I n d . E n g . Chem. Res. 1990,29, 896-901
896
Solidification at the Entrance to a Subcooled Tube Y. Leong Yeow* Department of Chemical Engineeering, University
of
Melbourne, Parkuille, Victoria, Australia 3052
David T. Gethint and Roland W. Lewis1 Department of Mechanical Engineering and Department of Civil Engineering, University College of Swansea, Singleton Park, Swansea SA2 8PP, U.K.
T h e Galerkin finite element technique has been applied to solve the equations of motion and energy transport for laminar flow in tubes where the tube wall is at a temperature below the melting point of the liquid. T h e axial variation in the thickness of the crust formed on the tube wall and the pressure drop in the tube are presented for a range of the dimensionless parameters that describe this problem. T h e conditions for the onset of complete blockage of the tube due to freezing are explained in terms of the shape of the pressure drop versus flow rate plot. In many heat-transfer operations, heat transfer takes place in the presence of solidification or melting. Solidification of a metal casting or of a continuous melt spun polymer fiber and the deposition of a crust on the wall of a tube held below the melting point of the liquid flowing through it are examples of industrial processes in which heat transfer is accompanied by phase change. The analysis of such problems is considerably more complicated than similar problems in which there is no phase change. The metal casting problem, for example, can be treated as a transient heat conduction problem, but it is necessary to take into account the enthalpy of solidification and to keep track of the solid-liquid interface as it moves across the casting. Such problems are referred to as moving boundary problems. The melt spinning problem and the crust deposition problem can both be modeled as a steady-state problem with a stationary solid-liquid interface. Here, the complications are the shape and location of the interface, which are not known beforehand and have to be determined during the solution of the problem. These problems are known as free boundary problems. Free and moving boundary problems are discussed in great detail by Crank (1984). Recently, special finite element techniques have been developed to solve industrially important moving boundary problems. See, for example, Lewis and Roberts (1987). In comparison, the equally important free boundary problems have received less computational attention. In this paper, standard finite element computation is applied to a particular free boundary problem-determination of the steady-state profile of the crust deposited by a liquid in laminar flow on the inside wall of a subcooled tube. This problem has been investigated by Zerkle and Sunderland (1968). Starting from a simplified fluid mechanical model of the process, these authors were able to obtain series solutions for the crust thickness and pressure drop. In this paper, some of their simplifications are relaxed and attention is focused on the entrance region of the subcooled tube where these simplifications are most likely to break down. The main aim of this paper is to demonstrate the application of standard finite element computation, now widely available in the form of commercial software packages, to solve a problem of considerable practical interest and to draw physical conclusions from the numerical results.
Governing Equations Figure 1 is a schematic sketch of the problem under investigation-a liquid of constant physical properties undergoes steady laminar flow through a circular tube whose wall is kept at a constant temperature, Tw,below the melting point of the liquid, T M .Prior to entering the tube, the liquid is assumed to have a uniform temperature To,where To > TM. It is further assumed that, at the entrance to the tube, the liquid has the parabolic velocity profile of fully developed Poiseuille flow. These upstream conditions are consistent with the usual experimental arrangements of putting the test liquid through a long, thermally insulated calming section before entering the freezing section (Zerkle and Sunderland, 1968). As the liquid flows through the tube, a solid crust is formed on the tube wall. Initially, because of the large thermal driving force, the crust thickness increases rapidly. This gradually slows down as the liquid temperature TL( r , z ) approaches T,. The position of the solid-liquid interface, r s L ( z ) ,is determined by the rate of heat transfer between the laminar liquid core and the solid crust. TL(r,z) is given by the energy transport equation,
where p , C,, and kL are the density, heat capacity, and thermal conductivity of the liquid, respectively. The velocity components, u, and u,, are given by the NavierStokes and contunity equations; Le.,
i -(ru,) a du, +=0 az
-
r ar
The temperature in the solid crust, Ts(r,z),is obtained by solving the Laplace equation
* Author to whom correspondence should be addressed. 'Department of Mechanical Engineering. Department of Civil Engineering.
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6 1990 American Chemical Society
Ind. Eng. Chem. Res., Vol. 29, No. 5, 1990 897 SOLID CRUST
T
SOLID CRUS1
Figure 1. Crust deposition on a subcooled tube.
