Barlow, R. E., Proschan, F.. Hunter, L. C.. "Mathematical Theory of Reliability", Wiley, New York, N.Y., 1965. Bass, L., Wynholds, H. W.. porterfield, W. R., "Fault Tree Graphics", presented at the Conference on Reliability and Fault Tree Analysis, University of California, Berkeley, Sept 1974. Batts, J. R., E€€ Trans. Reliab., R-20, 88 (1971). Bazovsky. I., "Reliability Theory and Practice", Prentice-Hall, Inc., Englewood Cliffs, N.J., 1961. Caceres, S., M.S. Thesis, University of Houston, Houston, Texas, Dec 1974. Crosetti, P. A,, /E€€ Trans. Nucl., Sci., NS-18, 465 (1971). Crosetti. P. A,, Bruce, R. A,, Ann. Reliability Maintainability, 9, 230 (1970). Fussel, J. B.. "Fault Tree Analysis-Concepts and Techniques", in "Generic Techniques in Reliability Assessment", E. J. Henley, J. Lynn, Ed., Nordhoff Publishing Co.. Leyden, The Netherlands, 1974a. Fussel. J. E.. "MOCUS-A Computer Program to Obtain Minimal Sets from Fault Trees", ANCR-1156, 1974b. Fussel, J. E., Powers, G. J., Bennets. R. G., I€€€ Trans. Reliab., R-23, 51 (1974~). Gandhi, S., Ph.D. Thesis, University of Houston, Houston, Texas, 1973. Green, E. A., Bourne, A. J., "Reliability Technology", Wiley, New York, N.Y.. 1972. Haasl. D., "Advanced Concepts in Fault Tree Analysis", Systems Safety
Symposium, sponsored by the University of Washington, Seattle, Wash.. 1965. Henley, E. J.. Williams, R. A,, "Graph Theory in Modern Engineering", Academic Press, New York, N.Y.. 1973. Inoue, K., Henley, E. J., "Computer Aided Reliability and Safety Analysis of Complex Systems", iFAC World Congress, Boston, Massachusetts, Parr. Ill, D, 1, IFAC. Secretariat, Pittsburgh, Pa., Sept 1975. Messinaer. M.. Shooman, M. L., Proc. /€€E Ann. Symposium on Reliability, 292 11967). Misra, K. E., Rao, T. S.M., /E€€ Trans. Reliab., R-19,19 (1970). Nelson, A. C., Bans, J. R., Beadles, R. L., /E€€ Trans. Reliab., R-19, 61 (1970). Powers, G. J., Tompkins, F. C., Jr.. AlChE J., 20, 376 (1974). Semanderes. S. N., /€€E Trans. Nucl. Sci., 126 (Nov 1970). Shooman. M. L.. /E€€ Trans. Reliab., R-19, 74 (1970). Vesely, W. E., "PREP and K I T , Computer Codes for the Automatic Evaluation of a Fault Tree", IN-1340, Aug 1970.
Received for review June 25, 1975 Accepted January 26,1976
The Effect of Coalescence on the Average Drop Size in Liquid-Liquid Dispersions Michael A. Delichatsios and Ronald F. Probstein" Deparfment of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02 139
It is shown that the increase of drop sizes with the fraction by volume of all drops (holdup) in agitated liquidliquid dispersions cannot be attributed entirely to turbulence damping caused by the dispersed phase. A new model, based on the role of coalescence of the dispersed phase, is suggested to account for the observed drop size behavior. The coalescence frequency resulting from binary drop collisions is equated to an effective breakup frequency to yield a semiempirical relation for the increase in drop sizes with holdup. The relation explains some differences among reported experimental results and correlates the data from systems with different degrees of chemical destabilization.
