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Znd. Eng. C h e m . R e s . 1991, 30, 277-279
on Figure 1 is the true process value. The “measurement” can be described as a second-order autoregressive integrated moving average (ARIMA) (MacGregor, 1988) with superimposed step and ramp changes. The line labeled “filtered process variable” is the filtered value calculated by the self-tuning filter (M= 5) with E,, = 4 units. A t the bottom of the graph are curves that indicate the standard deviation of the process variable as calculated by (15) and the filter time constant as calculated by (18). Occasionally events cause 7fto be extremely large, so there is a ceiling on 7f of 50T for visual graphical convenience. Note that the estimated opvis close to either the true 8 units or 2 units in spite of the presence of autoregressive drift, ramp, and step changes. Note that the filter time constant changes only in response to process variance in spite of the other transient effects. (While the period from the 1st through the 300th sample was intended to have a standard deviation of 8 units, the confluence of random numbers at the 20th sample caused large measurement deviations that the self-tuning filter detected. I t adjusted accordingly.) Note that the “filtered process variable” curve lags behind the “actual” process variable. This is an expected effect of the first-order filter. Also note that where the process variance is high the lag is greater than when the process variance is low. This is the desired effect of the self-tuning filter. A t low process variance, the filter time constant is low and the filtered value is responsive to process changes. A t high process variance the filter time constant is high and uses more data to obtain the average. At all cases the 95% confidence interval, E,,, was 4 units.
mentary statistical concepts. Toward developing an easy to implement and computationally inexpensive method, the author has incorporated simplifications grounded in this experience. The method has been demonstrated on a wide range of simulated conditions and has two advantages over standard practice. First, the user specifies the desired 95% confidence interval for the filtered value instead of a secondary parameter, the filter time constant. Second, the method automatically adjusts the filter time constant as the process variability changes to minimize filter lag while maintaining the desired accuracy. The method assumes that the sampling period is small in comparison to real changes in either process level or variability and that the noise is a random Gaussian fluctuation with a mean of zero.
Conclusion and Critique A method for automatic adaptation of a time constant for a first-order filter has been developed by use of ele-
Received for reuieu April 20, 1990 Revised manuscript received September 25, 1990 Accepted October 4, 1990
Literature Cited Bethea, R. M.; Rhinehart, R. R. Applied Engineering Statistics; Marcel Dekker: New York, NY, in press. Box, G. E. P. Jenkens, G. M. Time Series Analysis: Forecasting and Control, revised ed.; Holden-Day: Oakland, CA, 1976. Dixon, W. J.; Massey, F. J., Jr. Introduction to Statistical Analysis, 4th ed.; McGraw-Hill: New York, NY, 1983. MacGregor, J. F. On-Line Statistical Process Control. Chemical Engineering Progress; AIChE: New York, NY, 1988; Vol. 84, No. 10.
R. Russell Rhinehart Department of Chemical Engineering Texas Tech University MS-3121, Lubbock, Texas 79409-4679
Effect of Turbulence Damping on the Steady-State Drop Size Distribution in Stirred Liquid-Liquid Dispersions With the help of a recent model for the drop size distribution in stirred liquid-liquid dispersions, the effects of turbulence damping and coalescence, individually, on the characteristic drop sizes are investigated. The outcome of this work, which is based on an unstable dispersion where both coalescence and breakup occur simultaneously due to mixing, lead us to conclude that drop sizes in such systems are predominantly governed by turbulence damping, with coalescence playing an indirect role. Introduction A simple model for predicting the steady-state drop sizes in stirred systems has been recently proposed (Cohen, 1990). The approach considers a large number of drops dispersed in a continuous fluid being stirred rapidly. Due to stirring, the drops continually coalesce and break up, and as a result, a steady-state drop size distribution is observed after a period of time. Everyone of these drops is assumed to be composed of a certain number of primary droplets, each having a diameter, dmin,where the subscript min refers to “minimum”. Thus,the primary droplets are the smallest units that exist in the dispersion. In relation to the theory proposed by Hinze (1955), it is important to note that dminis the most stable droplet size detemined by the Weber number, We, and the dispersed-phase volume fraction or holdup, rp (Godfrey et al., 1989 and references within). We may consequently assume that for a given We and 4, any droplet larger than dmincould break down to form smaller droplets, while the minimum droplet size in the dispersion 0888-5885/91/2630-0277$02.50/0
remains fixed at dmin.Very briefly, therefore, the model is based on the assumption that (1)drops larger than dmin result from the coalescence of smaller droplets, or the breakup of larger drops, and (2) while the primary droplets of size dmhcan coalesce to form larger drops, they cannot break down to form smaller droplets. Avoiding the details of the derivation, the Sauter mean diameter, d32,of the drops in the suspension is shown to be given by (Cohen, 1990) d32/d,in Z1J3 (1) where 2 = In No- In (In No- In In No) (2) and (3) Note that N o is simply the total number of the primary droplets in the suspension having volume V, should the 0 1991 American Chemical Society
278 Ind. Eng. Chem. Res., Vol. 30, No. 1, 1991 lo-’
t
2.5
1
eo
10.21
3
. 1 1 Narsimhan cr ol (1980) (.$ = c o o l )
2 w .E
Ross er ai (1978) (005 5 0 5 0 2)
10 10
10
Theoretical d,,,
10
0.0 0.0
I
Figure 1. Experimental d, obtained from Narsimhan et al. (1980) and Ross et al. (1978) versus the theoretical dminobtained from eqs 1-3.
