Experimental and predicted performance of pulsatile fluidic pumps

Minka, C.; Myers, A. L. AIChE J. 1973, 19, 453. Morbidelli, M.; Storti, G.; Carra, S.; Niederjaufner, G.; Pontoglio, A. Chem. Eng. Sci. 1984, 39, 383...
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I n d . Eng. Chem. Res. 1987, 26, 2258-2263

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where two adjustable parameters appear. Ragistry No. 3-H3CC6H4CH3,108-38-3; 4-H3CC6H4CH3, 106-42-3;CeH5CH3, 108-88-3; C~H~CH(CH~)Z, 98-82-8. Literature Cited Broughton, D. B.; Neuzil, P. W.; Pharis, J. M.; Brearly, C. S. Chem. Eng. Prog. 1970, 66, 70. De Vault, D. J. Am. Chem. SOC. 1943,65, 532. Hildebrand, J. H.; Prausnitz, J. M.; Scott, R. L. Regular and Related Solutions; Van Nostrand Rheinhold: New York, 1970; pp 107-109. Minka, C.; Myers, A. L. AIChE J . 1973, 19, 453. Morbidelli, M.; Storti, G.; Carrl, S.; Niederjaufner, G.; Pontoglio, A. Chem. Eng. Sci. 1984, 39, 383. Morbidelli, M.; Santacesaria, E.; Storti, G.; Carri, S. Ind. Eng. Chem. Process Des. Dev. 1985a, 24, 83. Morbidelli, M.; Storti, G.; Carrl, S.; Niederjaufner, G.; Pontoglio, A. Chem. Eng. Sci. 1985b, 40, 1155. Morbidelli, M.; Storti, G.; C a d , S. Ind. Eng. Chem. Fundam. 1986a, 25, 89.

Morbidelli, M.; Paludetto, R.; Storti, G.; Carrl, S. "Adsorption from Chloroaromatic Vapor Mixtures on X Zeolites: Experimental Results and Interpretation of Equilibrium Data", paper presented at the Second International Conference on Fundamentals of Adsorption, Santa Barbara, CA, May 4-7, 198613. Myers, A. L. AIChE J . 1983,29, 691. Myers, A. L. "Molecular Thermodynamics of Adsorption of Gas and Liquids Mixtures", paper presented at the Second International Conference on Fundamentals of Adsorption, Santa Barbara, CA, May 4-7, 1986. Myers, A. L.; Prausnitz, J. M. AIChE J . 1965, 11, 121. Paludetto, R.; Gamba, G.; Storti, G.; Morbidelli, M. Chem. Eng. Sci. 1987, in press. Santacesaria, E.; Gelosa, D.; Danise, P.; Carri, S. Ind. Eng. Chem. Process Des. Dev. 1985, 24, 78. Sircar, S.; Myers, A. L. AIChE J. 1971, 17, 186. Storti, G.; Santacesaria, E.; Morbidelli, M.; Carrl, S. Ind. Eng. Chem. Process Des. Dev. 1985,24, 89.

Received for review September 11, 1986 Revised manuscript received July 8, 1987 Accepted July 22, 1987

Experimental and Predicted Performance of Pulsatile Fluidic Pumps James G. Morgan* and William D. Holland Fuel Recycle Division, Oak Ridge National Laboratory,' Oak Ridge, Tennessee 37831

As part of the Consolidated Fuel Reprocessing Program (CFRP) a t the Oak Ridge National Laboratory (ORNL), top-loading and bottom-loading pulsatile fluidic pumps were tested for possible use in a nuclear fuel reprocessing facility. A procedure was developed to obtain a calibration curve for a particular pump. A predictive model based on the calibration curve was found to adequately predict pump performance over the design range of interest. The model takes into consideration system resistance, density and viscosity of the pumped fluid, and air motivation pressure. 1. Introduction Fluidic pumps have been under development for some time and have found application in nuclear fuel reprocessing operations, especially in the United Kingdom (Tippetts, 1978, 1979). Having no moving parts, these pumps are maintenance free and do not dilute or heat the pumped fluid as do steam jets. Air is not entrained in the fluid as it is in air lifts. The pumps might be used in nuclear reprocessing plants as product tank mixers, as transfer pumps in accountability tanks, and as supplier pumps to metering devices that feed contador banks. The flow requirements for these pumps vary from about 8 to 300 L/h. The purpose of this investigation was to demonstrate prototypic fluidic pump applications and to develop a model to predict pump performance under different conditions. 2. Fluidic P u m p Description