Introduction I t has recently been argued (Doulah, 1975) that the increase of the steady-state average drop size with dispersed phase volume fraction or holdup in an agitated liquid-liquid dispersion can be adequately explained by increased damping of the turbulence. An alternative mechanism proposed by a number of authors (see, e.g., Mlynek and Resnick, 1972) is that the holdup behavior represents an effect of coalescence. In what follows we shall show that only coalescence could account for the magnitude of the observed holdup effect. We then derive an expression for the drop size for a model in which coalescense is a dominant mechanism. Average drop sizes in agitated liquid-liquid dispersions can be correlated by the relation (see, e.g., Mlynek and Resnick, 1972)
where C1 is a constant of order 1; t is the rate of turbulent energy dissipation per unit mass; u is the surface tension of the dispersed phase relative to the continuous phase; p is the density of the continuous phase, assumed not much different than the density of the dispersed phase; 4 is the volume fraction of the dispersed phase, Le., the holdup; and 134
Ind. Eng. Chem., Fundam., Vol. 15, No. 2, 1976
f(4) is the holdup function, expressed empirically by the linear relation
f(4) = 1 + c24
(2)
There does not appear to be any general agreement on the value of Cz, the reported range varying between 2.5 and 9 (Mlynek and Resnick, 1972). Recent experimental results on two-phase jet flows, reported in the Russian literature (Laats and Frishman, 1974) have shown that the damping of turbulent intensities can be approximated by
+ +
u' 1 0.24 =uo' 1 4
(3)
where u' is the root-mean-square turbulent fluctuating velocity in a dispersion of particles and uo' is the respective value in a free particle fluid. Using eq 3 and the relation (Batchelor, 1960) u'3 €--
L
(4)
where the macroscale of turbulence L is essentially determined by the physical dimensions of the apparatus, it follows that the available turbulent energy is damped by the factor (u'/uo')3.
From the functional dependence of drop size on the rate of turbulent energy dissipation per unit mass (eq 1) it follows that we can expect a drop size with a finite holdup which is a t most (1 4/1 0.24)6/5greater than for the case where 4 is negligibly small. Provided 4 is not too large, say less than Y4, this would imply a holdup function f(4) zz 1 C24 with Cz = 0.96. Comparison with eq 2, where the constant C2 is empirically determined t o range between 2.5 and 9, makes it clear that the turbulence damping effect would, by itself, be insufficient to account for the experimental results, a t least for small holdup. If coalescence of drops is held to be responsible for the drop size dependence on the dispersed phase fraction, then the steady-state drop size is defined in terms of an equilibrium between the rate of coalescence and the rate of breaku p of the drops. We consider a dilute, spatially homogeneous, polydisperse system of spherical drops, and assume that coalescence is the result of binary collisions with each breakup resulting in the production of two drops of equal volumes. Of course, should the drop break up into many small drops, the analysis which follows would have to be substantially modified. With u a drop volume and f ( u ) the drop volume probability density (particle size distribution) defined such that the number of drops per unit volume between u and u Av is f ( u ) Av, we may write for the time rate of change of the particle number density
+
+
+
+
n m
with the particle number density n and volume fraction defined, respectively, by
n=
Jm
f ( u ) du;
=
La
u f ( u ) du
(6)
Here, o ( u , u ’ ) is the volume coalescence frequency (cm3/s) between drops of volume u and u’, and p ( u ) is the breakup frequency of drops of volume u. The first term on the righthand side of eq 5 represents the decrease in number of drops by coalescence and the other term the corresponding gain associated with breakup. In the limit of very dilute systems this latter term goes to zero. At steady state dnldt = 0 and the resulting integral equation defines the steadystate drop size. Coalescence F r e q u e n c y Using simple mean free path concepts the volume coalescence frequency w(u,v’) has been estimated to be (Delichatsios and Probstein, 1975) w(u,u’) =
+ u’1/3)7/3t1/3
1.41A(u1l3
(7)
which is essentially an expression of the effective collision cross section times the relative collision speed. Here, the “coalescence coefficient” A has been introduced to account for the fact that not all collisions will result in coalescence (see, e.g., Shinnar, 1961). Furthermore, in writing eq 7 it is assumed, consistent with practical observations (Hinze, 1955), that the drop size lies within the inertial subrange. This condition is defined by the inequality L >> d >> 7 , where L is the length scale of the energy containing eddies (usually identified with the scale of the tank or the agitator) and 7 is the Kolmogorov microscale (the scale of the smallest eddies of the turbulent motion). We recall that 7 = ( ~ ~ / t ) lwhere / ~ , u is the kinematic viscosity. In general, the value of the coalescence coefficient is affected principally by the surface charge on the drops and
