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Break-up of a drop in a stirred tank
189
Break-up of a Drop in a Stirred Tank Zerfall eines Tropfens in einem Riihrbehtilter VLADISLAV
HANCIL
and VLADIMfR
ROD
Institute of Chemical Process Fundamentals, (Czechoslovakia) (Received
December
Czechoslovak
Academy
of Sciences,
16502 Prague 6
21, 1987)
Abstract The drop break-up mechanism was studied in a stirred tank containing two immiscible liquids. The daughter drops formed by break-up of a single drop of known size were recorded photographically. From the experiments at constant agitator speed the following results were obtained. There is a critical drop size under which drops do not break up under given conditions. The break-up frequency increases approximately linearly with increase in drop volume. The number of daughter drops, v, is a random variable with a mean v > 2 which increases with the volume of the mother drop. The relative volume of a daughter drop has a p-distribution.
Kurzfassung In einem mechanischen durchrfihrten Gefal3, das zwei nicht mischbare Fliissigkeiten enthielt, wurde der Zerfallmechanismus von individuellen Tropfen untersucht. Die Tochtertropfen, die sich bei dem Zerfall eines isolierten Tropfens bekannter GriiDe bilden, wurden photographisch registriert. Aus Versuchen, die bei konstanter Riihrerdrehzahl durchgefiihrt werden, ergaben sich folgende SchluDfolgerungen: bei vorgegebenen Bedienungen existiert eine kritische TropfengriiDe, unterhalb welcher es zu keinem Zerfall des Tropfens kommt. Die Frequenz des Zerfalles wachst praktisch linear mit dem Tropfenvolumen. Die Anzahl der sich bildener Tochtertropfen, v, ist eine ZufallsgrBBe mit einem Mittelwert grb&r als 2, der mit dem Volmnen des urspriinglichen Muttertropfens ansteigt. Die relativen Volumina der Tochtertropfen weisen eine /I-Verteilung auf.
Introduction
Experimental
The mathematical description of agitated dispersions requires information about the number and sizes of daughter drops formed by break-up of a mother drop of a given size. It has usually been supposed [l-4] that only binary break-up occurs and that the resulting drop volume distribution can be approximated either by a truncated normal or by a uniform distribution. Some authors [S, 61 describe the breakup process as a sequence of a number of binary breakages resulting in a specific daughter drop size distribution. With regard to the lack of experimental data in this field, we have made an attempt to find direct information from observation of the break-up of single drops. In the method used, a drop of known size was inserted in the continuous phase agitated in a vessel at a given paddle impeller speed. The lifetime of this mother drop and the number and sizes of the daughter drops resulting from its break-up were recorded.
Apparatus
0255-2701/88/$3.50
The apparatus for the drop break-up study (Fig. 1) consisted of the following elements: a cylindrical vessel of inner diameter 50 mm and height 80 mm, a paddle-type impeller, a device for organic-phase mother drop formation, a precise controller of impeller rotational speed and a photo-stand (which is not shown). The vessel was made from a precisely cut and ground thick walled glass tube of inner diameter 50 mm, to which a bottom of transparent optical glass was glued with acrylate instant cement. To avoid preferential wetting of the glue by the organic phase, the vessel was treated with chromic acid and 20% oleum before the experiments. The two-blade impeller of diameter 40 mm with a paddle width of 12 mm was located at the centre of the vessel. A pump driving unit (Master-flex C 7546) was used to control the impeller revolutions. The device for mother drop formation consisted of a 200 mm long
Chem. Eng. Process., 23 (1988) 189-193
0
KlsevierSequoia/Printed in The Netherlands
Fig. I. Schema of apparatus: 1, rotational speed controller; 2, paddle mixer; 3, continuous phase; 4, glass tube; 5, optical glass; 6, burette; 7, valve; 8, capillary of inner diameter 1.5 mm; 9, shielding glass tube; 10, Hamilton syringe; 11, Viton tubing.
glass capillary
diameter 1.5 mm connected = 2 mm) to a burette. The free end of the capillary was shielded by a glass tube of inner diameter 10 mm extending 10 mm beyond the open end of the capillary. The working stand enabled reproducible positioning of the~vessel with respect to the fixed mixer, and provided a holder for the device for mother drop formation. The photo-stand ensured precise location of the vessel with respect to the camera (SLR Exakta) while taking photographs. All pictures were taken from the top of the vessel with a flash covered with a ground screen and located under the bottom of the vessel.
with
Viton
of inner
tubing
(I.D.
