limit after the third or fourth experiment, and subsequent experiments did Mtle to improve the precision. The parameters of the actual experimental work showed convergence characteristics which are similar to the u = 0.001 curves for Y and E and u = 0.0005 for rn and n. The experimentally determined standard deviation, ue = 0.001, iq reasonably consistent with these observations. It is apparent that to obtain more precise parameter estimates of Y and E , in particular, a substantially smaller experimental error than that of the actual experimental work would be required.
G
rn Y n p
R r s
Conclusions
T
The convergence of the sequential experimental design was rapid and uniform for all experimental errors except for the largest error considered, u = 0.003. The parameters for this case did not converge to final values, but fluctuated even up the eighth hypothetical experiment. It should be noted, however, that the standard error was so large that the true values of the parameters were within the limits of the standard error of the predicted values even though there were large differences between the true and predicted values. The parameters calculated from the hypothetical experiments for smaller errors, though they converged uniformly, also differed from the true values according to the magnitude of the imposed experimental error. Though the estimated parameters deviated from their true values by substantial amounts, the predictive model, using the parameters obtained from the hypothetical experiments, gave results within the limits of the experimental error, even for the largest error considered. It is apparent from the foregoing that one must be concerned with the magnitude of the experimental error if accurate values of the parameters are important objectives of the work. Obviously, these conclusions are strictly valid only for the model considered. It is expected, however, that large experimental error would certainly affect adversely the parameter estimation for other models.
y
Nomenclature
F
=
the F distribution with p and N-p degrees of freedom a t the a significance level
=
matrix defined in Equation 1
gut = partial derivative of the dependent variable, y , with
t
D yt7,,
respect to ith parameter a t the uth set of experimental conditions and the set of parameter estimates of Y. E , m , and n determined after the N t h experiment = exponent of NbOC13 concentration term = number of observations = exponent of COC1, concentration term = number of parameters in Equation 2 = gas constant = reaction rate = mean square about the regression = reactor temrserature, O K = reaction time, sec = ratio of moles of COZ to moles of COz, COCl,, and Ar from the reactor calculated for each hypothetical experiment = ratio of moles of C02to moles of Cog. COCl,, and Ar from reactor estimated from Equation 2 and some set of parameter estimates = ratio of moles of COa to moles of Cor,COCI,, and h r from reactor estimated from Equation 2 and the true parameter estimates
GREEKLETTER E
=
experimental error of y
literature Cited
Box, G. E. P., Inst. Int. Statist. Bull., 36,215-25 (1957). Box, G. E. P., Hill, W. J., Technometrics, 9, 57-71 (1967). Box, G. E. P., Hunter, W.G., ibid., 7 , 23-42 (1965). Box, G. E. P., Lucas, H. L., Biometriba, 46,77-90 (1959). Graham, R. J., Stevenson, F. D., Ind. Eng. Chem. Process Des. Develop., 11, 160 (1972). Hill, W.J., Hunter, W.G., Wichern, D. W.,Technornetrics, 10, 145-60 (1968). Hunter, W.G., Atkinson, A. C., Chern. Eng., 73, 159-68 (1966). Hunter, W.G., Kittrell, J. R., Mesaki, R., Trans. Inst. Chem. Eng., 45, T146-Tl52, (1967). Hunter, W.G., Reiner, A. 11.)Technornetrics, 7, 307-23 (1965). Kit'trell, J. R., Hunter, W. G., Watson, C. C., AIChE J., 12,5-10 (1966). Kowalczyk, K . J., unpublished 11s thesis, Iowa State Universit'y, Ames, Iowa (1967). RECEIVED for revierr June 25, 1970 ACCEPTED September 3, 1971 Work was performed in the Ames Laboratory of the US.Atomic Energy Commission.