Two types of pulsatile fluidic pumps, top-loading and bottom-loading, were studied. Both types operate submerged in a tank of the liquid to be pumped with only a discharge line and an air line leading to the surroundings. The top-loading pump is easier to fabricate and is shown in Figure 1. At the beginning of the pump stroke, the chamber of the pump is full of liquid, and pressurized air is forced into the chamber through a three-way control valve. The liquid stream passes through the nozzle and 'Operated by Martin Marietta Energy Systems, Inc., for t h e

U S . Department of Energy. 0888-5885/87/2626-2258$01.50/0

is directed into the diffuser and up the discharge tube. The amount of liquid flowing into or out of the refill port depends on the resistance to flow in the discharge system. At the end of the pumping stroke, when the level of liquid in the chamber has fallen to the bottom of the discharge line, the air in'the chamber is exhausted to the atmosphere. The refill cycle begins with liquid entering the chamber through the r e f i port. A column of liquid in the discharge tube above the diffuser also falls back into the chamber when the air pressure is released. The top-loading pump used in these tests had a diameter of 6 in. and was 15 in. in height. A bottom-loading pump is shown in Figure 2. The pump and refill cycles are similar to the top-loading pump, except the pump chamber refills through a port located at the bottom. This type of design allows the host tank containing the pump to be almost completely emptied. A prototypic bottom-loading product tank pump was made of 4-in. schedule 40 pipe and was 4 f t in height. These dimensions allowed the pump to fit into a critically safe tank and also allowed a longer, more easily controlled pump time. A reverse-flow diverter (RFD) is a generic name for a device that redirects flow in one of its inlets. The design of the nozzle-diffusers used in this investigation was based on earlier work by Smith and Counce (1984a) who characterized flat-walled, venturi-like RFDs and later investigated &symmetric RFDs (198413). They found that the characteristic curves for RFDs were similar over a range of nozzle-diffuser throat diameters of 0.37-0.73 in. Nozzle-included angles ranged from 14' to 26', and diffuser angles ranged from 4' to 8'. The gap between the 0 1987 American Chemical Society

Ind. Eng. Chem. Res., Vol. 26, No. 11, 1987 2259 AIR IN

LIQUID

+ I

n

.IQUID .EVEL

Figure 1. Top-loading pump.

nozzle and diffuser was found to have little effect on performance between 0.5 and 1.5 gap ratio. The gap ratio is the gap width divided by the nozzle diameter. The nozzle-diffuser dimensions for the current investigation are given in Figure 3. These dimensions were chosen to conform to the above criteria and also fit conveniently into the flow system using standard size tubing. 2.1. Pump Calibration Method. Calibration data were obtained by immersing the pump in a tank filled with water and measuring the delivered volume during a complete pumping cycle as a function of motivation pressure and system resistance to flow. The system resistance was varied by incrementally closing a valve mounted in the output line. The pump output pressure was measured with a pressure cell located just above the pump. When delivered volume is plotted against pump output pressure, a series of curves result, each curve representing a different motivation pressure (see Figure 4). These data were obtained on the bottom-loadingpump with an 8 f t refill head. Following Smith and Counce (1984a), the data were normalized to obtain a single calibration curve (see Figure 5). The calibration curve is a plot of Q versus P, where

Q = Qo/Qi

DIFFUSEF

)ZZLE

P ~ R T

Figure 2. Bottom-loading pump.

Nozzle (cml Di

Dn Toploading

Bottom-loading

0.94 2.79 0.89 a

Diffuser (cm)

L,

h

Do

3.81

0.94 0.89

1.68

a

b

8.71 1.43 4.06

. N O U ~ ~is trumpet-shaped, decreasing from 1.80cm dim-to 0.89cm diam over a 0.89cm length, a8 shown in FQwe 2.

Figure 3. Nozzle-diffuser dimensions.

