Spray Formation from Pressure Cans by Flashing Eran Sher and Chaim Elata’ Department of Mechanical Engineering, Ben-Gurion University of the Negev, Beer-Sheva, lsrael
The formation of spray through flashing, from containers pressurized by volatile propellants dissolved into the spray liquid, is studied. A model for the flashing process is proposed. While the pressurized solution is discharged from the container into the atmosphere, vapor bubbles are produced. These bubbles grow rapidly through evaporation of the propellant. When they touch each other, flashing is assumed to occur-the bubbles “explode” and an aerosol is formed. The energy contained in these exploding bubbles is, in part, transformed into surface energy of the droplets of the aerosol. On the basis of this model of the flashing process, a mathematical derivation is presented, expressing the average spray droplet diameter in terms of the physical properties of the binary fluid system. An experimental program was carried out in which the droplet size distribution of an aerosol generated by a flashing process was measured. The average droplet diameter was correlated with the pressure in the container and with the relative concentration of the propellant in the mixture. Experiments were carried out at different temperatures. Comparing the experimental results with the proposed theory, it was found that deviations from the thermodynamic equilibrium have to be taken into account. Doing so, incorporating deviations from equilibrium for water vapor, the experlmental data fitted the theoretical prediction for the average drop size quite well over the range of experimental pressures and temperatures.
Introduction Aerosols, generated from spray cans by flashing, have wide household utilization as insecticides, deodorants, hair sprays, etc. The literature concerning the physics of the flashing process appears to be scarce. In this work, an attempt is made, based on a proposed physical model for the process of spray formation, to correlate the average drop size in the aerosol with the pressure in the container and the physical properties of the fluids involved. In spray cans, a liquidized propellant, usually a halocarbon, is dissolved into the liquid to be dispersed. This spray liquid usually has a vapor pressure less than the ambient pressure. Due to the high vapor pressure of the propellant, the saturation pressure of the solution, a t the prevailing temperature, will be greater than the ambient pressure. When the nozzle of the container is opened, the difference between the internal and ambient pressure forces the solution out of the can. The solution is discharged into the atmosphere and part of the propellant is transformed, seemingly spontaneously, into a gas. Depending on the physical properties of the liquid solution, this may result in flashing, which produces an aerosol. The liquid propellant seemingly “explodes” and the surrounding liquid is transformed into small drops. If this does not occur, foam may be generated. The flashing process results in primary atomization of the spray liquid. The size of the droplets may further decrease as a result of hydrodynamic break-up, due to the relative velocity between the droplets and the surrounding air and due to the further evaporation of the propellant and the spray liquid. In the following analysis, these latter effects are assumed to be negligible, an assumption that is subsequently corroborated by the experimental results. General observations concerning the drop size in an aerosol produced by flashing were presented by Wiener (1958) and later summarized by Sanders (1972). Wiener argued, on the basis of thermodynamic considerations, that the smaller the latent heat of the propellant, the more of it will flash with greater intensity, so that therefore, the average drop size will be smaller. On the other hand, the amount of propellant which will flash is limited, since when part of it evaporates, the temperature of the remaining solution will drop and may reach the equilibrium temperature of the propellant at the ambient pressure. For the same reasons, smaller droplets will result when the specific heat of the solution is higher and the boiling
temperature of the propellant is lower. Wiener also found that the average drop size becomes smaller for a larger density of the spray liquid and a smaller density of the propellant. It was also suggested that with less surface tension of the spray liquid, a greater quantity of smaller droplets will be formed having a larger total surface area. A clue to the flashing mechanism may be deduced from the work of Brown and York (1962). They presented an empirical correlation between the rate of growth of the bubbles produced when the pressure of a liquid is released suddenly and the size of the drops generated through flashing. Their experiments with water and Freon 11show the average drop size to be inversely proportional to the growth rate coefficient of the bubbles and in general to increase with Weber number.