In their analysis, Zerkle and Sunderland assumed that u, remains parabolic in the converging liquid core. This is justified as long as the reduction in the radius is graudal (it is therefore not valid near the upstream end where the crust thickness increases rapidly). With this assumption, it is possible to obtain, from the continuity equation alone, expressions for u, and u,. The Navier-Stokes equation is not solved explicitly. In their derivation of the series solutions, Zerkle and Sunderland also ignored the axial conduction terms in eqs 1 and 3. These simplifications are valid for high Peclet number flow in eq 1 and follow as a consequence of the slowly convergent liquid core assumption in eq 3. In the present work, these simplifications were not made; the results thus obtained should therefore be valid for the upstream end. The boundary conditions of the problem are summarized below. At the entrance to the freezing section (see Figure 11, u, = 2a[1 - (r/ro)21
coincide with the actual interface, the two thermal boundary conditions cannot be satisfied simultaneously. This is used as the convergence criterion of the iterative procedure for locating the interface. Before embarking on the finite element computation, it is convenient to convert the governing equations and the associated boundary conditions into dimensionless form. The tube radius (ro)is taken as the length scale and the average velocity (a)at the entrance plane as the velocity scale. Dimensionless temperatures in the solid and liquid domains are defined by
Ts* = (Ts - Tw)/(To- Tw) TL* = (TL - Tw)/(To- Tw) Following from these, the dimensionless melting point is given by TM* = ( T M- T w ) / ( T o- Tw)where clearly 0 C TM* < 1.0. With the exception of temperature, upper-case symbols are used to denote dimensionless variables. The dimensionless equivalents of eqs 1-9 are
aTL*
+
dTL*
t(R,)
aR
azTL*
+ a22 (la)
u, = 0
TL = To
(4)
u is the average upstream
axial velocity and ro is the tube radius. Along the center line, (5)
On the tube wall, clearly, Ts = Tw
A t the exit plane it is assumed that the axial gradients of u,, u,, TL, and Ts all vanish. The pressure, p , is taken to be zero a t r = 0 on the exit plane. In all the results presented below, the exit plane is taken to be two tube radii from the entrance plane (Le., it is a short tube). The freezing section has been kept relatively short partly because of the need to keep the finite element computation time under control and partly to stop the complete feezing over the liquid core for a wide range of subcoolings. Although the freezing section is only twice the tube radius, it can be as long as 10-12 times the final radius of the liquid core. Finally, the boundary conditions on the solid-liquid interface rsL(z) are u, = u, = 0
U, = 2 ( 1 - R 2 )
ur = 0
(44
TL* = 1.0
u r =au, - - - -aTL* - -0 aR
aR
(7)
and
where n is the unit normal on the interface pointing out of the liquid domain. The two thermal boundary conditions ensure the continuity of temperature and heat flux across the interface. The interface is not known, and an initial guess has to be made. Unless the guess happens to
In the above equations, the dimensionless pressure, P, is defined as p / [ p 2 / ( 4 p r o 2 ) ] .The independent parameters of the problem are Reynolds number, Re = 2pr,,ii/p; Peclet number, Pe = 2pC r0u/kL;ratio of the thermal conductivities, k s / k L ; an$TM*.