the phenomena of “film thinning” between the drops at separation distances less than a drop radius. I t is well known that any surface charge on the drops will inhibit coalescence due to electrostatic repulsive forces. In the present analysis we restrict our considerations to destabilized emulsions where the surface charge effects are absent or neutralized. Although numerous studies have been reported in the literature (see, e.g., Lang and Wilke, 1971) on the film thinning time between two dolliding drops or a colliding drop and a surface under relatively weak external forces such as gravity, so far the phenomenon has not been studied in any detail in the presence of the relatively strong external forces characteristic of turbulent flow. For the turbulent flows considered here the mean-square pressure fluctuations between the particles during coalescence are of the same order as the resistive forces to deformation (crld) (see, e.g., Hinze, 1955), so that the characteristic film thinning time will be much smaller than the characteristic turbulent coalescence time. I t follows that film thinning cannot be the rate-controlling mechanism of coalescence for the systems considered here. This is partly confirmed by previous experimental work (Shinnar, 1961). I t was observed in this work that film thinning was important only when a specific agent was dissolved in the continuous phase. Finally, we would note that mass transfer between dispersed and continuous phases does not take place in the systems under consideration, so that mass transfer effects on coalescense need not be taken into account. I t follows from the above discussion that the coalescense coefficient A should be a number of the order of 1 for completely destabilized dispersions of immiscible liquids under strongly turbulent conditions. Breakup Frequency The breakup frequency @ ( u )has recently been evaluated based on considerations of the statistical and unsteady nature of the pressure fluctuations across a drop diameter (Delichatsios, 1975). With U b the local turbulent velocity across a particle necessary to break it, it was shown that @ ( u ) was equal to twice the probability distribution function for the relative turbulent velocity difference between two points a t a distance of a drop diameter [ p ( U h ) ] , multiplied by the mean positive acceleration difference between the two points, when the corresponding velocity difference is Ub. Assuming that the relative acceleration can take on all possible statistical values independently of U h , we may identify the acceleration with its root-mean-square value. The drop breakup frequency, is then given by
With uT denoting the mean-square velocity difference between two points a t a distance of a drop diameter, we may write (Rotta, 1972)
-
(U2)1/2
-
(UT) d
(9)
A constant of proportionality of the order of 1 must be determined. From the experiments of Gibson and Massielo (1972) and Van Atta and Park (1972) the probability density distribution in the inertial subrange to good approximation can be written as a Gaussian with variance u 2 and cutoff velocity u,. I t follows from eq 8 and 9 that P ( u ) =-
d
where B is a “breakup coefficient”, which is the proportionality constant in eq 9 multiplied by ( 2 / x ) l / ’ . The velocities in eq 10 are defined as follows. From the Ind. Eng. Chem., Fundam., Vol. 15, No. 2, 1976
135
experiments of Van Atta and Park (1972) the cutoff velocity is found to be given approximately by
-
uc = 3(u2)1/2
(11)
For drop sizes within the inertial subrange L Rotta, 1972)
>> d >> 7 (see
-
u2 = 1 . 8 8 ( ~ d ) ~ / ~
exp[-4.51) (12)
Finally, from the work of Taylor (1949), in which he treated a drop as a vibrating spring-mass system, the breakup velocity is given approximately by
Here and elsewhere, the drop diameter is related to the volume through the relation u=-
locities (eq 11-13). In terms of our dimensionless variables, eq 15 can then be written in the form
rd3 6
W dE
(21)
Measurements of drop size distributions a t steady state (Sprow, 1967, Grossman, 1972) have shown the following: (i) the size distribution’is approximately symmetric; (ii) the ratio of the average size to minimum size is about 4; and (iii) the relative variance ranges from 0.3 to 0.7. With these considerations in mind the following observations can be made. First, the exponential function appearing in the integrand of eq 21 is of the order of 1 near the mean drop size. Assuming a Gaussian particle size distribution, this fact enables an asymptotic expansion of the integrand to be carried out about the mean size do. The result correct to the order of the square of the relative variance is