Procedure
In all experiments, the drops of organic phase, comprising a mixture of toluene and carbon tetrachloride, were formed in the water phase, a 0.001 M solution of Na,PO, containing 0.1% of polyvinyl alcohol. This additive was used to prevent drop coalescence. The density of the organic phase was carefully balanced with the density of the water phase by adjusting the Ccl, concentration to ensure very slow settling of the drops on the bottom of the vessel. The organic phase was coloured with dithizon to increase the contrast in the photographs of the drops. The vessel was carefully washed and rinsed four times with distilled water before starting the experiments. It was then filled with 100 ml of the water phase, located at the working stand and the agitator was switched on. All experiments were carried out at an impeller frequency of 3.2 s-l. The capillary for mother drop formation was fixed to the working stand with the open end upwards and was then filled with the water phase by opening the burette valve (Fig. 1). By means of a suitable type (l&50 ~1) of Hamilton syringe, a mother drop of a chosen size was formed at the syringe needle inserted in the capillary near its open end. The syringe was removed and the shielded capillary with the drop inside was immersed in the agitated water phase in the vessel. The drop was then displaced from the capillary through the shielding tube into the vessel by a controlled water-phase
Fig. 2. Photographs of daughter drops formed from a 10 ~1 mother drop.
flow from the burette. The time was measured from the instant the drop had left the shielding up to its visually determined break-up. As soon as the drop had broken up the mixing was switched off. When the daughter drop had settled on the bottom in a cluster, the vessel was transferred to the photo-stand and a picture was taken. Four of the pictures of the clusters of daughter drops are shown in Fig. 2.
Data treatment Each experiment yielded the following data: volume and lifetime of the mother drop and a photographic record of the daughter drops. The number and sizes of the daughter drops were determined from the picture of the record projected on the screen of a microfilm reader. Because of the nearly balanced densities of the phases, the drops were strictly spherical, so that their volumes could be calculated from their diameters, and the volume balance checked. The volumes of the daughter drops from each set of experiments with the mother drop of a given size were combined in six histogram bins. The average lifetime and the average number of the daughter drops in each set were calculated as arithmetic means of the corresponding values. It can be shown that the average number of the daughter drops per breakage, v, equals the reciprocal value of the mean of the distribution of the relative daughter drop volume, B(u/Q,). The volume balance for S experiments and B histogram bins can be written as s SVO =
C i=l
B C k=l
B
ni,kvk
=
C k=l
(1)
nkvk
where i relates to the experiment
and k to the bin.
191
Expressing
the average number
of daughter
drops
as v=(l/s)
5 k=
(2)
nk 1
and substituting
for S in eqn. (2) from eqn. ( l), we get
B vO v=-
c k=l
5 k=l
=
nk
nkVk
[,$,: Q)]-’
The slight difference obtained in the values of v calculated from eqns. (2) and (3) was caused by some information loss by data binning.