system comprise a monodisperse group of the primary units of size dmin. It therefore follows from eqs 1-3 that given 4 and V for a particular experimental set, and also determining d32 experimentally, the one remaining parameter, dmh,can be predicted theoretically. This procedure was carried out (by iteration) by using the data of Narsimhan et al. (1980) and Ross et al. (1978), where experimental drop size distributions, providing both dmi,and d32, are available. The result of these calculations is summarized in Figure 1where the experimental dminis compared with the theoretical dmi, obtained from the above equations. It is important to mention that the experimental values for dmincollected from the above-mentioned works were taken as the smallest diameters available from the steady-state distribution data. Furthermore, the d32)swere extracted from the volume distribution data of Narsimhan et al. (1980) after considering that they should approximately coincide with a cumulative volume of 0.5 (Cohen, 1990). Overall, accounting for the fact that 4 varies over 2 orders of magnitude in Figure 1 and that the present model does not contain any adjustable parameter, we note that the results are in satisfactory agreement with each other.
Effect of Holdup, 4 The existing data pertaining to drop formation by vigorous mixing is generally presented in the following empirical form d32/D = K j ( 4 ) (4) where K 1 is a proportionality constant and f ( 4 ) is the holdup function, which shall be discussed shortly. Equation 4 is valid for drop sizes larger than the Kolmogorov microscale. Upon defining the Weber number as pN2D3/u, where p and u, respectively, are the density and interfacial tension and N and D, respectively, represent the impeller speed and size, we rewrite eq 4 as [ P N2D3/ulo.6d32/D= K i f ( # ) (5) It also follows that
where ~ l ~ ~ ( 4 -isOthe ) Sauter mean diameter at infinite dilution. Regarding the holdup function, f(4), a popular representation is the linear relation f(4) = 1 + A 4 (7)
0.2
0.1
(cm)
Holdup,
0.3
0
Figure 2. Holdup function, f(4),versus holdup, 6. Experimental points are reduced from the data of Ross et al. (19781, and the solid curve is the best fit straight line.
t
- 1 -3
n 0.0 4 0.0
0.1
0.2
c
0.3
Holdup, @ Figure 3. Damping function, g(b), versus holdup, 9.The points are reduced from the theoretical dminvalues obtained from eqs 1-3 by using the data of Ross et al. (1978) for d3*. The solid curve is the best fit straight line.
in which A is an empirical constant ranging between 2.5 and 9 (Godfrey et al., 1989). Assuming that Hinze’s -0.6 power law, which appears in eq 4, is applicable to the data of Ross et al. (1978), we have, as an example, carried out the necessary calculations for obtaining the ratio d32/ d32(4-*0), which is just equal to f ( 4 )by virtue of eq 6. The result of this, which is displayed in Figure 2, yields a value of A approximately equal to 3.33.
Effect of Turbulence Damping on the Characteristic Drop Size It is necessary now to discuss the effect of turbulence damping due to the dispersed phase on the characteristic drop size. We choose to address this issue because it has been questioned a number of times and subsequently investigated repeatedly throughout the years. We must recall however that the case considered here is a dispersion in which both coalescence and breakup of the drops occur simultaneously; i.e., the suspension is “unstable”. Related experiments have consistently shown that an increase in holdup causes the mean drop diameter or cluster size to also increase (Mlynek and Resnick, 1972; Ross et al., 1978; de Boer et al., 1989; among many others). A number of investigators attribute this behavior to turbulence damping (Doulah, 1975 and references within), while others ascribe it mainly to coalescence (Delichataim ad Probstein, 1976). Moreover, some believe that both coalescence and damping are equally responsible for the observed trends (Godfrey et al., 1989 and some references within). Nevertheless, it appears that the approach proposed by Cohen (1990) can be used to explain the role of
Ind. Eng. Chem. Res., Vol. 30, No. 1, 1991 279 each mechanism (coalescence and turbulence damping) in determining the resulting mean drop diameter. Since the model, which seems to predict related experimental results rather satisfactorily (i.e., see Figure 1, and also Cohen (1990)), is based on the assumption that all drops larger than dminresult from the coalescence of the smaller droplets, then, essentially, dminmust be the characteristic drop size influenced most directly by the turbulence damping (in addition to the Weber number, of course). In other words, within, the context of the proposed model, this argument should prevail because dmh increases with 4, which is clearly evident in Figure 1. Furthermore, the fact that, for a given We, the primary droplets (size dmin)do not experience any further breakup has the important implication that dminis not a product of coalescence. Therefare, an increase in dminbrought about by 4 must be caused solely by turbulence damping. Hence, accordingly, it would be more appropriate to rewrite eq 4 in