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 t o generate theoretical drop size distributions, which were compared with the experimental results obtained from the liquid-liquid dispersions in agitated vessels. In a n attempt t o 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 t o 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 a t 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
2264 Ind. Eng. Chem. Res., Vol. 26, No. 11, 1987
a batch stirred vessel, the model becomes N A ( u ) [ g ( u )+ w ( u ) ] =
l:
1’-p(u’,v)v(u’)Ng(u’)A(u’) du’ +
X(u - u’,u’)h(u - u’,u’)NA(u - u’)NA(u’) du’
(1)
where w(u) =
lr-”
X(u,v’)h(u,u’)NA(u’) du’
(2)
A t steady state, drop breakage and coalescence are in dynamic equilibrium and the overall number of drops N in the vessel is constant. Equations 1and 2 were modified for drop diameter instead of drop volume by using the relationship A ( u ) du = A ( d p ) dd, (3) Breakage Frequency. Coulaloglou (1975) has developed a model for the deformation and breakage of a drop in a turbulent flow field and derived the functional form of the breakage frequency. The basic assumptions made in his derivation are that (1) the turbulence is locally isotropic and (2) the size d , of the breaking drops is within the size range of the inertial subrange eddies. Two mechanisms have been proposed for the breakage of drops. 1. The breakage of drops can be caused from a collision with a turbulent eddy. If the drop breaks up when the turbulent kinetic energy imparted from a collision with a turbulent eddy is greater than the drop surface energy, the breakage frequency is (Coulaloglou and Tavlarides, 1977) g(d,) = C I n
(
:)I3
ex.(
)
- pdn2D4/3d,5/3 “Ia
Table I. Systems Studied system dispersed phase continuous phase 1 5-cSt Dow Corning 200 fluid water 2 5-cSt Dow Corning 200 fluid 15% sucrose solution 3 kerosene water Table 11. Physical Properties at 23 O C P, g/cm3 system u, dyn/cm cont. disp. 1 42.5 1.000 0.920 2 32.0 0.920 1.087 3 41.9 1.000 0.787
N, (dyn s)/cm2 cont. disp. 0.010 0.0460
0.020 0.010
0.0460 0.0169
Coalescence Efficiency. When two drops collide, a film of the continuous phase is trapped between the two interfaces. It was assumed that the two drops will coalesce into one drop if the contact time is sufficient for the film to be drained out. Based on this assumption, the following expression for the coalescence efficiency of deformable drops has been derived (Coulaloglouand Tavlarides, 1977):
Sovov&(1981) has proposed an expression for the coalescence efficiency with the assumption that drop coalescence depends on the impact of the colliding drops; that is, if the turbulent energy of collision is greater than the total surface energy of the drops, the drops will coalesce:
(4)
2. If we consider that a drop will break if its turbulent kinetic energy is greater than its surface energy, the breakage frequency takes the form (Chatzi, 1983)
Breakage Distribution. The breakage of drops can be considered as a large number of independent random events. Therefore, it is reasonable to assume that the breakage distribution function is approximately normal. For a parent droplet of volume u’, the daughter droplets are distributed about a mean value as
where the variance a? is chosen so that 99.6% of the droplets formed lie within the volume range 0 to u’. If the number of daughter droplets is 7, eq 6 becomes
Collision Frequency. Coulaloglou and Tavlarides (1977) derived a model for collision frequency by assuming that the mechanism of collision in a locally isotropic turbulent field is analogous to the collisions between molecules as in the kinetic theory of gases. The collision frequency is given by h(dpl,dpz)= CIII(dp12 + dp22)(dp12/3 + dp22/3)1/2nD2/3 (8)
Experimental Methods Materials Used. The system chosen (Table I) for the dispersed phase was 5-cSt Dow Corning 200 fluid (Dow Corning Corp.) and kerosene. Distilled water and 15% sugar solution were chosen for the continuous phase. The physical properties of the systems are listed in Tables I and 11. A surfactant added was hydroxypropylmethylcellulose (HMC) supplied by Dow Chemical Co. The surfactant solution was prepared as described in Lee and Soong (1985). Apparatus. The mixing vessels were two sizes (16 and 29.2 cm in diameter) of flat-bottomed cylindrical vessels fitted with four equally spaced vertical wall baffles. The height of the liquid volume was always equal to the vessel diameters. A Model ELB experimental agitator kit (Bench Scale Equipment Co.) was used for mixing the liquids. The impellers chosen for this study were six-bladed flat turbines of two different sizes (7.6 and 10.2 cm in diameter). Drop Size Measurement. A microphotographic technique described by Hong and Lee (1983)was employed to take clear pictures of liquid-liquid dispersion during agitation with a magnification ratio of 3.87. The experimental procedure is described in Hong and Lee (1983). Results and Discussion Solution of the Population Balance Equation. Analytical solutions of eq 1 and 2 are not known. Numerical solutions can be obtained by employing an integration formula, as proposed by Valentas and Amundson (1966). The nonlinear integral population balance equation will be reduced to a set of nonlinear algebraic equations, which can be solved by an iterative technique, as described in detail by Chatzi (1983).