Results Experimental data on mother drop lifetimes and numbers of daughter drops are summarized in Table 1. The data in Table 1 reveal the dependences of the mother drop lifetime and of the number of daughter drops on the mother drop size. Since in the repeated experiments with 20 drops of volume 0.5 ~1 no more than two break-ups occurred in 600 s, this drop volume is approximately the critical one for break-up under the experimental conditions. The distributions of the relative volume of the daughter drops with respect to the mother drop volume are illustrated in the form of histograms in Fig. 3. The bin size in the histograms was chosen so as to minimize the uncertainties in determinations of the extreme drop sizes. It is evident from the data in Table 1 that a drop breaks up into a random number of daughter drops with a mean value v > 2 depending on the mother
TABLE
drop size. The experimentally found average numbers of daughter drops for various diameters of the mother drop are shown in Fig. 4 by points. The dependence was correlated by the empirical equation v =2+p(u&,,-
I)4
(4)
which fulfils the requirement v = 2 for a, = a,,. This empirical equation with its two adjustable parameters is supposed to be sufficiently flexible to describe the behaviour of any system. The parameters of the equation, p = 0.47 and q = 1S, were estimated by the least squares method, with the critical diameter a,, equal to 0.98 mm, which corresponds to a 0.5 ~1 drop. The correlation is represented in Fig. 4 by the solid line. If we want to describe the daughter drop size distridution obtained, we have to make some assumptions about the type of the distribution. The assumed form of the distribution should respect volume balance of the mother drop and the resulting daughter drops. This balance requires the mean relative volume of the daughter drops, P, to be equal to the reciprocal value of the average number of daughter drops per breakup, v-‘. This requirement is met by a p-distribution for the relative volume of the daughter drops with parameters 1 and v - 1: /+/vo)
= (v - I)( 1 - v/vo)“- *
(5)
and the mean p = l/v. The value of the parameter v was predicted for the known mother drop diameter from eqn. (4) and the distribution of the daughter drop calculated according to eqn. (5). The calculated distributions shown in Fig. 3 as solid lines are in excellent agreement with those found experimentally. The frequency of drop break-up, obtained as the reciprocal value of the average lifetime of the mother drop, is plotted against the drop volume in Fig. 5. The frequency of drop break-up increases from zero, corresponding to the drop of critical size, approximately linearly with the drop volume.
1. Lifetimes of mother drops and numbers of daughter drops
Mother drop volume (mm3)
No. of experiments
Lifetime T (s)
Average 39 29
106 40
20 10 5 3 2” 1” 0.5”
46 62 45 64 3 x 20 3 x 20 3 x 20
No. of daughter drops Std. dev.
Average
Std. dev.
2.70 3.39
2.7 2.5
6.0
6.6 _
4.39 11.11 13.33 26.7 120.0 10’ 104
2.0 5.2 7.6 6.3 _ _ _
3.7 3.7 2.4 2.5 -
2.0 1.9 0.7 0.7 _ _ _
-
“In these experiments 20 mother drops were prepared in the capillary and all of them were displaced into the mixed vessel. After 2, 5, and 10 mia, the cluster of the remaining drops with the daughter drops formed was photographed and, assuming Y = 2, 5 was estimated.
192
1
21 l-l
I
v.= 34
‘r \ -
1
0 Iri;i* 0 (4
a5
x (b)
0 3
2
1
0L 0
a5
x
1
(4 Fig. 3. Distributions of the relative volume of daughter drops for various mother drop volumes: the bins are experimental; the solid line is calculated from eqn. (5) with Y given by eqn. (4).
0.3 f,s” 0.2
.
0.1 0
0
2
a./a,, 4
Fig. 4. Average number of daughter drops versus mother drop dimensionless diameter.
Discussion
and conclusions
It was found that the average number of daughter drops is an increasing function of the relative mother drop size, uo/v,,. It could not be determined by visual observation whether the break-up process leading to a number of daughter drops greater than 2 is in fact a sequence of binary breakages, or whether the daughter drops are formed simultaneously. The first explanation could be accepted only with the assumption that the sequence of the binary breakages, triggered by the first one, is an extremely rapid one and hence the difference between the two descriptions of the break-up process is of no practical importance. The relative volume of the daughter drop can be approximated well by the /I-distribution (5) with the
.
‘/I
-0
10
20
30 v, ,PI
Fig. 5. Frequency of break-up
mother drop
mean decreasing with a,/a,,. It is worth mentioning that this is a single-parameter distribution, so that the whole family of distributions can be described by means of two constants in eqn. (4) correlating v with a,/a,,. In a particular case, the P-distribution with an integer v represents the result of break-up modelled by v - 1 cuts of the mother drop elongated in a cylinder. It is interesting to note that the preference for equal-size drop formation at binary break-up of drops of nearly critical size, usually assumed, was not found. The dependence of the number of daughter drops on the mother drop size can be explained by a spatial distribution of turbulence in a mixed vessel. It is well known that the turbulence intensity and energy dissipation rapidly decrease with the distance from the
193 impeller. In order that the drop can break up, it has to enter a region in the vessel where the energy dissipation is higher (the stream from the impeller) than a critical value necessary for its binary break-up. The smaller the drop, the smaller is the volume in which the drop can break up, and so is the opportunity for its break-up. As the break-up of a given mother drop into a high number of daughter drops requires higher energy than that required for a low number, the small drops have less chance of breaking up into more daughter drops than the large ones. The large drops have a good chance of entering the region with a turbulence intensity high enough to split them into several drops. This reasoning is supported by the dependence of the variances of v and t on mother drop size. Our experiments were performed at a relatively low energy input and for a high ratio of impeller to tank diameter. It can be expected that the values of parameters in eqn. (4) will depend on the experimental arrangement and on the system and that the values found are specific for our experimental conditions. The physical properties of the system affect a,, significantly and it has been stated by others that a,, is a function of the Weber criterion. On the other hand, eqn. (5) seems to be of general validity.
