at finite velocity using the procedures of the present paper [or the more precise ones suggested by Shor, et al. (1968a)l when the membrane is the ion-exchange type or the procedure suggested by Sheppard and Thomas (1970b) when the membrane is neutral; finally calculate c,/cl from eq 20. The present results confirm the previous analysis of Shor, el al. (1968a), in which linear extrapolation on a plot of In [(I - Robsd)/Robsd]us. v/us” was used to obtain the value of the intrinsic rejection a t infinite circulation velocity for ion-exchange type membranes. As shown by the approximate equations, the effect of the concentration dependence of intrinsic rejection is reflected primarily in the value of coefficient K of eq 16 and 18, and does not affect the value of the exponent on flux or velocity and hence the extrapolation to infinite circulation velocity on a plot of log [(l Robsd)/RobsdlUS. u/ua”. Nomenclature
C* cu ct
= =
= = = =
Zu* j
J K
KaPP
= = = = =
= = = =
= =
membrane capacity, equiv/kg of HzO in exchanger concentration at membrane-solution interface, mole/ kg of HzO concentration in turbulent core, mole/kg of HzO concentration of product, mole/kg of HzO equilibrium distribution coefficient, dimensionless hyperfiltration j-factor, dimensionless flux, gal/ft2.day coefficient in eq 1 coeficient in eq 18 moles of solute/kg of HzO eq7 Reynolds number, dimensionless Schmidt number, dimensionless observed rejection, dimensionless intrinsic rejection, dimensionless intrinsic rejection at velocity, u,dimensionless intrinsic rejection a t infinite circulation velocity, dimensionless
u
= axial velocity, cm/sec
u z
= flux, cm/sec = charge of ion with same sign as membrane fixed charge
GREEKLETTERS
re -yi* Y*
= YI:c/Yi = mean ionic activity coefficient of salt in membrane =
mean ionic activity coefficient of salt in solution
literature Cited
Brian, P. L. T., Proc. Int. Symp. Water Desalination, 1st 1, 349 (1965). Hamer, E. A. G., Officeof Saline Water Report 424 (1969). Hendricks, T. J., Williams, F. A., Desalination 9, 155 (1971). Johnson, J. S., Jr., Dresner, L., Kraus, K. A., in “Principles of Desalination,” pp 345-439, K. S. Spiegler, Ed., Academic Press, New York, N. Y., 1966. McAdams, W. H., “Heat Transmission,” 3rd ed, p 220, McGrawHill, New Xork, N. Y., 1954. Richardson, J. L., Segovia, G., Baerg, W., Anderson, M. L., Office of Saline Water Report 455, December 1968. Richardson, J. L., Segovia, G., Brodie, A. O., Office of Saline Water Report 576, December 1970. Rosenfeld, J., Loeb, S., Ind. Eng. Chem., Process Des. Develop. 6, 122 (1967). Sachs, S. B., Baldwin, W. H., Johnson, J. S., Desalination 6, 215 (1969). Sheppard, J. D., Thomas, D. G., Desalination 8, 1 (1970a). Sheppard, J. D., Thomas, D. G., Appl. Polym. Symp. KO.13, 121 (1970b). Sherwood, T. K., Brian, P. L. T., Fisher, R. E., IND. EXG.CHEM., FUNDAM. 6 , 2 (1967). Sherwood, T. K., Brian, P. L. T., Fisher, R. E., Dresner, L., IND. ENG.CHEM.,FUNDAM. 4, 113 (1965). Shor, A. J., Kraus, K. A., Johnson, J. S., Smith, W. T., Jr., IND. ENG.CHEM.,FLJKDAM. 7, 44 (1968a). Shor, A. J., Kraus, K. A., Smith, W. T., Jr., Johnson, J. S., J. Phys. Chem. 72, 2200 (196813). RECEIVED for review October 15, 1970 ACCEPTED February 24, 1972 Research sponsored by the Office of Saline Water, U.S. Department of the Interior, under Union Carbide Corporation’s contract with the U. S. Atomic Energy Commission.