Finite Element Meshes and Interface Iteration The principles of the Galerkin finite element method for solving equations such as (1)-(3) can be found in
898
[nd. Eng. Chem. Res., Vol. 29, No. 5 , 1990
Figure 2. Finite element meshes of the solid and liquid domains
textbooks such as that by Zienkiewicz and Morgan (1983) and will not be repeated here. In this investigation, physical variables, viz., U,, Uz,P, and T , were used in the finite element discretization. Alternative schemes, such as the vorticity-stream function formulation, can also be used. Provided the finite element mesh is properly designed and the interpolation functions have the necessary methematical properties (Zienkiewicz and Morgan, 1983), all these schemes can be expected to produce near identical results The finite element meshes used are shown in Figure 2. ABCD is the liquid domain, and AEB is the solid domain. AD is the entrance plane. BC and EB form the exit plane, CD is the central axis, and AE is the tube wall. AB is the solid-liquid interface to be determined by iteration. The liquid domain is divided into 200 8-noded quadrilateral elements, while the solid domain is divided into 70 8-noded quadrilateral elements and 4 6-noded triangular elements. The triangular elements were introduced to accommodate the inital increase in crust thickness. Following the normal procedure in finite element discretization, velocity components and temperature are approximated, within each quadrilateral element, by eight quadratic interpolation functions, while pressure is approximated by four linear functions. The corresponding numbers for triangular elements are six and three, respectively. The same elements were used for velocity, temperature, and pressure interpolations. In the case of linear interpolation, only the corner nodes of an element were utilized, while in quadratic interpolation both the corner nodes and the mid-side nodes were used. Notice, on the interface, that the nodes of the solid and the liquid domains coincide. Every time the solid-liquid interface is adjusted, the finite element mesh in each domain has to be regenerated. This was carried out in such a way that the number of elements in each domain remains unchanged. Furthermore, the same set of nodes always remain on the interface and their axial coordinates were kept fixed. Only the radial coordinates were changed. In order to keep the algorithms of mesh regeneration simple, within each domain, the nodes were equally spaced radially and axially. It was found that, for a wide range of flow and thermal conditions, these meshes were capable of providing the degree of accuracy required. The iterative scheme used to solve the free boundary problem starts with an initial guess of the interface AB. The governing equations of the liquid domain were then solved by the Galerkin finite element method, satisfying all the boundary conditions except the heat flux condition (eq 91, which was not imposed. Instead, the normal heat flux was computed, for each interfacial node, from the solution of (1)and (2). In the next step, the finite element solution of the d i d domain temperature was obtained
A -
F i g u r e 3. Isotherms in the solid and liquid domains. AB is the solid-liquid interface. (a) Pe = 37.5, Re = 5 X IO4, TM*= 0.5, k s / k L = 1.0; (b) Pe = 75.0, Re = 1 X 7'M* = 0.5, k s / k L = 1.0.
Here all the boundary conditions, except those on the interface, were imposed. On the int,erface,the normal heat flux obtained from the liquid domain was imposed. The resulting solid temperature on the interface, TsL*(Z 1, was in general not equal to the melting point, TM*. The deviation of T s L * ( Z )from TM*was used to guide the adjustment of the interface. The necessary radial adjustment o'R was computed according to i 10)
This adjustment was implemented at each interfacial node to yield the new interface for the next iteration cycle. The radial gradients required in (10) are already known and only need to be extracted from the liquid domain results. Experience has shown that, depending on the inital guess, convergence to the solid-liquid interface can be achieved in less than 10 iterations. Difficulties were encountered only when conditions are such that either AB is very close to AE or B gets very close to C. In such cases of very little freezing or close to complete freezing, it is clear that one or the other of the finite element meshes has become too degenerate to be able to yield physically meaningful solution. Such extreme conditions will not be discussed. The iterative scheme described here represents, probably, the simplest procedure for solving free boundary problems. It was found to be adequate for the present problem where the solid-liquid interface is a monotonic smooth function. Other more sophisticated iterative and noniterative methods for locating the free surface can be found in Crank 11984). Results and Discussion Typical results of the finite element computation are shown in Figure 3. This figure shows the isotherms in the solid and liquid domains. The interface is marked by AB. The value of the dimensionless parameters is indicated on the diagram. Isotherms obtained using a coarser mesh (100 elements in the liquid domain and 34 elements in the solid domain) are practically indistinguishable from those in Figure 3 except in the neighborhood of A where there i s
Ind. Eng. Chem. Res., Vol. 29, No. 5, 1990 899 L
0
0.5
01
U
I
0.2’
1.0 I
1,5 I
2.0
I
0.2
I
F i g u r e 4. Crust profile at different Pe. TM*= 0.5,Re = = 1.0.
ks/kL
F i g u r e 6. Crust profile at different k,/kL. Pe = 7 5 , Re = = 0.5.
TM*
i
0
0.5
1, o
1.5
2,o
0’
$
=
0.7;
0.2-
c
h P
F i g u r e 5. Crust profile at different TM*.Pe = 75,Re = = 1.0.
“C’
!