Theoretical Considerations Under ideal conditions, the formation of vapor nuclei in a liquid is due to thermal fluctuations. Boltzmann’s probability theory permits us to predict the number of nuclei generated. The formation of nuclei in a real flowing liquid is due to the wall roughness of the tube through which it flows and/or to the nonuniformity of its cross section. The generated nuclei, in equilibrium with the surrounding liquid discharging into the atmosphere, will be bubbles of radius
Ro = 2u/AP
(1)
where u is the surface tension of the liquid and A P is the difference in pressure of the pressurized liquid and the outside. When the pressure of the liquid decreases, the bubbles will grow due to the evaporation of the additional liquid at the bubble walls. For the case under consideration, it is assumed that from the solution of propellant into the spray liquid, only the propellant evaporates to fill the bubbles with its vapor. A theory for the growth of such bubbles under thermodynamic equilibrium was presented by Plesset and Zwick (1954) and is summarized as follows. Assuming the bubbles to be spherical and without translatory motion relative to the viscousless noncompressible liquid, while the inertialess vapor has a uniform pressure and temperature-and is thermodynamically at equilibrium-the equation of motion may be expressed by
R f i + ( 3 / 2 ) f i 2= ( l / p l ) ( P b - Pa)
(2)
Ind. Eng. Chem., Process Des. Dev., VoI. 16, No. 2, 1977
237
PRESSURE VESSEL
account. Although this effect was neglected in the following derivation, the resulting expression does seem to fit the experimental results well. It is assumed here that flashing will occur when the growing bubbles form a close-packed cubic array just touching each other, at which time they will burst. This process transforms the discharging mass from a continuous liquid phase with separate vapor bubbles, into a continuous gas phase with separate liquid droplets as shown schematically in Figure 1. The energy contained in the bubbles a t bursting will be transformed partly into surface energy of the droplets produced. In the assumed close-packed array fractional volumes occupied by vapor and liquid are a/6 and 1 - 7r/6, respectively. The mass of liquid a t bursting per unit volume of bubblecontaining fluid will thus be
FLUID EGRESSING F R O M ORIFICE
’
‘AMBIENT
P~ESSURE
CROSSSECTION OF EXPANSION J E T WITH BUBBLES JUST PREVIOUS TO sLF(STI
= pl(1 - a/6)
Figure 1. The proposed flashing mechanism.
where p1 is the liquid density, P b is the pressure inside the bubble and Pais the ambient pressure. For bubbles considerably larger than the original nuclei (R >> Ro), the radius will change with time t according to R = CtlI2
(3)
where C is a growth rate coefficient defined by
c = 2(3/a)1”$J(Cp/L)(T1 - Tb)(pl/pL,)D1”
(4)
where $ is a dimensionless coefficient, C, is the specific heat of the solution, L is the vapor heat of the propellant, T Iis the temperature of the liquid, T , is the saturation temperature a t ambient pressure, pV is the density of the vapor, and D is the thermal diffusion coefficient. (In the following T I- T , = AT.) According to the theory of Plesset and Zwick, $ = 1. Hooper and Abdelmessih (1966) carried out experiments to measure the rate of growth of water vapor bubbles at different temperatures above the boiling temperature at atmospheric pressure. For temperatures less than 15 O F above the boiling point, the value of $ = 1 was confirmed. A t higher temperatures, $ was found to decrease in value, due to deviations from thermal equilibrium, which was one of the basic assumptions in the theory of Plesset and Zwick. Utilizing the linearized Clapeyron equation, AT = (T/L) ( Upv) AP and assuming the vapor to follow ideal gas laws such that pv = MP/(R’T), eq 4 becomes
(7)
Assume that as a result of the flashing process, n droplets are generated per unit volume. Experiments show that the size distributions of such droplets are log normal of the first order (Herzka and Pickthal, 1961; Silverman, et al., 1971). Following the latter, the total mass of all n droplets may then be expressed by K-
,tt = n p i - d 6
a = n p l - djOi exp(4.5 In’ erg)
6
(8)
where d is the droplet diameter, djo is the number median diameter, and C T ~is the geometric deviation. This mass will be equal to the available mass expressed in eq 7. Equating eq 7 and 8, one obtains 1 - a/6
n = ( ~ / 6 ) d ; , ,exp(4.5 ’ In’ ug)
(9)