Drop Formation in Non-Newtonian Fluids Rajinder Kumar” and Yellamraju P. Saradhy Department of Chemical Engineering, Indian Institute of Science, Bangalore 16, India
Data have been collected for formation of drops in non-Newtonian fluids. An existing model has been modified to explain the results. The modified model proposed predicts the drop volumes satisfactorily. It i s found that consideration of non-Newtonian behavior i s important at low and medium flow rates, and its importance decreases with increasing flow rates.
A number of investigators (Hayworth and Treybal, 1950; Kumar, 1971; Null and Johnson, 1958; Rao, et al., 1966; and Scheele and Meister, 1968) have reported studies on drop formation from single nozzles submerged i n Newtonian fluids. However, the phenomenon of drop formation in non-Newtonian fluids has not as yet been reported. As these fluids are industrially important, an attempt has been made in the present investigation to study drop formation in these fluids.
Experimental Setup and Procedure
The setup employed for drop formation studies is of the same type as used by Rao, et al. (1966). Drops were formed by introducing the dispersed phase a t controlled flow rate through a capillary submerged in the non-Newtonian fluid. The drop volume was determined from the flow rate and frequency measurements. Frequencies up to 150 drops/min were observed visually. For higher frequencies a strobometer Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
307
Table 1. Range of Variables Studied in the Present Investigation Range Variables
Sodium silicate solutions
Concentration, w t % 2 Deiisity of continuous phase, g/cm3 1.01558 n, dimensioiiless 1.21137 k , g/cm sec2-, 0.001884 Interfacial tension dyn/cm 31.33 X,,, dimensionless 0.976 Diameters, cm 0.106
Sodium carboxymethylcellulose solutions
10
30
1.06372 1.14029 0.004279 30.81 0.982 0.134
1.23228 1.02885 0.045526 30.6 0.992 0.17
was used. Before each run, the two phases were mutually saturated. Experiments were conducted for different capillary diameters and continuous phases. Analar grade benzene of BDH (99.9% pure) was used as the dispersed phase in all the experiments. Carboxymethylcellulose (CMC), Carbopol, and sodium silicate solutions made in distilled water were employed as the continuous phases. The power law parameters were varied by changing the concentrations of the solutions. Carboxymethylcellulose solutions were shown to exhibit viscoelastic behavior (Kotaka, et al. 1959). The CMC solutions used in the present investigation, however, were tested for the viscoelasticity by performing a simple experiment (Bird, 1971) attributed to Philippoff. Spheres dropped in viscoelastic fluids were shown to fall and then oscillate up and down because of the pronounced elastic effects. The same test was performed on the C X C solutions used in the present study. Only those which failed to produce oscillations, indicating the absence of viscoelasticity, have been included here. The diameters of the capillaries a t the drop-forming end were measured by making use of a traveling microscope. Densities were measured through a specific gravity bottle. A du Suoy tensiometer was employed for the measurement of interfacial tensions. The range of variables covered is presented in Table I. The power law parameters were determined by a capillary viscometer.