Nomenclature a B
f
n S
diameter of a drop, m number of histogram bins frequency of break-up, s-’ number of drops number of experiments
V X
p P V 7
volume of a drop, m3 = v/vo, relative volume of daughter to mother drop
drop related
probability density mean volunie of daughter drop related to volume of mother drop average number of daughter drops per breakage lifetime of mother drop, s
Subscripts cr t 0
critical related related related
value to ith experiment to kth bin to mother drop
References K. J. Valentas, 0. Bilous and N. R. Amundson, Analysis of breakage in dispersed phase systems, Ind. Eng. Chem., Fundam., J(1966) 271. C. A. Coulaloglou and L. L. Tavlarides, Description of interaction processin agitated liquid-liquid dispersions, Chem. Eng. Sci., 32 (1977) 1289. G. Narsimhan, J. P. Gupta and D. Ramkrishna, A model for transitional breakage probability of droplets in agitated lean liquid-liquid dispersions, Chem. Eng. Sci., 34 ( 1979) 257. R. K. Bajpaj, D. Ramkrishna and A. Prokop, A coalescence redispersion model for drop size distribution in an agitated vessel, Chem. fing. Sci., 31(1976) 913. M. Molag, G. E. H. Joosten and A. H. Drinkenburg, Droplet breakup and distribution in stirred immiscible two-liquid systems, Ind. Eng. Chem., Fun&n., 19 (1980) 275. M. Konno, Y. Matsunaga, K. Arai and S. Saito, Simulation model for breakup process in an agitated tank, J. Chem. Eng. Jpn., 13 (1980) 67.
Break-up of a Drop in a Stirred Tank Zerfall eines Tropfens in einem Riihrbehtilter VLADISLAV
HANCIL
and VLADIMfR
ROD
Institute of Chemical Process Fundamentals, (Czechoslovakia) (Received
December
Czechoslovak
Academy
of Sciences,
16502 Prague 6
21, 1987)
Abstract The drop break-up mechanism was studied in a stirred tank containing two immiscible liquids. The daughter drops formed by break-up of a single drop of known size were recorded photographically. From the experiments at constant agitator speed the following results were obtained. There is a critical drop size under which drops do not break up under given conditions. The break-up frequency increases approximately linearly with increase in drop volume. The number of daughter drops, v, is a random variable with a mean v > 2 which increases with the volume of the mother drop. The relative volume of a daughter drop has a p-distribution.
Kurzfassung In einem mechanischen durchrfihrten Gefal3, das zwei nicht mischbare Fliissigkeiten enthielt, wurde der Zerfallmechanismus von individuellen Tropfen untersucht. Die Tochtertropfen, die sich bei dem Zerfall eines isolierten Tropfens bekannter GriiDe bilden, wurden photographisch registriert. Aus Versuchen, die bei konstanter Riihrerdrehzahl durchgefiihrt werden, ergaben sich folgende SchluDfolgerungen: bei vorgegebenen Bedienungen existiert eine kritische TropfengriiDe, unterhalb welcher es zu keinem Zerfall des Tropfens kommt. Die Frequenz des Zerfalles wachst praktisch linear mit dem Tropfenvolumen. Die Anzahl der sich bildener Tochtertropfen, v, ist eine ZufallsgrBBe mit einem Mittelwert grb&r als 2, der mit dem Volmnen des urspriinglichen Muttertropfens ansteigt. Die relativen Volumina der Tochtertropfen weisen eine /I-Verteilung auf.