Assuming the number of nuclei generated originally per unit volume of fluid to be m and the radius of the uniform spherical, touching bubbles at bursting R*
m (1 - ~ / 6 ) ( 2 R *=) ’1
(10)
The “availability” Et, for the irreversibly bursting bubbles was taken to be equal to the mechanical work which could be performed isothermically. The availability may be derived following ideal gas laws, assuming the bubble pressure to be close to the ambient one and the surface energy to be neglectable, both assumptions to be justified in the discussion. Eh
= m (1 -
5)
J,R*
Pi,4aR“ dr
(5)
where P is the average value of the absolute pressure inside and outside the can, T is the absolute average temperature of the liquid and the saturation temperature at ambient pressure, R’ is the universal gas constant, M is the molecular weight of the propellant, and 9is the difference in pressure which depends on the physical characteristics of the liquid (mixture), as defined previously. Equation 3 substituted into eq 2 will give Pb
- Pa = (1.8)pl(Cs/t)
(6)
The liquid solution will be discharged from the pressurized container into the atmosphere. The rapidly growing bubbles will take up an increasing part of the total volume of the expanding fluid. Under those conditions bubble growth may be restricted through mutual interference. The theory of Plesset and Zwick, derived for single bubbles, did not take this into 238 Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 2 , 1977
The surface energy required to generate a spherical droplet of diameter d from a continuous liquid phase is u r d ? . The energy required to generate n droplets whose diameters have a log normal distribution will be Ed
= n u a d 2 = noad& exp(2 In’ u g )
(12)
The efficiency ‘7 may be defined as the ratio between the energy required to generate the surface of the droplets to the energy contained in the bubbles and may be obtained by dividing eq 12 by 11 giving
PROPELLANT V A P O 6 PRESSURE GAGE
IMARY EXPANSION
VALVE I
SPRAY
/+ /
, SAMPLING SLIDE
SHUTTER
Figure 2. T h e experimental setup.
(A]
COMUERUPL NOZZLE
NIW
( 8 ) EXPERMENTAL NOZZLE
’
Figure 3. Configuration of nozzles used in experiments.
Substituting eq 9 into eq 13, we may extract d50
d- = 4 . 5 ~ exp(-2.5 ln2 C T ~ ) a0 rlmxR*3(Pb* - P a ) Equation 14 is an expression for the average droplet diameter produced by the flashing process, as a function of the internal pressure and radius of the bubbles, just previous to bursting. Substituting eq 3 , 5 , 6 , and 10 into eq 14 results in
WRING
SLIDE
where CY = 1.226/qrn2l3. This equation expresses the relationship between the average droplet diameter produced by flashing from a solution supersaturated due to a pressure difference /1p, as a function of the physical properties of the spray liquid and propellant, ug constituting a criterion for the uniformity of the droplets, the efficiency of energy conversion 7, the volume density of vapor nuclei m, and the dimensionless coefficient Ic,.
Figure 4. Spray sampling shutter system.
Experimental Section In order to confirm the theoretical derivation of the mean drop size and the underlying assumed physical model, a series of experiments was carried out. A model spray can was filled with a mixture of toluene and Freon 22. When the mixture was released into the atmosphere, it flashed and the resulting spray drops were sampled and their diameters were measured. The experimental setup consisted of a spray system, schematically shown in Figure 2. The spray can, in the upright position, was filled with a predetermined weight of toluene. After evacuating the air, the can was filled with Freon 22 from a pressurized supply bottle. The added amount of Freon 22 could be determined by the increase in weight. The pressure calculated from Dalton’s equation was found to confirm with the pressure measured inside the container, within the range of experimental error. Thus, the mixture was found to behave as an ideal solution. The vapor pressure of toluene was found to be sufficiently small, so that its influence on the vapor pressure of the solution could be neglected. When the valve in the outlet tube was opened, the liquid was discharged through an orifice and an aerosol was formed by flashing. In a commercial spray can, the fluid exits through a nozzle, as shown in Figure 3a. The liquid enters an expansion chamber, where vapor nuclei are created (preflashing). The fluid, with bubbles, then egresses through the actuator orifice.
