ0 ) initial drop diameter in the dispersion, m τ ) interfacial coalescence time, s τ0 ) initial interfacial coalescence time, s ψ ) interfacial coalescence rate, m/s ψ0 ) initial interfacial coalescence rate, m/s ψi ) interfacial coalescence rate at the inflection point, m/s

Kynch, G. J. A theory of sedimentation. Trans. Faraday Soc. 1952, 48, 166-176.

Literature Cited

Nadiv, C.; Semiat, R. Batch settling of liquid-liquid dispersion. Ind. Eng. Chem. Res. 1995, 34, 2427-2435.

Aris, R.; Amundson, N. R. Mathematical Methods in Chemical Engineering; Prentice-Hall, Inc.: Englewood Cliffs, NJ, 1973; Vol. 2, p 232. Derjaguin, B.; Kussakov, M. Anomalous properties of thin polymolecular films. Acta Physicochim. U.R.S.S. 1939, 10, 25. Hartland, S.; Vohra, D. K. Koaleszenz in vertikalen dichtgepackten Dispersionen. Chem. Ing. Tech. 1978, 50, 673. Hartland, S.; Jeelani, S. A. K. Gravity Settlers. In Liquid-Liquid Extraction Equipment; Godfrey, J. C., Slater, M. J., Eds.; John Wiley & Sons Ltd.: Chichester, England, 1994.

Jeelani, S. A. K.; Hartland, S. Effect of interfacial mobioity on thin film drainage. J. Colloid Interface Sci. 1994, 164, 296-308. Kumar, A.; Hartland, S. Gravity settling in liquid-liquid dispersions. Can. J. Chem. Eng. 1985, 63, 368-376.

Vrij, A.; Overbeek, J. Th. G. Rupture of thin liquid films due to spontaneous fluctuations in thickness. J. Am. Chem. Soc. 1968, 90, 3074.

Received for review August 4, 1997 Revised manuscript received November 21, 1997 Accepted November 24, 1997 IE970545A