Introduction
Experimental
The mathematical description of agitated dispersions requires information about the number and sizes of daughter drops formed by break-up of a mother drop of a given size. It has usually been supposed [l-4] that only binary break-up occurs and that the resulting drop volume distribution can be approximated either by a truncated normal or by a uniform distribution. Some authors [S, 61 describe the breakup process as a sequence of a number of binary breakages resulting in a specific daughter drop size distribution. With regard to the lack of experimental data in this field, we have made an attempt to find direct information from observation of the break-up of single drops. In the method used, a drop of known size was inserted in the continuous phase agitated in a vessel at a given paddle impeller speed. The lifetime of this mother drop and the number and sizes of the daughter drops resulting from its break-up were recorded.
Apparatus
0255-2701/88/$3.50
The apparatus for the drop break-up study (Fig. 1) consisted of the following elements: a cylindrical vessel of inner diameter 50 mm and height 80 mm, a paddle-type impeller, a device for organic-phase mother drop formation, a precise controller of impeller rotational speed and a photo-stand (which is not shown). The vessel was made from a precisely cut and ground thick walled glass tube of inner diameter 50 mm, to which a bottom of transparent optical glass was glued with acrylate instant cement. To avoid preferential wetting of the glue by the organic phase, the vessel was treated with chromic acid and 20% oleum before the experiments. The two-blade impeller of diameter 40 mm with a paddle width of 12 mm was located at the centre of the vessel. A pump driving unit (Master-flex C 7546) was used to control the impeller revolutions. The device for mother drop formation consisted of a 200 mm long
Chem. Eng. Process., 23 (1988) 189-193
0
KlsevierSequoia/Printed in The Netherlands
Fig. I. Schema of apparatus: 1, rotational speed controller; 2, paddle mixer; 3, continuous phase; 4, glass tube; 5, optical glass; 6, burette; 7, valve; 8, capillary of inner diameter 1.5 mm; 9, shielding glass tube; 10, Hamilton syringe; 11, Viton tubing.
glass capillary
diameter 1.5 mm connected = 2 mm) to a burette. The free end of the capillary was shielded by a glass tube of inner diameter 10 mm extending 10 mm beyond the open end of the capillary. The working stand enabled reproducible positioning of the~vessel with respect to the fixed mixer, and provided a holder for the device for mother drop formation. The photo-stand ensured precise location of the vessel with respect to the camera (SLR Exakta) while taking photographs. All pictures were taken from the top of the vessel with a flash covered with a ground screen and located under the bottom of the vessel.
with
Viton
of inner
tubing
(I.D.
Procedure
In all experiments, the drops of organic phase, comprising a mixture of toluene and carbon tetrachloride, were formed in the water phase, a 0.001 M solution of Na,PO, containing 0.1% of polyvinyl alcohol. This additive was used to prevent drop coalescence. The density of the organic phase was carefully balanced with the density of the water phase by adjusting the Ccl, concentration to ensure very slow settling of the drops on the bottom of the vessel. The organic phase was coloured with dithizon to increase the contrast in the photographs of the drops. The vessel was carefully washed and rinsed four times with distilled water before starting the experiments. It was then filled with 100 ml of the water phase, located at the working stand and the agitator was switched on. All experiments were carried out at an impeller frequency of 3.2 s-l. The capillary for mother drop formation was fixed to the working stand with the open end upwards and was then filled with the water phase by opening the burette valve (Fig. 1). By means of a suitable type (l&50 ~1) of Hamilton syringe, a mother drop of a chosen size was formed at the syringe needle inserted in the capillary near its open end. The syringe was removed and the shielded capillary with the drop inside was immersed in the agitated water phase in the vessel. The drop was then displaced from the capillary through the shielding tube into the vessel by a controlled water-phase
Fig. 2. Photographs of daughter drops formed from a 10 ~1 mother drop.
flow from the burette. The time was measured from the instant the drop had left the shielding up to its visually determined break-up. As soon as the drop had broken up the mixing was switched off. When the daughter drop had settled on the bottom in a cluster, the vessel was transferred to the photo-stand and a picture was taken. Four of the pictures of the clusters of daughter drops are shown in Fig. 2.