In our experimental system, in addition to the commercial one, a nozzle made up of a uniform discharge tube (with valve) and a simple orifice, as shown in Figure 3b was used. Orifices with hole diameters of 1.1and 1.6 mm were used. In this setup the valve acted as a source for the nuclei formation; without it, flashing did not occur. A droplet sampling system was constructed, as shown in Figure 4. It consisted of a flat cover plate with an aperture in the form of a triangle. A separate shutter plate with a similarly shaped opening could be made to move parallel to the fixed plate. The apex of the triangle was positioned on the axis of symmetry through the nozzle. By quickly pushing the shutter plate from position 1to 2, the apertures in the two plates became aligned for a fraction of a second. Underneath these apertures, and parallel to them, standard glass slides could be positioned. These slides were covered with magnesium oxide powder. The spray, dyed to facilitate measurement, discharged from the pressurized container and was collected on the cover plate. When the shutter was opened, the spray droplets passed through the apertures and became imbedded in the powder layer on the slides (see Figure 5). The fast shutter action prevented the slides from becoming saturated. The individual droplets could easily be distinguished and their diameters measured, with the aid of a microscope with calibrated
RKlTlON 1 APERTMS
Ind. Eng. Chern., Process Des. Dev., Vol. 16,No. 2, 1977
239
100
c
80 70
-
60 50 -
(j/ 20
Figure 7. Values of averace droo size as tunctwn ot oressure ditterenee.
Figure 5. Typical slide with dyed droplets inbedded in magnesium
oxide powder.
dsol
dkIr t
,001
0.5
0.20.0 0.1 2
PERCENT*%
bP.l.3
a 30 30
OFOROPII
x)
s+
\ *'
010
37.0
a
0
0
9B 9911 99.9
'Ot I
BELOW A a i m SIZE
ClAL
I
d
\.
Figure 6. Typical drop size distributions. 50
background screen (using the experimentally verified, uniform correction factor of 8/91. This measurement technique utilized is applicable for droplet sizes of d > 5 wm only. The drop sizes actually determined in the experiments were substantially larger, with always less than l%of the drops smaller than d = 15 wm. The experimental results, as expected from the collective experimental evidence (Silverman et al., 1 9 7 0 , showed that the drop size distribution was log-normal of the first order. Figure 6 shows two typical results of the drop diameter normalized with dso, vs. the percentage of drops helow a given size. The geometric standard deviation for all experiments carried out was found to be cS = 1.31. Experiments were carried out with the sampling plane a t different axial distances from the orifice. The results showed no difference in drop size. From this it was concluded that secondary atomization or evaporation could be neglected. Experiments were carried out over a range of pressures (1-5.5 eta). Each pressure was due to a particular mixture of propellant and spray liquid. Therefore, the range of pressures represented a corresponding range of relative concentrations. 240
Ind. Eng. Chem.. ProcessDes. Dev., Vol. 16, No. 2, 1977
40t
,i
10
10 20 30 40 50 60 X R 22 % Figure 8. Values of average drop size as function of Freon concentration.
Changing the operating spray temperature changed the relationship between relative concentrations and pressure. Thia d s m indanmdont s.ceqsment. nf t.he ...., n l l n u r __.. _r"___l___.^"lllI ...I.._ ~. ...-infliienre of pressure and relative concentration on drop sme. ' Each experiment was carried out three tilnes. In the following, the average values of the measured drop sizes are ~
Table I. Physical Properties of Freon 22 and Toluene
A 4-
15*C 4 0 0 ~ PRESENT EXPERIMENTAL RESULTS 60.C EXPERIMENTAL RESULTS FOR WATER
I
Property
Units
Freon 22CHClF2'
TolueneC7H8'
M 86.47 92.14 P, (20 "C) ata 9.01 0.029 T , (boiling point, OC -49.9 110.6 1 ata) T (melting point, O C -160 -95 1 ata) Pl(20 "C) g/crn" 1.16 0.88 c, (20 "C) cal/g-"C 0.304 0.425 L (boiling point, cal/g 56.0 1 ata) D (20 "C). cm"s 5.7 x 10-4 1.1x io-:< u (20 " C ) dyn/cm 9.6 26.3 Matheson (1971). (Relevant properties corrected to 20 "(2.1 h Weast (1972). (Relevant properties corrected to 20 "C.) Calculated from basic properties.