Atomization and Drop Size of Polymer Solution Kuo-hui Wang' and Chi Tien' Department of Chemical Engineering and Materials Science, Syracuse Cniversity , Syracuse, .V Y 1321 0
T h e atomization of liquids into fine drops constitutes one of the essential operations for many important engineering processes, which include spray drying, spray coating, and fuel injection. Information on the size distributions of the atomized liquid drops is of fundamental importance, because the heat and mass transfer rates t o (or from) these drops depend, t o a very large extent, on the drop sizes and the drop size distributions. In the past three decades, a tremendous amount of research work on liquid atomization has Present address, Shell Development Co., Houston, TX.
* To whom correspondence shoiild be addressed.
been performed, primarily with Newtonian fluids. Most of these researches aere motivated by the need for understanding and improvement of fuel combustion in internal or rocket engines. Experimental results are very scarce, however, for the atomization of nonNewtonian fluids which are encountered more frequently than Xewtonian fluids in many industrial processes, such as the spray drying of detergents and food products. Because of the enormous increase in high-polymer production and their use, polymer solutions which are often nonNewtonian fluids have been widely used in industrial operations and commercial products. For example, C l I C and Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 2, 1972
169
Carbopol have been used in coating formulations to suspend and deflocculate the pigments and to bind the coating more firmly to the surface. They have been used also in insecticides and paints to stabilize the emulsions. The ever increasing use and importance of nonSewtonian liquids in the chemical processing and related industries, makes the need for a study of the atomization of such materials urgent. The results of such research would provide information on the size distribution function of droplets in terms of the physical properties of the nonSewtonian fluids and the operating conditions of the atomizing process. I n order that the experimental results could be of practical use, we have performed the experiments using aqueous solutions of commercial polymers and standard types of nozzles over a wide range of operating conditions. Also, we have devised a simple method of estimating the proper values of the apparent viscosities of the polymer solutions. The physical variables that we have used in the correlations of mean drop sizes and standard deviations are the operating pressure, the physical properties of the fluid, and the nozzle dimensions. The correlations which we have obtained are, therefore, general enough to include both Newtonian and nonNewtonian fluids, and are sufficiently simple to be of immediate practical value in design work. To compare our correlations with those obtained by previous workers, we have also taken data on two Sewtonian fluids in our experimental work and our final correlations, incorporating published experimental data obtained under comparable conditions to our own, are applicable and, therefore, serve to unify the treatment of the body of experimental data. Literature Survey
Drop size correlations for spray particles obviously depend on the type of atomizers. Generally speaking, there are three basic types of liquid atomizers-centrifugal, pneumatic, and pressure. An excellent review on atomization involving the use of these t,hree types of atomizers can be found in the monograph by Marshall (1954). For the present investigation, a particular type of pressure atomizer was used-the grooved-core swirl nozzle atomizer. I n the following, a brief review on past studies which employed the use of this kind of nozzle for Newtonian liquid, is presented. The first investigation apparently was made by Turner and hloulton (1953) who sprayed molten naphthol and benzoic acid a t room temperature. Empirical correlations were established relating the mean drop size and the standard deviation with operating variables as well as physical properties of the liquid. A later investigation on the spraying of water was made by Tate and Marshall (1953), and established t h a t the size distribution of liquid droplets from a grooved-core swirl nozzle obeys the square-root normal distribution. Subsequent' studies by Darnel1 (1953) and XcIrvine (1953) provided additional data on the atomization of liquid with grooved-core nozzles, and measured the air core diameter by photographing water flow in transparent plastic nozzles. Darnel1 also confirmed the square-root normal distribution of droplets in t'he atomization process. I n a n attempt to simplify the experimental procedure for the determination of droplet size, Choudhury (1955) developed the liquid nitrogen freezing method. This method was later modified by Nelson (1958) and Nelson and Stevens (1961). The work of Nelson and Stevens perhaps represents the most comprehensive work on liquid atomization from grooved-core nozzles to date. Their result's support the square170 Ind. Eng. Chem. Process Des. Develop., Vol. 1 1, No. 2, 1972