Holdup Function Using the expressions for the volume coagulation frequency (eq 7) and breakup frequency (eq lo), the equation defining the steady-state drop size is given from eq 5 (with dnldt = 0) by
P ( $ ) %5 1
(2%)
The result of eq 22b is obtained by expanding eq 17. The holdup function is defined by (cf. eq 1) (23)
exp(-uc2/2~)1$(E)dt (15)
From eqs 22 we then obtain the following form for the holdup function
Here, we have introduced the dimensionless variables (Swift and Friedlander, 1964) where The factor P($) on the left-hand side of eq 15 is defined by c4
(17) I t is a measure of the effect of drop polydispersity on the coalescence rate, and is equal to 1 for a monodisperse system (see, e.g., Delichatsios and Probstein, 1975). Equation 15 is important because it expresses quantitatively the effect of holdup on drop size. Consider an agitated liquid-liquid dispersion with a constant rate of turbulent energy dissipation per unit mass and different values of the dispersed phase holdup. For extremely small values of the holdup the left-hand side of eq 15 is negligibly small and the drop size is given from the condition p ( u ) = 0 (eq 10) or
-
Ub2
= Uc2 = 9U2
(18)
From the definitions of the various velocities (eq 11-13) we find (cf. eq 1)
(3
dmin= 0.18~-~/’
(19)
3/5
I t is important to note that this result for zero holdup gives the minimum drop size. Increasing holdup leads to an increased drop size. Using eq 19 we may rewrite the breakup velocity appearing in the exponential of eq 15 in terms of dmin,i.e. (20) Again, we have introduced the definitions of the various ve136
Ind. Eng. Chem.. Fundam., Vol. 15, No. 2, 1976
A = 3(8)
The constant C4, which depends only on the ratio of the coalescence to breakup coefficients, must be determined empirically. However, if the assumptions we have made are meaningful we should expect C4 to be a constant of order 1 with its exact value dependent on the stability of the dispersion.
Data Correlation The present theory may actually explain the reported divergence among the measured values of the coefficient C2 in the linear holdup function (eq 2). For sufficiently small 4 our result also reduces t o the linear form of eq 2 but with C2 C4, which is in turn proportional to the “coalescence coefficient” A . This coefficient must, as we have already noted, depend on the stability of the dispersion which we can expect to have been somewhat different between the various experiments. For this reason, of course, C4 cannot be expected to be a “universal” constant but will be smaller the more stable is the dispersion. These considerations are brought out more clearly in Figures 1 and 2. In Figure 1 we have plotted the experimental results of Sprow (1967) for a dispersion of isooctane in water, and Mlynek and Resnick (1972) for a dispersion of isooctane plus carbon tetrachloride in water. The experimental values of the drop size for 0 are taken to be the experimental values given for the very dilute dispersions. Both sets of experiments used similar agitated tank systems and preparation, and the dispersions were essentially destabilized. The ordinate is the holdup function (cf. eq 2) and the
-
+
20
i
I 0
I 0.2
0.1
I j
I
0.3
0.4
9 Figure 1. H o l d u p function dependence a s given by experiment, empirical linear relation of Mlynek a n d Resnick (1972), a n d semiempirical relation of present analysis.
IN WATER
0 CCI,
]
0 150-OCTANE IN WATER
10
I 02
I
01
VERMEULEN ET A L 119551
I 03
1 1
04
9
Figure 2. H o l d u p function dependence a s given by experiments of Vermeulen e t al. (1955) a n d semiempirical relation of present analysis.
abscissa is the fractional holdup. The curve in the figure represents eq 24 with the empirical constant c4
(26)
=1
as determined from a best fit of the equation to the data. We note that the results are consistent with the analysis, in that Cq turns out to be around 1. We have also shown in Figure 1 the empirical linear holdup function (1 5.44) suggested by Mlynek and Resnick (1972). I t can be seen from the figure that for holdups less than 0.1 both the linear form and the holdup function of eq 24 represent the experimental data fairly well. However, for higher holdups our semiempirical function gives values in better accord with the experimental results. Figure 2 is a similar plot of the earlier and somewhat less accurate (averaged) data for Vermeulen e t al. (1955). Our theoretical relation (eq 24) again correlates the data very well, but with C4 = 0.68. As before, C4 is of order 1 but jts exact value is somewhat lower than that which best correlates the data of Mlynek and Resnick (1972) and Sprow (1967). According to our theory this indicates a somewhat more stable dispersion, a fact which could be ascribed to different degrees of purity between the different experiments (Calderbank, 1958).