I

I

MOTlVdTlON PRESSURE

(1)

and

p = (Pz- P3)/(P1- P3)

(2) The quantity Q is called the “split” because the pumped volume, Qi, is split into two parts: Q,, which reaches the delivery piping system, and (Qi - Q,, which flows back through the refill port to the supply tank. A similar calibration curve for the top-loading pump is presented in Figure 6. Calibration curves are used with the predictive model discussed later. 2.2. Top-Loading Pump Demonstration. The first fluidic pump tested was the small top-loading pump with the RFD mounted in the discharge tube at the top of the pumping chamber. Conductivity probes near the top and

I

I

0

10

O 20 PUMP O U T P U T PRESSURE lprigl

Figure 4. Bottom-loading pump output.

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2260 Ind. Eng. Chem. Res., Vol. 26, No. 11, 1987 11

MOTIVATION PRESSURE

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25~1s

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-0 6

Q

O5I

04

il

r- 1 I

11

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4 6 PUMP TIME ( 5 )

10

8

Figure 7. Pump times, top-loading pump.

e -

01-

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1 2

6o

t

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e w

30 f t

50

20 f l

MOTIVATION 0 32 prig

ERUS S ERP e l

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-

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0-0

I 30

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DELIVERED VOLUME IL cycle1

Figure 8. Top-loading pump output.

7 O

02

;

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P

06

i

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1 1 2-1n

l

diam P

A

T

Figure 6. Calibration curve, top-loading pump.

bottom of the pump used to signal when the refill and pump cycles had ended. This could be confirmed visually by observing the refill liquid advancing up the vented plastic air line and, at the end of the pump stroke, noting air bubbles emerging from the nozzle-diffuser port. The instrument control setup allowed the pump to be operated automatically from the probe signals or by timers set for specific pump and refill times. The effect of motivation pressure on the time to empty the pump chamber is shown in Figure 7. The pump was tested while immersed in a 55-gal drum with about a 1-ft refill head. It was operated to deliver its output at different heights through 3/8-in.-i.d. tubing. The volume of liquid in the pump chamber was 5.8 L, and the amount of fallback at the 30-ft height was 0.6 L. Results are shown in Figure 8, where delivered volume is plotted against motivation pressure. The curves represent

6 25 i t

L-w 11 F03

u 4 1 1 FO1

Figure 9. Fluidic pump test.

the delivered volume of water at lo-, 20-, and 30-ft elevations, respectively. When a zinc bromide solution (specific gravity = 2.12; viscosity = 8.2 cP) was used as the pumped fluid, the pumping rate decreased compared to water. 2.3. Bottom-Loading Pump Demonstration. The prototypic bottom-loading pump was tested in a pilot plant facility simulating a product pumping situation. As shown

Ind. Eng. Chem. Res., Vol. 26, No. 11, 1987 2261 in Figure 9, the 4-ft fluidic pump rested in the bottom of the 3000-L host tank, llF03. A nitric acid (0.3 M) and uranyl nitrate solution (specific gravity = 1.47) was used as the pumped fluid at an operating temperature of 40 OC. The discharge line reached an elevation of 16 f t and ran nearly horizontally another 9 f t before turning down and emptying into a stand pipe on receiving tank, llF01. The stand pipe was installed to prevent siphoning between the tanks. Except for manually setting the motivation pressure, the entire operation of the experiment was conducted from a process control room. Starting with the host tank filled to the 7-ft level, the pumping cycles began using previously determined refill times which were changed with each l-ft refill height decrease. The pump times for each motivation pressure had been previously determined, but as an added control, a microphone, attached to the host tank, signaled when air exhausted through the port. The pump time was then decreased by 1 s to operate without blowout. The run continued until the level of the host tank reached the top of the pump. The fluid in the receiving tank was then transferred to another area, and the test continued until the pumping essentially stopped. These results are compared in section 3.3 to calculated results by using the predictive model.

:

o

3. Predictive Model 3.1. General Description. The performance of a fluidic pump is highly dependent on the configuration of the output system. If one has available a normalized calibration curve and a description of the output piping arrangement (lengths and diameters of pipes, heads, and fittings), it is possible to estimate the performance of a particular fluidic pump. As mentioned earlier, the normalized calibration curve consists of a plot of the fraction of the fluid in the chamber which is delivered to the output system, Q = Qo/Qi, versus a dimensionless pressure ratio, P = (P2- P3)/(P1- P3).This curve is unique to a given pump geometry over a wide range of fluid properties. The method of solution of the equations governing the performance of the pump is iterative. The procedure is as follows. a. A value of the split, Q, is estimated. b. The volume of fluid transferred to the output system where Qi is the known volume of fluid per cycle Q, is QiQ, in the pump. c. The volume flow rate of fluid in the piping system is Q,/tl, where tl is the pump time. d. Velocities in the piping system are calculated by dividing the volume flow rate by the cross-sectional areas. e. The Bernoulli equation, which relates pressure drop in the piping system to flow rates, skin, and other frictional effects and static heads, is used to calculate P2, the pressure drop through the piping system. f. PIis the operating pressure of the pump, and P3is the static pressure at the RFD caused by the refill head. g. Knowing P1,P2, and P3,one may calculate P = (P2 - P3)/(P1- P3)and compare this value to that obtained from the Q versus P calibration curve. Agreement of the two values of P indicates the correct Q has been assumed. The procedure is repeated until satisfactory agreement between assumed and resultant splits is obtained. A computer program was written to accomplish the calculations described above (Morgan and Holland, 1986). The program is user interactive and allows the user to provide assumed splits; the program calculates splits based on a polynominal approximation of the calibration curve for comparison. A second-order a proximation was found to fit the experimental Q versus data to an accuracy of

J

MOTIVATION PRESSURE

r

PUMP TIME 111

10

12

14

16

Figure 10. Prototypic pump time.

about 1% for values of P less than 0.725. All the experimental pumping runs were carried out within this range of P values. Once agreement between the assumed and calculated splits is achieved, the flow per cycle is calculated. In addition, since the refill time and pumping time are known, the program calculates the expected mean flow rate in liters per hour. This value is corrected for fallback. The resulting value gives an estimate of expected pump performance in the particular flow configuration. 3.2. Pump Cycle Times. The total pump cycle time is the refill time plus the pumping time. The refill time for the prototypic bottom-loading pump is 4-7 times longer than the pumping time, depending on the refill head. A refill time can be calculated using an experimentally determined orifice coefficient of 0.73 and taking into account the level change as the chamber fills: tf = TD2[H01/2- (H, - H,)1/2]/(CoS,(2g))

(3)

Values of refill time were determined experimentally and fitted to the above equation to obtain the appropriate value for C,. Pumping time was determined experimentally as a function of motivation pressure and, in the case of the prototypic bottom-loading pump, the level in the pump chamber. A plot of fluid level in the pump versus pump time results in a series of straight lines with different slopes for different motivation pressures as shown in Figure 10. These data were reduced to an equation accounting for the change of slope for different motivation pressures. The pump time for this pump could be expressed as tl = H1(0.001571P12- 0.1453P1 + 5.751)

(4)

If no data are available, the pump time may be calculated by dividing the liquid volume in the pump chamber by the volumetric flow rate through the nozzle. The volumetric flow rate through the nozzle may be obtained from standard fluid mechanics equations assuming a discharge coefficient of unity. Fallback after each pump stroke is calculated in the model. In the pilot plant experiment, it was experimentally determined that the entire volume of the delivery line (except for -200 cm3) fell back and so must be subtracted from the predicted delivered volume per cycle. This volume, 1.7 L, although small compared to a full pump chamber, becomes important and is the limiting value when pumping a nearly empty host tank. 3.3. Predictive Model Application. Results of the top-loading pump test with water at 10-ft delivery elevation were compared with values generated by using the predictive model. This comparison is shown in Table I. Good agreement was obtained over the pressure range 11-43 psig.

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2262 Ind. Eng. Chem. Res., Vol. 26, No. 11,1987 Table I. Comparison of Experimental and Predicted Values Obtained by Using a ToD-Loadiniz Pumo output flow of water, L/cycle motivation press., psig exptl. predicted 11 2.06 1.99 2.23 2.34 16 2.36 2.50 21 2.45 24.5 2.56 2.55 2.58 28 2.56 2.59 32 2.62 2.59 36 2.58 2.59 40.7 2.60 2.60 43 Table 11. Summary of Pilot Plant Submerged Tests for the Bottom-Loading Prototypic Pump av flow, L / h motivation av host tank press., psig exptl. predicted level, f t 25 333 365 6.4 287 311 5.6 241 255 4.8 382 6.4 30 370 324 5.1 304 4.4 256 264 406 6.7 35 382 344 326 5.8 278 267 4.4 413 6.7 40 388 349 5.6 343 25 1 4.5 256

EXPERIMENTIL

0

0

0

20

30

PRE3ICTED

e

50

\ l O T l V 4 T l O k PRESSURE 4 0 % g

Figure 11. Flow rates for prototypic pump.