(HOOPER
02
1
01
-
AND ABDELMESSIHI
L 2 -
20
presented. The measured average drop diameters d m are presented as a function of the pressure difference AP in Figure 7 for different nozzle configurations, distances from the nozzle, and a t different spray temperatures. In general, it was observed that the average drop size decreases wit1 increasing pressure difference. At small pressure differences, this decrease is quite steep, while at larger values, the influence of change in AP on d50 is quite small. The differences in nozzle configurations did not have an observable effect on the average drop size. Increases in the spray temperatures resulted in a slight decrease in average drop size, especially in the range of higher pressure differences. An alternative and more lucent way of presenting the data is shown in Figure 8, where average drop size is correlated with the relative concentration of propellant in the solution. From this figure, which shows a similar trend as the previous one, the effect of changes in spray temperature can be deduced more clearly.
Discussion of Results The experimental results presented in Figures 7 and 8 can now be compared with the theoretical expression in eq 15. The physical properties of toluene and Freon 22 are listed in Table I. The relevant physical properties of the solution, appearing in eq 15, depend on the relative concentrations of the components in the binary solution and on temperature and were calculated according to the usual procedure. The value of ug was taken as a constant equal to 1.31 as appeared from all the experiments. The coefficient a is unknown. Assuming CY to be independent of AP, eq 15 was made to coincide with the experimental data a t some intermediate value of AP.It can be immediately seen that the variation of d50 with the -4th power of AF' does not fit the general trend of the data at all. It was pointed out already that an experimental investigation of Hooper and Abdelmessih (1966) has shown that the coefficient $ is not constant but decreases monotonously with increases in AI' (see also Abdelmessih (1969). The results of Hooper and Abdelmessih are presented in Figure 9. $(AT) represents the deviation from thermal equilibrium due to temperature gradients inside the growing bubbles and across the bubble surface. This results in restrictions to the rate of evaporation. A similar effect, as measured directly by these authors, on growing bubbles should affect the growth rate coefficient C defined in eq 4, used in the present theoretical development. Let it be assumed that Hooper and Abdelmessih's mea-
60
40
80
iU0
120 A T
OC
Figure 9. The growth rate correction coefficient vs. superheat.
.
1.0
I
-
-
I5.C 4 m / P R E S E N T EXPERIMENTAL 60.C
lRESULTS
EXPERIMENTAL RESULTS FOR WATER (HOOPER AND ABMLMESSIH)
F R E O N - 2 2 WITH TOLUEUC
I
1
2
3
4 0 t a S A P 6
1
-
7
Figure 10. The growth rate correction coefficient vs. pressure dif: ference. surements of $ ( A T ) are generally valid. On the basis of the experimental data, by substituting $(AT) and AP = (P,L/ T ) A T ,the average value of CY was found to be 0.60. By reversing this procedure, using the value of a and the experimental data substituted in eq 15, the relation $ ( A T )of Hooper and Abdelmessih was found. In this manner, the present data of average drop diameter and those of Hooper and Abdelmessih for bubble growth were made to coincide. This is shown in Figure 9. A justification for this procedure and the underlying assumption that $ is independent of spray temperature (i.e., relative concentration) is found in the collapse of the present data carried out at different spray temperatures on a single curve. Using the transformation from A T to AP,$(AI') is found as shown in Figure 10. Since ,,LIT is different for water and for Freon 22, the resulting curves fitting the data of Hooper and Abdelmessih and the present data diverge. Substituting $(AP)for Freon 22 and the value of CY into eq 15,curves through the data are obtained as shown in Figures 7 and 8 for different values of the spray temperature. An order of magnitude evaluation may be made of the relInd. Eng. Chem., Process Des. Dev., Vol. 16, No. 2, 1977
241