+
Conclusions The analysis has shown that increased drop size with higher fractional holdup can only be accounted for by al-
lowing for coalescense, turbulence damping playing a secondary role. I t was necessary to take into account the finite time for a drop to break. Previously reported experiments indicate that the steady-state drop size is given by an expression equal to the drop size for very small holdup multiplied by a function of the holdup that increases with increasing holdup (eq 24). The undetermined coefficient in this equation is of the order of 1 but its exact value depends on the stability of the dispersion. The expression developed explains the divergence in results among several sets of reported data.
Nomenclature A = coalescence coefficient, eq 7 B = breakup coefficient, eq 10 C1 = constant in drop size relation for 4 = 0, eq 1 C2 = constant in linear holdup function, eq 2 Cs = constant related to cutoff velocity in probability density function, eq 25a C4 = constant proportional t o . ratio of coalescence to breakup coefficients, eq 25b d = dropdiameter,cm &in = minimum drop diameter, cm f ( u ) = drop volume probability density f(4) = holdup function, eq 2, 23 L = length scale of energy containing eddies, cm n = drop number density, number/cm3 J I ( U b ) = velocity probability density P ( $ ) = polydispersity factor, eq 17 u' = root-mean-square turbulent velocity, cm/s uo' = root-mean-square turbulent velocity in particle free fluid, cmls U b = difference in turbulent velocities across drop diameter to cause breakup, cmls uc = cutoff velocity in probability density distribution, -cmls u 2 = average difference in the mean square of the velocities across drop diameter, cm2/s2 k = difference in turbulent accelerations across drop di-ameter a t breakup, cm/s2 k 2 = average difference in the mean square of the accelerations across drop diameter, cm2/s4 u,u' = drop volume, cm3
Greek Letters p ( u ) = breakup frequency of a particle with volume u , s-l
TJ
= rate of turbulent energy dissipation per unit mass, erglg s = Kolmogorov microscale of turbulence, cm
IJ
= kinematic viscosity, cm2/s
t
( = dimensionless drop volume, eq 16 p
= density of continuous phase, g/cm
u = surface tension, dynlcm
4 = volume fraction of dispersed phase (holdup) $(() = dimensionless drop volume probability density, eq
16 w(u,u') = volume coagulation frequency between two particles with volumes u and u', cm3/s
Literature Cited Batchelor, G. K.. "Homogenous Turbulence," p 103, Cambridge University Press, Cambridge, 1960. Calderbank, P. H., Trans. lnst. Chem. Eng., 36, 443 (1958). Delichatsios, M . A,, fhys. Fluids, 18, 662 (1975). Deiichatsios, M . A , , Probstein, R . F., J. Colloid lnferface Sci., 51, 394 (1975). Doulah, M. S.,lnd. Eng. Chem., Fundam., 14, 137 (1975). Gibson, C. H.. Masiello, 0. J., "Statistical Models and Turbulence," M. Rosenblatt and C. Van Atta, Ed., p 427, Springer-Veriag. Berlin, Heidelberg, New York, N.Y.. 1972. Grossman, G., lnd. Eng. Chem., Process Des. Dev., 11, 537 (1972). Hinze, J. O., A.I.Ch.E.J., 1, 289 (1955). Laats, M. K., Frishman, F. A,. Fluid Dynamics (translated from Russian), 8, 304 (1974). Lang, S. B., Wilke, C. R., lnd. f n g . Chem., Fundam., 10, 329; 341 (1971). Mlynek, Y . . Resnick, W., A.l.Ch.E. d., 18, 122 (1972). Ind. Eng. Chem., Fundam., Vol. 15, No. 2, 1976
137
Rotta, J. C., "Turbulente Stromungen." p 96, B. G. Teubner, Stuttgart. 1972. Shinnar, R.. J. FluidMech., I O , 259 (1961). Sprow, F. E., Chem. Eng. Sci., 22, 435 (1967). Swift, D. L., Friedlander, S . K., J. ColloidSci., 19, 621 (1964). Taylor, G. I., "The Scientific Papers of Sir Geofrey lngram Taylor," G. K. Batchelor. Ed.. D 457. Cambridoe Universitv Press. Cambridae. 1963. Van Atta, C:, Park, J., '"Statistical Models and Turbulence,"-M: Rosenblatt and C. Van Atta, Ed., p 402, Springer-Verlag, Berlin, Heidelberg, New York, N.Y.. 1972.