Ind. Eng. Chem. Res., Vol. 26, No. 11, 1987 2265 Table 111. Theoretical Models for Breakage a n d Coalescence function model breakage 1. drop breakup if the turbulent frequency, g(d) kinetic energy from a collision with a turbulent eddy is greater than the drop surface energy. 2. drop breakup if its turbulent kinetic energy is greater than its surface energy. no. of daughter drops, u ( u )
160,
,
I ~
eu
4
I 120
n
1
coalescence efficiency, 1. film drainage model (deformable drops) UdPl,dP2) 2. impact of colliding drops
4
I
5
I
constant and equal to 2, 3, 4, 5, or 7
breakage, distribution, normal distribution P(dPl'dP2) collision frequency, drops whose sizes are in the inertial subrange h(dPl,dPP)
---B
-A
6
I
I
\
I /
/ I
0.01
8 9 10
I
\
002
d,,
;---,
d
0 03
cm
Figure 2. Effect of the number of daughter drops on the theoretical drop size distribution. Curve A: v(v) = 7 . Curve B u(v) = 2. Other parameters are the same for both curves: CI/CII! = IO4, CII = 0.08, CIV= log. (The system and the operating conditions are the same.)
140.----7--~120
1
120 I
-
I
la\
100 r 100-
-
5 80-
U
601 4
* OO+
0 01
0 02
0 03
d p , cm
Figure 1. Theoretical size distributions predicted with two different models of breakage frequency. Curve A: CI/CIII = 1.43 X lo5, CII = 0.10, CN = 1.87 X lo7, v(v) = 5 (eq 4). Curve B: C$/CIII = 1.43 X lo5, C',, = 0.08,CIv = 1.87 X lo3, u(u) = 5 (eq 5 ) . (The physical properties of the system and the operating conditions for the simulation are the same for both cases.)
In an attempt to choose a model which would best fit our experimental results, various alternatives (Table 111) have been considered. Two adjustable constants, CI and Cn, are involved in the breakage process. CI influences the breakage time and C, the fraction of breaking drops. Two other constants, CIIIand CIv, are involved in the coalescence process. CIIIinfluences the collision frequency and CIvthe coalescence efficiency. To simplify the trial-anderror procedure in selecting proper values of the constants to fit the experimental drop size distributions, the constants CI and CIIIhave been combined into one constant, CI/CIII.
Breakage Frequency. The theoretical drop size distribution predicted by employing the two alternative models for the breakage frequency (eq 4 and 5) provided an almost identical curve (Figure l), even though the values of the parameters for the two models to predict the curves were not the same. Therefore, we can use either eq 4 or 5 for the solution of the population balance equation. Number of Daughter Drops. Previous investigators (Coulaloglouand Tavlarides, 1977; Sovovd, 1981) assumed that the number of daughter drops formed per breakage was two, regardless of the drop size. Konno et al. (1983) observed the breakups in the mixing vessels with highspeed cinematography. They found that the dispersed drops were gradually elongated such that they resembled
in 0 01
0 02
d,,
003
cm
Figure 3. Experimental and theoretical size distributions predicted with two different models of coalescencefrequency. Curve A C,/CIn = 1.43 X lo5, CII = 0.05,CIv = 1.83 X lo3 (eq 9). Curve B: CI/CIII = 1.43 X lo5, CII = 0.08, C',, = 1.87 X lo3 (eq 10). (System 2; $C = 0.05,n = 4.7 rps; D = 10.1 cm; T = 29.2 cm.)
two fluid lumps separated by an elongated thread. By breakage of the drop, two drops corresponding to the lumps and several drops corresponding to the thread are produced. Therefore, it is more realistic to assume that the average number of daughter droplets is larger than 2. The effect of the number of daughter drops per breakage, u(v),on the theoretical drop size distribution is shown in Figure 2. Curve A is the case when u(u) = 7, and curve B is for v(u) = 2. When v(v) = 2, the drop size distribution predicted has the shape of a normal distribution curve. As the value of u(v) increased, the average drop size decreased and the shape of the distribution was skewed toward the smaller drop size. Experimental observations of the drop size distribution typically gave a skewed or log-normal distribution (Hong and Lee, 1983). Coalescence Efficiency. Two alternative models have been proposed for the coalescence efficiency (Table 111). Figure 3 shows the predicted drop size distribution according to the film drainage model (curve A) and the impact of the colliding drop model (curve B). The two theoretical drop size distributions are at the same drop size range, but curve B is narrower and shows a higher maximum than curve A. Comparison with Experimental Data. The drop size distributions of turbulently agitated liquid-liquid dispersions in batch vessels were measured experimentally. The data were compared with the theoretical distributions
2266 Ind. Eng. Chem. Res., Vol. 26, No. 11, 1987
,
120 r
?------
dsBp: 00171
cm
d$$:”= 00181 cm
-
80
-
U
0-
YU -.