!
ks/kL
a discontinuity in the thermal boundary condition. This is taken as an indication that further mesh refinement is not warranted. It is relatively simple to carry out numerical investigations of the effects of the dimensionless parameters such as Pe,ks/kL, and TM*. Figure 4 is a plot of the solid-liquid interface for different Pe keeping all the other parameters constant. A large Pe means that a high proportion of the enthalpy of the liquid is convected downstream instead of being lost, by radial conduction, to the heat sink along the tube wall. This is reflected by the decrease in crust thickness with increasing Pe in Figure 4. For the lowest Pe in Figure 4, the solid crust has grown to approximately 90% of the tube radius in the short freezing section. A further decrease in Pe would lead to complete freezing of the tube. Figure 5 is a plot of solid-liquid profiles for different TM*. From its definition, it can be seen that large TM* corresponds to a high degree of subcooling and hence a thick crust. This trend is observed in Figure 5. All the results presented so far assume that k s / k L = 1.0. In real materials this ratio can be larger or smaller than unity. The effects of changing k s / k L are summarized in Figure 6. The trend exhibited by these profiles is again in agreement with physical expectation. A situation of considerable practical interest is the complete blockage of the flow as a result of freezing. The onset of complete blockage can be investigated by plotting the pressure drop across the freezing section as a function of, say, flow rate or Re. Figure 7 is a plot of dimensionless ], Re. The most pressure drop, A P = A p / [ p 2 / ( 4 p r o 2 )against outstanding feature of this plot is the existence of a critical Re at which AP is a minimum. A simple explanation for this can be found. For a given liquid, low Re also implies low Pe and hence extensive freezing in the tube. This is responsible for the high pressure drop below the critical Re. An increase in Re is accompanied by an enlargement of the liquid core and a reduction in the pressure drop. The subsequent increase in AP with Re is the normal phenomenon observed in laminar flows. Below the critical
Figure 7. Pressure drop vs Reynolds number. TM*= 0.5,k s / k L = 1.0, Pr = C , p / k = 7.5 x lo‘.
Re,an increase in pressure drop (the driving force of the flow) is accompanied by a reduction in flow rate. This section of the flow curve has a negative slope and is unstable. Above the critical Re the slope becomes positive, and the flow there is stable. These observations can be arrived at by the following simple physical argument. The dimensionless viscous drag, F , acting on the molten liquid flowing through the tube is a function of Re,i.e., F = F(Re). Under steady-state conditions, this viscous drag is balanced by the applied pressure difference hP. When the steady-flow is perturbed so that Re changes to Re + 6, the viscous drag in the perturbed flow is given, approximately, by F + (dF/dRe)G. If dF/dRe is positive, the resulting imbalance between the viscous drag and the applied pressure difference will cause the flow to move in the opposite direction of the perturbation. This implies that the flow is stable to small disturbances. Conversely, a negative d F / d R e means that the perturbation will increase in size, leading to flow instability. An unstable flow is not physicall realizable. Any disturbance will cause a point on the unstable portion of the flow curve to be pushed toward the Re = 0 axis. This can be interpreted to mean that below the critical Re the tube will be completely frozen. Thus, the computation of the critical Re serves to identify the conditions for the onset of blockage due to freezing. The critical Re is clearly dependent on, among other factors, the length of the freezing section. Figure 7 is for a relatively short freezing length, too short to be of great practical interest, and is included here merely to illustrate a phenomenon. Following the simplified analysis of Zerkle and Sunderland, Cheung and Epstein (1984) were able to produce flow curves for freezing length as large as 100 radii. Their results also showed the existence of a minimum in the pressure drop. These authors also discussed the practicallv important problem of freezing in tubes when
900 Ind. Eng. Chem. Res., Vol. 29, No. 5 , 1990
ture-dependent viscosity and the finite element meshes modified to include a short length of the calming section so as to yield a more realistic entrance velocity profile. Other possible improvements are, as mentioned before, a longer freezing section and a higher range of Reynolds numbers. All these can be implemented with no foreseen difficulties but a t the expense of greatly increased computation time.