1.5 1.00199 0.7647 0.7371 30.8 1.251
3.0 1.0068 0.75974 4.57417 30.15 1.257
4.0 1.01583 0.66327 30.4517 29.15 1.266
Corbopol solution
1 1.00018 0.7939 0.8089 21.2 1.222
been reported extensively (Slattery, 1962; Tomita, 1959; Wallick, et al., 1962; and Wassermann and Slattery, 1964) and is given by
in which X , is a function of power law parameter n and N R ~ ' is the modified Reynolds number defined as
From the definitions of CD (Bird, et al., 1960) and v (Kumar and Kuloor, 1967) and eq 1 and 2, the drag force F D becomes
X, is not rigorously known but its upper and lower bounds have been reported by Wasserman and Slattery (1964). In the present investigation, the mean value has been employed for calculation purposes. Expressing the various forces quantitatively as done by Kumar (1971) along with the drag expression from eq 3, the first-stage equation becomes
Unified Model Extended to Non-Newtonian Fluids
An attempt has been made to explain the data collected in the present investigation through the extension of the unified model of Kumar (1971) to non-Newtonian fluids. This model was chosen because of its application over a wide range of variables to both drop and bubble formation phenomena. Drop is considered to form in two stages and the final drop volume is the sum of the volumes added during the two stages. During the expansion stage, various forces acting on the forming drop are upward buoyancy force, upward kinetic energy force, downward inertial force, downward viscous drag, and downward interfacial tension force. All forces other than viscous drag acting during drop formation in non-Newtonian fluids are same as for h'ewtonian fluids. The drag term must be replaced by an equivalent which takes into consideration the drag in power law fluid. The viscous drag experienced by a sphere during its creeping motion in a power law fluid has 308
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
VE is calculated by trial and error. The Harkins and Brown (1919) correction has not been employed during calculations for the same reason as given by Kumar (1971). Thus the present equation cannot be employed when the flow rates are vanishingly small, where eq 4 becomes
Following Kumar's analysis, the equation for the detachment stage is written as
motion of the drop is negligible compared to drag due to expansion. Then eq 6 reduces to
where dx
=QdT T = VE Qt v
+
241rX,k A = 2n+l(3/,,)(n-Z)/3
B =
(Po
dzx - - Kit dtz
- Kz
Qps
- Pd)g QP~
The above equation is solved with the boundary conditions a t t = 0, v = dx/dt = 0, and x = 0 to obtain Klt3 - K2tZ- 62 Ps = Pd
f
11/16Pc
The above equation is the same one as used by Kumar (1971) except for the drag term which has been modified. The actual drag term which has been used for modification in the second stage is
Equation 6 is solved by Runge-Kutta method with the boundary conditions dx/dt = 0 and x = 0 a t t = 0 (or T = VE), for obtaining the final d.rop volume VF. Simplified Equation
At the cost of some accuracy eq 6 is linearized with the following three assumptions. (i) Mass is constant and is equal to the mass of the drop a t the end of first stage, except in the buoyancy force. (ii) The drag due to expansion remains constant and is equal to the drag on the drop a t the end of expansion state. (iii) The additional drag due to the upward
=
0
(9)
From eq 9 the time of detachment t c is calculated and then the final drop volume is found using the equation
VF
VE
+
00) It was found during computation that eq 6 converges very slowly. The simplification of eq 6 not only resulted in giving an analytical solution but also helped in reducing the computation time by more than a hundred times. The resulting error introduced during the process of simplification is found to be only 18% in predicting the final drop volumes. A typical comparison is made in Table I1 between drop volumes found from experiments and those found from eq 6 and eq 10 through eq 4 and 9. =
Qtc
Results and Discussion
Figures 1-3 show a typical set of data collected and compared with the model. I n all the figures the drop volumes are plotted against the flow rate taking the diameter of the capillary as a parameter. The points are experimental and the solid lines are those calculated from eq 6. I t is evident that the model predicts the drop volumes satisfactorily. The influence of power law parameters n and k could not Ind Eng. Chem. Fundom., Vol. 11, No. 3, 1972
309
0.01: O.O!
ti
0.04
t
% Y
z
’
0.03
0.02
0 0
A
THEORETICAL
EXPERIMENTAL
I
0.012
0.1
0.I
now
RATE
cmVsoc
0.3 -c
0.4
C
Figure 3. Benzene injected into 30% sodium silicate solution
be studied independently for the reason that these could not be varied without varying the other properties. However, the influence of flow index n is predictable from the model. A comparison is made in Figure 4 between the drop volumes calculated from eq 6 for n = 1, n = 0.63, and n = 1.211. I t may be seen that the difference between the two curves n = 1 and n = 1.211 goes on reducing with the increase in flow rate and finally the two merge together. The reason for this is while the drag force increases with the flow rate, so 310 Ind.
Eng Chem. Fundam., Vol. 1 1 , No.