0. 021_-
-L -__
004
006
dp,
008
0 10
cm
301
002
0 03
Figure 4. Theoretical and experimental drop size distributions for d p , cm system 1 (& = 0.10,n = 3.6 rps; D = 10.1 cm; T = 29.2 cm; C I / C ~ ~ ~ Figure 6. Theoretical and experimental drop size distributions for = 1.43 x 105; cII= 0.20; cIV = 1.83 x 107; u(u) = 7). system 2 (& = 0.05; n = 5.8 rps; D = 10.1 cm; T = 29.2 cm; CI/CIII = 1.43 X lo5; C,I = 0.13; CIV = 1.83 X lo7; u ( u ) = 7).
‘20-
j
d s p z 00149 cm
-
80
-
d$=
-
I;
00162 cm
E
u 80-
- _ U
40
20 L
001
0
0 02 dp,
001 dp,
cm
Figure 5, Theoretical and experimental drop size distributions for system 2 (4 = 0.05;n =: 4.7 rps; D = 10.1 cm; T = 29.2 cm; CI/CIII = 1.43 x 105; cII= 0.09; cIv= 1.83 x 107; u ( u ) = 7).
predicted by the solutions of the population balance equation (eq 1) with various models for breakage and coalescence as summarized in Table 111. Any combination of the models in Table I11 could predict the drop size distribution resonably well with adjusted values of the constants CrCIV. However, in the case of the number of daughter droplets, when the number was assumed to be 7, the predicted distribution curve gave the better fit of the experimental data than when it was 2. This finding agrees with the experimental observation made by Konno et al. (1983). Figures 4-6 show the theoretical and experimental drop size distribution for the system without surfactants. The models used to solve eq 1 are eq 4, 6, 8, 9, and v ( u ) = 7. The values of the constants chosen for the best fit based on five different runs (Chatzi, 1983) are CI/CIII = 1.43 X lo5, Cn = 0.09-0.20, and CIv = 1.83 X 10’. The agreement between the experimental and theoretical drop size distributions and Sauter mean diameters can be considered satisfactory. However, to be able to use the model to predict the size distributions, the values of constants should be universal. The values of CI/CIII and CIv were the same for all five runs we compared, but the Cu value was not constant. It is noteworthy to see the effect of the constant CII on the breakage frequency. If you decrease the value of CII, the breakage frequency g(d ) in eq 4 will increase. Therefore, the decrease of C I wilfenhance ~ the breakage frequency so that the average drop size will decrease. The inconsistent value of CII may be due to the oversimplification in formulating the breakage frequency model and experimental errors.
0 03
cm
Figure 7. Theoretical and experimental drop size distributions for system 3 with a surfactant (a = 17.23 dyn/cm; 4 = 0.20; n = 5.8 rps; D = 7.62 cm; T = 16 cm; surfactant = HMC; CI/CIII= 1.43 X lo5; cII= 0.12; cIV= 1.83 x 109;v(u) = 7).
Coulaloglou and Tavlarides (1977) modified eq 4 by introducing the “damping” effects on the local turbulent intensities at high holdup fractions 4 as follows:
(11)
However, the use of eq 11 instead of eq 4 did not improve the inconsistency of C’$I values. Figure 7 shows the theoretical and experimental drop size distribution for the kerosene-water system with the addition of a surfactant (HMC). In this instance, the constant CIValso had to be increased to generate a curve which agrees with the experimental data. The increase of the C N value decreases the coalescence efficiency according to eq 9 and results in smaller drop sizes. Recently, Lee and Soong (1985) investigated the effects of surfactants on the mean drop size of liquid-liquid dispersions in agitated vessels. They reported that the Sauter mean diameter of the contaminated system is 47% smaller than the Sauter mean diameter predicted by the correlation developed by Hong and Lee (1985). The latter was based on experiments with five systems without surfactants. The coalescence is extremely sensitive to the presence of traces of impurities (Sovovii, 1981). When a drop collides with another drop, a film of the continuous phase is trapped between the two interfaces. The film has to be drained out before coalescence may occur. It was reported (Hodgson and Lee, 1969) that the drainage time, one measure of the difficulty of the coalescence, was delayed significantly with minute quantities of surfactant and
I n d . Eng. Chem. Res. 1987,26, 2267-2274
2267
d , = droplet diameter, cm d , = Sauter mean drop diameter, cm D = impeller diameter, cm g(u) = breakage frequency of drops of having volume u, s-l h(u,u’) = collision frequency of drops of volume u and u’, s-l n = impeller speed, rps N = total number of drops at time t T = tank diameter, cm u = volume of droplet, cm3 IJ = average volume of droplet, cm3
further delayed with the increase of the concentration of surfactant. The delay was explained in terms of interface movement and a double-layer repulsion. Therefore, the increase of the CIv value is due to the decrease of coalescence efficiency in the presence of a surfactant whose effect was not considered in the coalescence efficiency model (eq 9).