Data treatment Each experiment yielded the following data: volume and lifetime of the mother drop and a photographic record of the daughter drops. The number and sizes of the daughter drops were determined from the picture of the record projected on the screen of a microfilm reader. Because of the nearly balanced densities of the phases, the drops were strictly spherical, so that their volumes could be calculated from their diameters, and the volume balance checked. The volumes of the daughter drops from each set of experiments with the mother drop of a given size were combined in six histogram bins. The average lifetime and the average number of the daughter drops in each set were calculated as arithmetic means of the corresponding values. It can be shown that the average number of the daughter drops per breakage, v, equals the reciprocal value of the mean of the distribution of the relative daughter drop volume, B(u/Q,). The volume balance for S experiments and B histogram bins can be written as s SVO =
C i=l
B C k=l
B
ni,kvk
=
C k=l
(1)
nkvk
where i relates to the experiment
and k to the bin.
191
Expressing
the average number
of daughter
drops
as v=(l/s)
5 k=
(2)
nk 1
and substituting
for S in eqn. (2) from eqn. ( l), we get
B vO v=-
c k=l
5 k=l
=
nk
nkVk
[,$,: Q)]-’
The slight difference obtained in the values of v calculated from eqns. (2) and (3) was caused by some information loss by data binning.
Results Experimental data on mother drop lifetimes and numbers of daughter drops are summarized in Table 1. The data in Table 1 reveal the dependences of the mother drop lifetime and of the number of daughter drops on the mother drop size. Since in the repeated experiments with 20 drops of volume 0.5 ~1 no more than two break-ups occurred in 600 s, this drop volume is approximately the critical one for break-up under the experimental conditions. The distributions of the relative volume of the daughter drops with respect to the mother drop volume are illustrated in the form of histograms in Fig. 3. The bin size in the histograms was chosen so as to minimize the uncertainties in determinations of the extreme drop sizes. It is evident from the data in Table 1 that a drop breaks up into a random number of daughter drops with a mean value v > 2 depending on the mother
TABLE
drop size. The experimentally found average numbers of daughter drops for various diameters of the mother drop are shown in Fig. 4 by points. The dependence was correlated by the empirical equation v =2+p(u&,,-
I)4
(4)
which fulfils the requirement v = 2 for a, = a,,. This empirical equation with its two adjustable parameters is supposed to be sufficiently flexible to describe the behaviour of any system. The parameters of the equation, p = 0.47 and q = 1S, were estimated by the least squares method, with the critical diameter a,, equal to 0.98 mm, which corresponds to a 0.5 ~1 drop. The correlation is represented in Fig. 4 by the solid line. If we want to describe the daughter drop size distridution obtained, we have to make some assumptions about the type of the distribution. The assumed form of the distribution should respect volume balance of the mother drop and the resulting daughter drops. This balance requires the mean relative volume of the daughter drops, P, to be equal to the reciprocal value of the average number of daughter drops per breakup, v-‘. This requirement is met by a p-distribution for the relative volume of the daughter drops with parameters 1 and v - 1: /+/vo)
= (v - I)( 1 - v/vo)“- *
(5)
and the mean p = l/v. The value of the parameter v was predicted for the known mother drop diameter from eqn. (4) and the distribution of the daughter drop calculated according to eqn. (5). The calculated distributions shown in Fig. 3 as solid lines are in excellent agreement with those found experimentally. The frequency of drop break-up, obtained as the reciprocal value of the average lifetime of the mother drop, is plotted against the drop volume in Fig. 5. The frequency of drop break-up increases from zero, corresponding to the drop of critical size, approximately linearly with the drop volume.
1. Lifetimes of mother drops and numbers of daughter drops
Mother drop volume (mm3)
No. of experiments
Lifetime T (s)
Average 39 29
106 40
20 10 5 3 2” 1” 0.5”
46 62 45 64 3 x 20 3 x 20 3 x 20
No. of daughter drops Std. dev.
Average
Std. dev.
2.70 3.39
2.7 2.5
6.0
6.6 _
4.39 11.11 13.33 26.7 120.0 10’ 104
2.0 5.2 7.6 6.3 _ _ _
3.7 3.7 2.4 2.5 -
2.0 1.9 0.7 0.7 _ _ _
-
“In these experiments 20 mother drops were prepared in the capillary and all of them were displaced into the mixed vessel. After 2, 5, and 10 mia, the cluster of the remaining drops with the daughter drops formed was photographed and, assuming Y = 2, 5 was estimated.