terms of dminrather than d32; i.e., dmin/D= Kg(4)We4,6 (8) where K, is again a proportionality constant and the new holdup function, g(4),is a factor that represents only the effects of turbulence damping; thus, we choose to name it the "damping function". d3,, on the other hand, is related to dminthrough eqs 1-3. In view of this, therefore, we see that d3*is indirectly influenced by damping since these effects are passed on by dmin. Similar to the approach taken earlier, we now express eq 8 as dmin/dmin(4+) = g(4) (9) and proceed to investigate the behavior of the damping function, g(4). This is possible since dmhcan be calculated from eqs 1-3, once the experimental values for d32 are known. The calculations were carried out by using the data of Ross et al. (1978), and the result is plotted in Figure 3 as g(4) versus 4. In analogy to f($), we find that g(4) is representable by the linear function (10) g(4) = 1 + B4 where the empirical constant B is found to be approximately equal to 3.30 for this particular set of data. Comparison of the damping function, g(4), with the holdup function, f(4),for the experiments of Ross et al. (1978) clearly indicates that the two are nearly identical with each other. Obviously, the almost 1% difference that exists between the empirical constants, A and B (recall that A = 3.331, must be due to coalescence, if not experimental scatter. Because of this little difference between g(4) and f(@),dividing eq 4 by eq 8 gives the ratio -dB-2 --Klf(4) (11) dmin Kg(4) to be almost constant in 4. This result is clear from the experimental points included in Figure 7 of Cohen (1990) where a 2 order of magnitude variation in No causes little change in the ratio dB2/dmin. It is also interesting to mention that applying the same to the data of Mlynek and &nick (1972) leads to a similar conclusion. Consequently, we are able to conclude from the analyzed data that the influence of coalescence is secondary, so that the predominant cause for the existence of the holdup function, f ( @ ) ,is turbulence damping.
Conclusions On the basis of a recently developed model for predicting the steady-state drop size distribution in a stirred sus-
pension, we have investigated the effects due to coalescence and turbulence damping, individually, on the mean drop diameter. From this work, we conclude that for suspensions where both coalescence and breakup of the drops occur simultaneously, the holdup function, f(#), is predominantly governed by the turbulence damping induced by the dispersed phase. The contribution of coalescence to f(4) is found to be rather insignificant. Moreover, the smallest sized droplets of diameter dmin absorb all the damping effects, and these are, in turn, transferred to the larger drops by the process of coalescence. Thus, the larger drops, including the mean-sized drops, are indirectly affected by turbulence damping by being dependent on the size of the primary droplets. Finally, it is important to keep in mind that, within the framework of the model, the arguments presented above remain valid for stirred but "unstable" suspensions only and that they may not be applicable to "stable" dispersions.
Nomenclature A,B = empirical constants appearing in the holdup and damping functions, respectively (see eqs 7 and 10) dmh = diameter of the smallest sized droplet (primarydroplet) in the dispersion d3, = Sauter mean diameter D = impeller size (see eq 5) f ( 4 ) = holdup function g(4) = damping function K1,KP= proportionality constants (see eqs 4 and 8) N = impeller speed (see eq 5) No = see eq 3 W e = Weber number Greek Symbols p = liquid density u = interfacial tension 4 = dispersed phase volume fraction (holdup) Literature Cited Cohen, R. D. Steady-state Cluster Size Distribution in Stirred Suspensions. J. Chem. Soc., Faraday Trans. 1990, 86, 2133-2138. de Boer, G. B. J.; De Weerd, C.; Thoenes, D. Coagulation in Turbulent Flow: Part 11. Chem. Eng. Res. Des. 1989, 67, 308-315. Delichatsios, M. A.; Probstein, R. F. The Effect of Coalescence on the Average Drop Size in Liquid-Liquid Dispersions. Znd. Eng. Chem. Fundam. 1976, 15, 134-138. Doulah, M. S. An Effect of Holdup on Drop Sizes in Liquid-Liquid Dispersions. Ind. Eng. Chem. Fundam. 1975, 14, 137-139. Godfrey, J. C.; Obi, F. I. N.; Reeve, R. N. Measuring Drop Size in Continuous Liquid-Liquid Mixers. Chem. Eng. Prog. 1989,85, 61-69. Hinze, J. 0. Fundamentals of the Hydrodynamic Mechanism of Splitting in Dispersion Processes. AIChE J . 1955, I , 289-295. Mlynek, Y.;Resnick, W. Drop Sizes in an Agitated Liquid-Liquid System. AIChE J. 1972, 18, 122-127. Narsimhan, G.; Ramkrishna, D.; Gupta, J. P. Analysis of Drop Size Distributions in Lean Liquid-Liquid Dispersions. A IChE J. 1980, 26,990-1000. Ross, S. L.; Verhoff, F. H.; Curl, R. L. Droplet Breakage and Coalescence Processes in an Agitated Dispersion. 2. Measurement and Interpretation of Mixing Experiments. Ind. Eng. Chem. Fundam. 1978, f7,101-108.
Ruben D.Cohen Department of Mechanical Engineering & Materials Science Rice University Houston, Texas 77251 Received for review May 21, 1990 Revised manuscript received August 27, 1990 Accepted September 8,1990