Conclusions A population balance equation based on the breakage and coalescence models proposed by Coulaloglou and Tavlarides (1977) and S O V O(1981) V ~ 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 (Table 111) have been considered. 1. The theoretical drop size distribution predicted by employing the two alternative models for the breakage frequency (eq 4 and 5) provided an almost identical curve. 2. When the number of the daughter droplets was assumed to be 7, the predicted distribution curve gave better fit of the experimental data than when it was 2. This finding agrees with the experimental observation made by Konno et al. (1983). 3. The predicted drop size distributions according to the film drainage model or the impact of the colliding drop model were in the same drop size range, but the latter model results in a narrower distribution than the former. 4. Any combination of the models tested (Table 111) could predict the drop size distribution reasonably well. 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. More parameters may need to be adjusted to fit the data for a wider range of operating conditions than tested. Therefore, more studies are needed to formulate more realistic and reliable models for the breakage and coalescence phenomena of liquid-liquid dispersion.
Greek Symbols @(u’,u) = number fraction of droplets with volume u to v dv formed by breakage of a drop of volume u’ X(u,u’) = coalescence efficiency of drops of volume u with drops of volume u’
+
= viscosity, (dyn s)/cm2 u(v) = number of drops formed per breakage of drop of volume U
= density, g/cm3 u = interfacial tension, dyn/cm :u = variance 4 = fraction of the dispersed phase w ( u ) = coalescence frequency out of v to v p
+ du, s-l
Subscript d = dispersed phase
Literature Cited Chatzi, E. M. S. Thesis, Cleveland State University, Cleveland, OH, 1983. Coulaloglou, C. A. Ph.D. Thesis, Illinois Institute of Technology, Chicago, 1975. Coulaloglou, C. A.; Tavlarides, L. L. Chem. Eng. Sci. 1977,32,1289. Curl, R. L. AZChE J. 1963,9,175. Hodgson, T. D.; Lee, J. C. J. Colloid Interface Sci. 1969,30, 94. Hong, P. 0.; Lee, J. M. Znd. Eng. Chem. Process Des. Dev. 1983,22, 130. Hong, P. 0.;Lee, J. M. Znd. Eng. Chem. Process Des. Dev. 1985,24, 868. Konno, M.;Aoki, M.; Saito, S. J. Chem. Eng. Jpn. 1983,16, 312. Lee, J. M.;Soong, Y.Znd. Eng. Chem. Process Des. Dev. 1985,24, 118. Rietema, K. Adv. Chem. Eng. 1964,5,237. SovovB, H. Chem. Eng. Sci. 1981,36,1567. Valentas, K. J.; Amundson, N. R. Znd. Eng. Chem. Fundam. 1966, 5,533. Valentas, K. J.; Bilous, 0.;Amundson, N. R. Znd. Eng. Chem. Fundam. 1966,5,271.
Nomenclature A(u) = probability density of droplet size u in vessel CrCIv = constants d,, = size of the largest stable drop in the dispersion, cm
Received for reuiew September 11, 1986 Revised manuscript received July 20, 1987 Accepted August 5, 1987
Bilinear Model Predictive Control Yeong K. Yeot and Dennis C. Williams* Department of Chemical Engineering, Auburn University, Auburn, Alabama 36849
Bilinear model predictive control is defined for single-input-single-output systems. Offset compensation is provided to correct for the effects of unmeasured disturbances and model inaccuracies. Controller tuning is accomplished through the adjustment of a single parameter. T h e control algorithm is explicit, which allows implementation without requiring large computer memory or rapid computational speed. A filter with a single parameter is used to correct for the effects of an incorrect model. Model predictive control is the most important development in chemical process control in the last several
* Present address: HIMONT Research and Development Center, Wilmington, DE 19808. ‘Present address: Division of Chemical Engineering and Polymer Technology, KAIST, Dong Dae Mun, Seoul, Korea. 0888-5885/87/2626-2267$01.50/0
years. Model predictive controllers were developed independently a t ANDREA/GERBOIS in France as Model Predictive Heuristic control (MPHC) (Richalet et al., 1978; Richlet, 1980) and a t Shell Oil Company in the United States as Dynamic Matrix Control (DMC) (Cutler and Ramaker, 1980). Many successful applications have been reported, including a furnace (Cutler and Ramaker, 1980)) 0 1987 American Chemical Society