i ?C-
2
I
>:
I _ _ _ -
Figure 8. Comparison of crust profile. (i) Zerkle and Sunderland for all Pe. (ii) Finite element computation, Pe = '75. (iii) Finite element computation, Pe = 25. In all curves TM*= 0.5, k s / k L = 1.0. and R P =
the flow is no longer laminar. The results of the finite element computation can be checked against that from the analysis of Zerkle and Sunderland by comparing the crust profile, RSL(Z),under different conditions. Figure 8 is a plot of RSL(Z) against Z l P e for TM*= 0.5 and k S / k L= 1.0. According to Zerkle and Sunderland, plotting R S L ( Z )against Z l P e will result in a single master curve valid for all P e . Curve i is the master curve replotted from Figure 3 of Zerkle and Sunderland (1968). Curves ii and iii are typical finite element results for the combination of Pe = 75, Re = loT3,and Pe = 25, Re = respectively. As expected, the agreement between curve i and the finite element results improves as Pe becomes larger. In view of the major simplifications in the analysis of Zerkle and Sunderland, the discrepancies between their results and that of finite element computation should not come as a surprise. See below. It should also be pointed out that, in a more thorough comparison, it will be necessary to extend curve i to lower values of Z l P e and the finite element computation to longer freezing sections and a wider range of R e . All these will involve extensive additional computation and have not been attempted. The validity of the assumption that the radial velocity can be ignored made by Zerkle and Sunderland can be checked by evaluating the radial velocity component as a fraction of the magnitude of the local velocity. From the finite element results, typically, near the entrance plane, this fraction can be as large as 0.3-0.4. But as Z becomes larger, this decreases rapidly to less than This shows that, apart from a small neighborhood of the entrance piane, the approximate solution of Zerkle and Sunderland is a very good one. Consequently, in investigating freezing in long subcooled tubes, it is only necessary to carry out finite element computation for a short entry zone and match this to the much simpler solution of Zerkle and Sunderland downstream of this zone. This situation is very similar to the matching procedure used by Trang and Yeow (1986) to study the free boundary problem associated with the extrudate swell of Newtonian fluids, where they matched the finite element solution of the swell region to the simple solution of the one-dimension approximation of the fully developed flow downstream of the swell. In the current model of flow through a subcooled tube, it was assumed that the flow is fully developed a t the entrance plane and the liquid viscosity is independent of temperature. These are probably the most drastic of all the assumptions made. The finite element scheme described above can be generalized to incorporate tempera-
Conclusion A finite element scheme has been developed to solve the free boundary problem describing the solidification of a liquid on the wall of a subcooled tube. The finite element results are in reasonable agreement with that from a simplified analysis in the literature. The finite element computation has revealed the presence of a critical Reynolds number below which any increase in pressure drop across the tube is accompanied by a reduction in flow through it. This anomalous situation can be taken as an indication of the onset of complete flow blockage due to freezing. Acknowledgment The mechanism of flow instability a t low Re was discussed with Christoph Seeling of the Department of Chemical Engineering, University of Melbourne. His input is gratefully acknowledged.
Nomenclature C, = specific heat k = thermal conductivity n = normal direction p = pressure P = dimensionless pressure Pe = Peclet number P r = Prandtl number r = radial coordinate R = dimensionless radial coordinate Re = Reynolds number T = temperature T * = dimensionless temperature u = velocity lJ = dimensionless velocity u = average velocity z = axial coordinate Z = dimensionless axial coordinate Greek Letters 6 = perturbation A = difference fi = viscosity p =
density
Subscripts L = liquid M = melt r = radial S = solid SL = solid-liquid interface U' = tube wall o = entrance plane z = axial
Literature Cited Cheung, F. B.; Epstein, M. Solidification Melting in Fluid Flow. In Advances in Transport Processes; Mashelkar, R. A,, Majumdar, A. S., Kamal, R., Eds.; Wiley Eastern: New Delhi, India 1984; Vol. 3. Crank, J. Free and Moving Boundary Problems; Oxford University Press: Oxford, U.K. 1984. Lewis, R. W.; Roberts, P. M. Finite Element Simulation of Solidification Problems. Appl. Sci. Res. 1987, 44, 61-92.
I n d . E n g . C h e m . Res. 1990, 29,901-909
Trang, C. T.; Yeow, Y. L. Extrudate Swell of Newtonia and NonNewtonian Fluids-the Effect of Gravitational Body Force. J . Non-Newtonian Fluid Mech. 1986,20, 103-116.
Zerkle, R. D.; Sunderland, J. E. The Effect of Liquid Solidification on a Tube upon Laminar-flow Heat Transfer and Pressure Drop. J . Heat Transfer 1968, 90,183-190.
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Zienkiewicz, 0. C.; Morgan, K. Finite Elements and Approximation; Wiley: London, 1983.