3, 1972
does the kinetic energy force. As the flow rate is increased, the relative difference between these two forces decreases and finally the drop volume becomes essentially independent of the drag force. At this stage the lines for different values of n merge together. For n values less than 1.0, it is seen that n has insignificant influence and the drop volumes may as well be predicted with the model for Newtonian fluids. For a limited range, the model has been tested for bubble formation in non-Newtonian fluids also and is found to be
5
~~
~~
~
~~
~~
NRel
Table II. Comparison of Drop Volumes Obtained from Simplified Equation with Volumes Obtained Experimentally and with Those Found from Rigorous Equation; 27& Sodium Silicate Solution‘
Diameter,
cm
Flow rate, cm3/sec
Exptl
0.010 0.0743 0.102 0.0725 0.194 0.0560 0.134 0.010 0.0925 0.134 0.0900 0.227 0.0770 0. I7 0.06 0.1204 0.252 0.1150 0.495 0.0352 a n = 1.21137; k = 0.00188 g/cm 1.01558 g/cm3; pd = 0.87 g/cm3; y = 0.106
Drop volume, cm3 Calcd from Calcd from the rigorous the simplified equation equation
0.07521 0.07310 0.05835 0.09556 0.09268 0.08050 0.12150 0.11610 0.05200 sec2-%;X, = 31.33 dyn/cm.
0,07803 0.08569 0.06594 0.09794 0.10052 0.08648 0.13610 0.12274 0.05475 0.976; po =
applicable. When n = 1, the model reduces to the unified model which has already been tested for both bubble formation and drop formation in Newtonian fluids.
D
FD 9
k
KI, K2 n
= =
r
=
TE
VE
= = = = = = = =
BE
=
2
= =
t tc te
T 21
V
X,
modified Reynolds number, dimensionless volumetric flow rate, cm3/sec radius of the drop, cm radius of the force balance drop, cm time, sec time of detachment, sec time in the expansion stage, sec substitution as given in the text velocity, cm/sec volume of the drop, em3 volume of the force balance drop, cm3 volume of the final drop, cm3 distance from the capillary tip, cm function of the parameter, n, dimensionless
GREEKLETTERS = interfacial tension, dyn/cm = density of continuous phase, g/cm3 = density of dispersed phase, g/cm3 = Pd l l / ~ ~ ~ cg/cm3 ,
Y Pc
+
Pd P S
Literatue Cited
Bird. R. B.. Dersonal communication. 1971. Bird; R. B:, Stewart, W. E., Lightfoot, E. N., “Transport Phenomena,” p 182, Wiley, London, 1960. Harkins, W. D., Brown, F. E., J . Amer. Chem. Sac., 41, 499 (1919). Hayworth, C. B., Treybal, R. E., Ind. Eng. Chem., 42, 1174 (1950). Kotaka, T., Kurata, M., Tamura, M., J. A p p l . Phys. 30, 1705 (1 959 ).
Nomenclature
A , B , C, D,E CO
Q
substitutions as given in the text drag coefficient, dimensionless = diameter of the capillary, cm = drag force on the sphere moving in a power law fluid, g/cm sect = acceleration due to gravity, cm/sec2 = parameter in power law model, g/cm sec2-n = substitutions as given in the text = parameter in power law model, dimensionless
= =
KU&&,’R., Chem. Eng. Sci. 26, 177 (1971). Kumar, R., Kuloor, N. R., Chem. Tech., Leipzig 11, 657 (1967). Null, H. R., Johnson, H. F., A.I.Ch.E. J . 4,273 (1958). Rao. E. V. L. N.. Kumar. R.. Kuloor. N. R.. Chem. Ena. Sci. 21. 867 (1966). ’ Schekle, U.~F., Meister, B. J., A.I.Ch.E. J . 14, 9 (1968). Slattery, J. C., A.I.Ch.E. J . 8 , 663 (1962). Tomita, Y., Bull. Sac. Mech. Eng. 2, 469 (1959). Wallick, G. C., Sovins, J. C., Atterburn, D. R., Phys. Fluids 5 , 367 (1962). Wassermann, M. L., Slattery, J. C., A.I.Ch.E. J . 10, 383 (1964). RECEIVED for review December 28, 1970 ACCEPTEDMarch 2, 1972 I
,
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
31 1