192
1
21 l-l
I
v.= 34
‘r \ -
1
0 Iri;i* 0 (4
a5
x (b)
0 3
2
1
0L 0
a5
x
1
(4 Fig. 3. Distributions of the relative volume of daughter drops for various mother drop volumes: the bins are experimental; the solid line is calculated from eqn. (5) with Y given by eqn. (4).
0.3 f,s” 0.2
.
0.1 0
0
2
a./a,, 4
Fig. 4. Average number of daughter drops versus mother drop dimensionless diameter.
Discussion
and conclusions
It was found that the average number of daughter drops is an increasing function of the relative mother drop size, uo/v,,. It could not be determined by visual observation whether the break-up process leading to a number of daughter drops greater than 2 is in fact a sequence of binary breakages, or whether the daughter drops are formed simultaneously. The first explanation could be accepted only with the assumption that the sequence of the binary breakages, triggered by the first one, is an extremely rapid one and hence the difference between the two descriptions of the break-up process is of no practical importance. The relative volume of the daughter drop can be approximated well by the /I-distribution (5) with the
.
‘/I
-0
10
20
30 v, ,PI
Fig. 5. Frequency of break-up
mother drop
mean decreasing with a,/a,,. It is worth mentioning that this is a single-parameter distribution, so that the whole family of distributions can be described by means of two constants in eqn. (4) correlating v with a,/a,,. In a particular case, the P-distribution with an integer v represents the result of break-up modelled by v - 1 cuts of the mother drop elongated in a cylinder. It is interesting to note that the preference for equal-size drop formation at binary break-up of drops of nearly critical size, usually assumed, was not found. The dependence of the number of daughter drops on the mother drop size can be explained by a spatial distribution of turbulence in a mixed vessel. It is well known that the turbulence intensity and energy dissipation rapidly decrease with the distance from the
193 impeller. In order that the drop can break up, it has to enter a region in the vessel where the energy dissipation is higher (the stream from the impeller) than a critical value necessary for its binary break-up. The smaller the drop, the smaller is the volume in which the drop can break up, and so is the opportunity for its break-up. As the break-up of a given mother drop into a high number of daughter drops requires higher energy than that required for a low number, the small drops have less chance of breaking up into more daughter drops than the large ones. The large drops have a good chance of entering the region with a turbulence intensity high enough to split them into several drops. This reasoning is supported by the dependence of the variances of v and t on mother drop size. Our experiments were performed at a relatively low energy input and for a high ratio of impeller to tank diameter. It can be expected that the values of parameters in eqn. (4) will depend on the experimental arrangement and on the system and that the values found are specific for our experimental conditions. The physical properties of the system affect a,, significantly and it has been stated by others that a,, is a function of the Weber criterion. On the other hand, eqn. (5) seems to be of general validity.
Nomenclature a B
f
n S
diameter of a drop, m number of histogram bins frequency of break-up, s-’ number of drops number of experiments
V X
p P V 7
volume of a drop, m3 = v/vo, relative volume of daughter to mother drop
drop related
probability density mean volunie of daughter drop related to volume of mother drop average number of daughter drops per breakage lifetime of mother drop, s
Subscripts cr t 0
critical related related related
value to ith experiment to kth bin to mother drop
References K. J. Valentas, 0. Bilous and N. R. Amundson, Analysis of breakage in dispersed phase systems, Ind. Eng. Chem., Fundam., J(1966) 271. C. A. Coulaloglou and L. L. Tavlarides, Description of interaction processin agitated liquid-liquid dispersions, Chem. Eng. Sci., 32 (1977) 1289. G. Narsimhan, J. P. Gupta and D. Ramkrishna, A model for transitional breakage probability of droplets in agitated lean liquid-liquid dispersions, Chem. Eng. Sci., 34 ( 1979) 257. R. K. Bajpaj, D. Ramkrishna and A. Prokop, A coalescence redispersion model for drop size distribution in an agitated vessel, Chem. fing. Sci., 31(1976) 913. M. Molag, G. E. H. Joosten and A. H. Drinkenburg, Droplet breakup and distribution in stirred immiscible two-liquid systems, Ind. Eng. Chem., Fun&n., 19 (1980) 275. M. Konno, Y. Matsunaga, K. Arai and S. Saito, Simulation model for breakup process in an agitated tank, J. Chem. Eng. Jpn., 13 (1980) 67.