During the calibration test of the bottom-loading prototype, a run was made with an 8-ft refill head and a 6-ft delivered head. This resulted in the largest flow obtained in any of the tests. The results are shown in Figure 11 together with the predicted values using the model. A flow rate of over 750 L/h was reached under these conditions. Other experimental and predicted flow values for the bottom-loading pump are shown in Table 11. The motivation pressures are nominal set pressures. The actual pressure during pumping, as measured by the pressure cell, was about 5 psi lower. The actual pressures were used in the predictive model. The host tank level was pumped down in approximately 1-ft increments, at which time the refill time was increased. This procedure resulted in a lower value of average flow (L/ h) as the runs progressed, although the amount of liquid delivered each cycle remained nearly constant. The pump times for each of the three runs at a given motivation pressure had the same value, except for the third run at 40 psig which was 9 s instead of 10 s. This procedure resulted in the low flow value observed for that run. When the level in the host tank dropped below 4 ft, the pumping chamber became unsubmerged. With the level in the pump chamber also below 4 f t a t the beginning of the pump cycle, the pump times as well as the refill times decreased. The results of these unsubmerged tests are

Table 111. Summary of Pilot Plant Unsubmerged Tests for the Bottom-Loading Prototypic Pump motivation av flow, L / h av host tank press., psig exptl. predicted level, f t 25 198 190 3.6 167 167 3.1 144 131 2.5 116 99 1.9 55 33 1.5 7 1.3 30 144 133 3.6 154 147 2.7 87 58 1.6 51 16 1.2 13 0.9 35 184 186 3.0 146 141 2.3 140 105 1.9 66 34 1.4 17 1.1 40 220 214 3.3 184 165 2.6 83 49 1.6 31 0.9

shown in Table I11 together with the predicted values. As the host tank level and the level in the pumping chamber approach 1ft, the predicted values deviate considerably from the experimental values. Around 1f t and below, the model predicts no flow. At a motivation pressure of 30 psi and a pump chamber height of 0.9 ft, the output flow of 13 L/h corresponds to only 150 mL/cycle. Fallback of 1.7 L and a split loss of 20% during the pump cycle result in essentially no delivered volume. A word of caution in using the predictive model is necessary. The model was found to diverge considerably from experimental results at larger values of P. This situation occurs when the resistance to flow in the system is large and results in most of the fluid leaving the pump being diverted through the refill port. Much of the data (shown in Figure 8) for the top-loading pump were in this unreliable region where Q varies considerably with small changes in P. These data did not agree well with the predictive model. On this steep part of the calibration curve, a 5% uncertainty in the calculated resistance of the system can result in an output flow change of about 40%. The 10-ft elevation run using water had a lower system resistance and was more predictable. When one examines the ZnBr, solution runs, the resistance resulting from increased density has increased the P term value to 0.985, again on the steep part of the calibration curve. A well-designed fluidic pump should be made to operate in the region where P is less than about 0.7. Not only is the performance more predictable but also the pump is more efficient as less fluid bypasses the delivery system. 4. Conclusions The prototypic slab tank pump has been demonstrated to deliver the required average output flow for product mixing. It is also suitable for use in the other portions of the flow sheet, requiring feed delivery to stations metering flow to contactor banks. It was possible to confirm the validity of the predictive model and predict flows within 10% of actual experimentally determined values over the desired operating region.

Acknowledgment This work was performed for the Consolidated Fuel ReprocessingProgram located in the Fuel Recycle Division of the Oak Ridge National Laboratory, operated by Martin Marietta Energy Systems, Inc., for the United States

I n d . Eng. Chem. Res. 1987,26, 2263-2267

Department of Energy under Contract DE-ACOB840R21400. The research was sr>onsoredby the Office of Facilities, Fuel Cycle, and TesiPrograms:

Nomenclature C, = orifice coefficient (0.73) D = diameter of pump chamber, f t g = acceleration of gravity, 32.17 ft/s2 Hf = final height in pump chamber, f t H , = refill head, f t H1= pump chamber level, f t P, = motivation pressure, psig P2 = pressure drop through piping, psig = static pressure above the RFD, psig P = dimensionless pressure ratio Qi = volume of fluid initially in pumping chamber, ft3 Q, = volume of fluid delivered, ft3 Q = split So = orifice area, ft2