evant bubble sizes and times which are assumed to be the basis of the spray formation process. For the energy conversion efficiency 1 < 17 < lo%,m can be calculated to be lo6 < m < lo7 cm-3. According to eq 10,1.5 X < R* < 5 X cm. The order of magnitude of Ro in eq 1 is cm. The assumption of R >> Ro on which eq 3 is based in Plesset and Zwick's theory is thus validated, showing that the thermal diffusion dominates the rate of bubble growth. From a calculation of the pressure inside the bubble, following eq 6, it is found that 0.014 < Pb - Pa< 0.14 atm. It can be seen that the energy contained in the bubbles is due mainly to internal pressure and the surface energy is negligible as assumed in eq 11. The assumed physical model for the spray formation re~ be independent of the nozzle sulted in eq 15 and shows d s to configuration. This was verified by the experimental results, which seem to indicate that if ample opportunity for nuclei generation exists, as in the expansion chamber in the commercial nozzle or in the valve of the experimental nozzle, the number of bubbles created per unit volume is constant. Brown and York (1962), in their earlier mentioned experiments, did find a correlation of average drop size with orifice diameter. It seems that in their case, the nuclei were created at the orifice, whose roughness was also an important parameter. Returning to the observation of Wiener, it can easily be seen from eq 15 that a larger density and a larger specific heat of the spray liquid, a smaller surface tension, and smaller latent heat of the propellant will result in smaller droplets. The observation of Wiener that a smaller density of the propellant will have a similar effect is not clear from eq 15.
Conclusions The assumed physical model for the formation of spray by flashing seems to be corroborated by the comparison between the experimental results and the theoretical expression for the average drop size, although the mathematical derivation is based on many simplifying assumptions such as the Clausius-Clapeyron linearization and the neglect of the effect of interference of bubbles during their growth. The bubble growth is not in thermodynamic equilibrium. The divergence from equilibrium for the growth of Freon 22 bubbles is found to be similar to that of water vapor bubbles. Increasing the operating pressure in the spray can causes the average drop size to become smaller. At high pressures, the average drop size is less and less affected. The drop size distribution was found to be similar in all of the experiments. Nozzle configurations seem to have no appreciable influence on drop size as long as ample opportunity for generation of vapor nuclei is provided.
242
Ind. Eng. Chem., Process Des. Dev., Vol. 16,No. 2, 1977
Nomenclature C, = specific heat at constant pressure of the solution C = growth rate constant of bubbles D = thermal diffusion coefficient d = diameter ofdrops djo = the largest diameter of the smallest 50% of the droplets E = energy L = latent heat of propellant = totalmass M = molecular weight of propellant m = number of bubbles per unit volume n = number of drops created per unit volume of liquid P = absolute pressure LP= difference between inside and outside pressure R = radius of bubble R' = universal gas constant T = absolute temperature SIT = superheat, T I - T , t = time a = coefficient q = efficiency of energy conversion p = specific density = surface tension cg = geometric standard deviation $ = coefficient Subscripts a = ambient b = bubble d = drop 1 = liquid o = nucleus s = saturation v = vapor Superscripts = averagevalue * = value just previous to bursting
-
Literature Cited Abdeimessih, A. H., in "Cocurrent Gas-Liquid Flow", Rhodes and Scott, Ed., Plenum Press, New York, N.Y., 1969. Brown, R., York, J. L., AlChE J., 8, 149 (1962). Herzka, A., Pickthal, J., "Pressurized Packaging", Butterworths, London, 1961. Hooper, F. C., Abdelmessih, A. A,, "Proceedings of the Third International Heat Transfer Conference", Chicago, Ill., 1966. Matheson, "Matheson Gas Data Book", Matheson Gas Products, 197 1. Plesset, M. S., Zwick, S. A,, J. Appl. Phys., 493 (1954). Sanders, "Principles of Aerosol Technology", Van Nostrand, Princeton, N.J., 1972. Silverman, L., Billings, C. E.. First, M. W., "Particle Size Analysis in industrial Hygiene", Academic Press, New York, N.Y., 1971. Weast, R. C., "Handbook of Chemistry and Physics", 52nd ed. Chemical Rubber Publishing Co., Cleveland, Ohio, 1972. Wiener, M. V., J. SOC.Chem., 289 (1958).
Received for review August 10, 1976 Accepted December 28, 1976