Received for review June 8, 1989 Revised manuscript received December 26, 1989 Accepted January 25, 1990
Characteristics of Electric-Field-Induced Oscillations of Translating Liquid Droplets Timothy C. Scott,* Osman A. Basaran, and Charles H. Byers Chemical Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831
Aqueous droplets falling through 2-ethyl- 1-hexanol have been forced to undergo shape oscillations by pulsed electric fields. Frequency and decay factors for droplets with a radius in the range 0.0296-0.1243 cm were measured and found to be well described by appropriate theories. Frequencies were predicted within several percent by accounting for flow past an oscillating drop, for droplets with radii in the 0.06-0.1243-cm range. Satisfactory representation of the frequency (within 12%) is given by this theory, and a numerical solution of the dispersion equation that governs the oscillation of isolated drops in the 0.0296-0.06-cm range. Measured decay factors lay between the predictions of two theories. These measured natural frequencies coincide with the pulsed electric field frequency, which most easily causes droplet instability and rupture. Finally, the new results eliminate some discrepancies between theory and experiment in the literature, which previously have been incorrectly attributed to inaccuracies in bulk and interfacial properties.
Introduction Interaction of drops and bubbles with electric fields is prevalent in nature and is important in many technological applications. Situations displaying similar behavior can be found in such diverse areas as colloidal systems (Miller and Scriven, 1970), meteorology and cloud physics (Sartor, 1969), electrostatic spraying of liquids (Balachandran and Bailey, 19811, power engineering applications (Krasucki, 1962), nuclear physics (Bohr and Wheeler, 1939), aerosol science (Whitby and Liu, 1966), and many others. The behavior that links these seemingly unrelated situations together is the physics describing droplet formation, stability, oscillation, breakup, and coalescence. Interaction of droplets of a dispersed liquid phase with a surrounding continuous liquid phase represents the fundamental unit in liquid-liquid solvent extraction. Figure 1 is a schematic diagram representing this “basic building block” that can be used to characterize masstransfer behavior in extraction. In terms of this simple system, mass-transfer rates between the droplet and continuous phases depend upon the physicochemical properties of transferring species and solvents as well as on the hydrodynamic state of the system. Although one may envision circumstances under which electric field and charge effects may influence the physicochemical properties of the components, in general, the interactions between the field and liquid-liquid system are manifested as changes in droplet-continuum hydrodynamics. This applies in virtually all fluid-fluid systems of interest under applied electric (Melcher and Taylor, 1969) and magnetic fields (Rosensweig, 1985). Some of the most important effects of these changes can be understood in terms of the following types of interactions: altered droplet translation, deformation, stability, and oscillation (Scott, 1989). *Author to whom correspondence is to be addressed.
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Previous studies in our laboratory have involved the use of pulsing dc electric fields to cause oscillation of aqueous droplets that are surrounded by a nonconducting continuous phase (Scott, 1986; Wham and Byers, 1987; Scott, 1987; Scott and Sisson, 1988; Scott and Wham, 1989). In these studies, imposed droplet oscillations were generally carried out for one of two purposes: (1)the use of steady, stable oscillations for altering droplet hydrodynamics; or (2) the creation of high-amplitude oscillations that lead to droplet breakup and/or emulsification. The use of stable oscillations has the effect of changing the convection patterns within and around liquid droplets, thereby increasing the rate of heat or mass transfer in the liquidliquid system. Droplet breakup and emulsification in a transient electric field leads to the creation of vast amounts of interfacial surface area and lends possibilities for droplet size control and phase separation. A central issue to both of the aforementioned modes of operation involves knowledge of the so-called “natural oscillation frequency” of the droplet. If the transient electric field is modulated at this frequency, the droplet will undergo the maximum amount of distortion for a given field strength. In the case of stable oscillations, driving the droplet at this frequency will induce the largest changes in the associated droplet-continuum velocity fields. Previous observations on droplet rupture in pulsed fields (Scott, 1987) also indicate that forced oscillation of a droplet at a “preferred” frequency enables a relatively low field strength to be used to create instabilities. Because this preferred frequency is a strong function of droplet size, it was suggested that it may be possible to use pulsed electric fields to perform droplet size control in liquidliquid processing systems. In order to further investigate such behavior, a new series of droplet oscillation experiments has been carried out in which the method of natural frequency determination is based on the measurement of the aspect or amC 1990 American Chemical Society