2263

t f = refill time, s tl = pump time, s

Literature Cited Morgan, J. G.; Holland, W. D. ORNL/TM-9913, Feb 1986; Oak Ridge National Laboratory, Oak Ridge, TN. Smith, G. V.; Counce, R. M. Ind. Eng. Chem. Process Des. Dev. 1984a, 23, 295-299. Smith, G. V.; Counce, R. M. ”Performance Characteristics of Axisymmetric Venturi-Like Reverse-Flow-Diverters”,84-WA/DSC-9, Proceedings of the Winter Annual Meeting, American Society of Mechanical Engineers, New York, 1984b. Tippetts, J. R. “A Fluidic Pump for Use in Nuclear Fuel Processing”, Proceedings of the 5th International Fluid Power Symposium, New York, 1978. Tippetts, J. R. “Some Recent Developments in Fluidic Pumping”, Proceedings of the 6th Technical Conference of the British Pump Manufacturer’s Association, Canterbury, England, 1979.

Received for review May 5, 1986 Revised manuscript received July 28, 1987 Accepted August 17, 1987

Analysis of Interactions for Liquid-Liquid Dispersions in Agitated Vessels Eleni Chatzit and James M. Lee** Department of Chemical Engineering, Cleveland State University, Cleveland, Ohio 441 15, and Department of Chemical Engineering, Washington State University, Pullman, Washington 99164

A population balance equation based on the breakage and coalescence models was solved numerically to generate theoretical drop size distributions, which were compared with the experimental results obtained from the liquid-liquid dispersions in agitated vessels. In an attempt to choose a model which would best fit our experimental results, several alternatives have been considered. I t was found that any combination of the models tested could predict the drop size distribution reasonably well. Regarding the number of daughter droplets, when the number was assumed to be 7, the predicted distribution curve gave a better fit of the experimental data than when it was 2. However, one parameter out of four in the models has to be adjusted to fit the data for the liquid-liquid system without surfactant. For the system with a surfactant, two out of four parameters have to be adjusted. The liquid-liquid dispersions in mechanically agitated vessels are of major importance in a number of chemical processes such as suspension polymerization and liquidliquid extraction. Two immiscible liquids in an agitated vessel form a dispersion, in which continuous breakup and coalescence of drops occur simultaneously. If the agitation is continued over a sufficiently long time, a local dynamic balance between breakup and coalescence is established. The steady-state drop size distribution depends on the conditions of agitation. Several models have been proposed to describe the liquid-liquid interactions in an agitated vessel. The models can be classified into two categories: the nonhomogeneous interaction model (Rietema, 1964) and the homogeneous interaction model (Curl, 1963; Valentas et al., 1966; Valentas and Amundson, 1966),depending on whether or not they take into consideration the local variations of flow characteristics. The former is more realistic than the latter; however, its applicability is limited because it is difficult, if not impossible, to evaluate various parameters at various locations in a vessel. Cleveland State University. Present address: Department of Chemical Engineering, University of Thessaloniki, Thessaloniki, Greece. *Washington State University. 0888-5885/87/2626-2263$01.50/0

In this paper, the population balance equation of the homogeneous interaction model (Curl, 1963; Valentas et al., 1966; Valentas and Amundson, 1966) for a batch stirred vessel, based on the breakage and coalescence models proposed by Coulaloglou and Tavlarides (1977)and Sovovd (1981),has been solved numerically to generate theoretical drop size distributions. The objectives of this work are to analyze these models and to test their ability to predict the experimental results of the liquid-liquid dispersions in agitated vessels.

Theory Valentas and Amundson (1966) have proposed a model based on the assumption of statistical homogeneity of the contents of the vessel. This does not necessarily mean that the turbulence and energy conditions are distributed homogeneously throughout the vessel. In general, the impeller and circulation regions are clearly distinguished and the homogeneous interactions model is a very rough approximation. They considered both coalescence and redispersion as occurring at a finite rate and derived a population balance equation by introducing five functions describing the breakage distribution function, @(u’,u), the number of daughter drops formed per breakage, ~ ( u ) , breakage frequency,g(u), the coalescence efficiency, X(u,u’), and the collision frequency, h(u,u’). A t steady state for 0 1987 American Chemical Society