Vermeuien, T., Williams, G. M., Langlois. G. E., Chem. Eng. Prog., 51(2), 85-F (1955).
Receiued for review July 28, 1975 Accepted January 5, 1976 This research was supported by the Office of Naval Research under Contract No. N00014-67-A-0204-0057.
E XP E R IMENTA 1 TECHNIQ UES
Constant-Stress Rheology of Asphalt Cements Herbert E. Schweyer" and Fred Y. Kafka Department of Chemical Engineering, Universify of Florida, Gainesville, Florida 326 7 1
Constant shear rate rheometry in the range of 1 to 100 MP in capillary extrusion instruments requires costly instrumentation and is very time consuming for evaluations of the shear susceptibility of asphalt cements. The use of a constant-stress instrument powered by gas pressure provides a simple procedure with an almost instantaneous shear rate response. Such a rheometer has been developed; studies of repeatability and reproducibility are reported. The constant-stress results at 25 O C are compared with constant-shear data on a number of asphalts ranging from 19 to 77 in penetration. In addition, a limited number of results are shown for comparable viscosities run at 50 and 10 O C for paving grade asphalts and one roofing asphalt at 25, 60, and 150 O C . Pressure can increase the observed viscosity of asphalt cements. A procedure is presented for estimating the effect of pressure simultaneously from constant-stress measurements as evaluated from viscosity data at different stresses.
Introduction Studies in asphalt rheology over a number of years a t the University of Florida have emphasized capillary flow studies a t constant shear rates (Schweyer, 1972; Schweyer and Busot, 1971). However, exploratory studies in a constantstress mode indicated that considerably faster measurements could be made. This results because apparently there is no equilibration time needed to obtain a constantstress result compared to a long equilibrium time for constant-shear-rate methods. The sliding plate viscometer is a constant-stress device but has a limited upper range of stress. The cone-and-plate viscometer operates in a constant-stress mode but requires an equilibration period, and, in addition, it has low productivity and requires certain expertise. The apparatus and procedures reported here provide a method which meets many of the objectives of other methods in determining the absolute viscosities of asphalts with confidence over a wide range of temperatures from 5 to 60 "C or higher on either a routine or a research basis.
Apparatus and Procedure The essential components of the apparatus are shown in Figure 1 and a photograph of the setup is shown in Figure 2. Details of the sample tube, capillary, support ring, and plunger are shown in Figures 3 and 4. The apparatus consists of a set of interchangeable capil138
Ind. Eng. Chem., Fundam.. Vol. 15, No. 2, 1976
laries and sample tubes which permits selection of a suitable combination to meet any requirements of available loads for the viscosity range of interest which depends upon the test temperature. The assembled sample tube capillary and plunger as a unit is readily placed in the support ring and the assembly is placed on the lower plate. The drive plunger has a hemispherical adapter for alignment that fits over the plunger. The drive cylinder supplies the load and is supplied by compressed nitrogen a t the desired pressure ranges. The arrangement shown has a multiplying factor of 5 so that a gas pressure of 2.1 MPa (approximately 300 psi) provides a high range load of 6700 N (approximately 1500 lb of force). Application of a fixed load, through controlled gas pressure, results in a constant velocity movement of the plunger which is recorded by means of an LVDT transducer. I t requires an average of 2 to 4 min to obtain a velocity measurement for each setting of the load which means that a rheogram of shear stress vs. rate of shear can be obtained in about 10 min ( 3 points) to 25 min (6 or 7 points) depending upon the confidence required for the result. Actually if desired it would require only one reading after set up to obtain a viscosity value at a given load level for a standardized test. This procedure would require only about 5 min if the sample tube assembly has been preconditioned for the test temperature. However, more than one point is necessary to evaluate shear susceptibility and to compare the viscosity of different samples a t a common rate of shear.