root normal distribution. However, it was necessary to obtain separate correlations for the organic liquid and water data. The volumetric mean drop size for the organic liquid was found to be
Y
=
0.0811 X 2
+ 0.124 X - 0.186
(la)
where
Y
= loglo
(~/DZ)
The standard deviation correlation is
Y
=
0.150 X 2 - 0.359 X
+ 0.686
(24
where
The velocity term used in defining N R and ~ N w e is taken as the average axial velocity a t the orifice based on liquid film thickness. This quantity, together with the spray cone angle, 8 , can be obtained only from experimental measurements. For this reason, Nelson and Stevens used the results of Darnell's work. Experimental
Selection of Freezing Process. There are three basic methods for determining the size distribution of liquid droplets produced by a spray nozzle: photographic q e t h o d , electronic-scanning methad, and freezing method. This last method was chosen in the present work primarily because of the following three considerations: all the particles produced in a spray are collected and processed, and, therefore, the results are believed to represent more closely the true distribution; coalescence and secondary atomization of the spray particles can be minimized by promoting very rapid freezing through the use of sufficiently low temperatures; and the method is relatively simple and rapid. Choudhury (1955) appears t o have been the first to use liquid nitrogen as the freezing medium for spray particles. In his work, the sample fluids were sprayed directly into liquid nitrogen and the screening of the frozen particles was accomplished in a cold room. Nelson and Stevens (1961) modified Choudhwy's method and eliminated the necessity of a cold room by screening the frozen particles within an insulated sieve-container mounted on a giant shaker. Liquid nitrogen was poured intermittently over the container t o keep the particles from melting. The experimental method used in the present work is essentially based on the one described by Nelson and fjtevens. Experimental Equipqent. The equipment consists of five main units: a pressurizing system, a swirl nozzle, a collecting unit for spray drops, a sieve shaker, and a weighing box. The pressurizing system includes a stainless steel, highpressure vessel and a stainless steel pipe line. The vessel is a cylinder rated for a working pressure of 180Qpsi and a capacity of 11/2gal. The pipe line is made of '/An. stainless tubing and Ferulok fittings. Flow in the pipe line is controlled by a normally closed solenoid valve. The system is pressurized with compressed nitrqaen from a commercial cylinder and system pressure is controlled by a high-pressure regulator.
Experimental doto on the atomization of nonNewtonian liquids b y a grooved-core nozzle were reported. The drop size distribution determined b y sieve analysis was found to obey the square-root normal distribution over the range of mean drop size of 40 to 370 p. Unlike previous studies, the mean drop size and the standard deviations were successfully correlated in terms of nozzle dimensions, operating pressure, and physical properties of the liquid. Consequently, predictions con b e made without knowledge of the spray cone angle, liquid sheet thickness, ond other variables which a r e often difficult to obtoin.
The collecting unit is composed of an upper and a lower section. A sketch of the collecting unit is shown in Figure 1. The upper section with 20-in. i d . and in height is a cyclonelike cold chamber made of galvanized iron sheet and fiber glass. Liquid nitrogen can he introduced into the chamber and distributed uniformly over the wall surface by means of two copper rings situated a t the top of the collecting chamber and connected individually t o liquid nitrogen tanks. The chamber has an insulated cover made of galvanized iron sheet and fiber glass. There is an &in. opening at the center of the cover through which a sample fluid is sprayed from the nozzle into the chamber. Covering the opening is a stainless steel housing for a sliding cut-off drawer which is operated manually to control the spray time. The spray time is recorded on an electric timer triggered by the motion of the drawer. T h e spray nozzle is placed in the drawer housing
soil particles. This was found t o be unsuccessful because of the excessive weight of the insulated sieve container. The weighing box is a large insulated chest designed for weighing the sieves and for containing Dry Ice t o maintain low temperatures. The weighing balance is placed on the top of the box and the weighing pan is suspended inside the box by a nylon cord through a small opening on the box. The atomization nozzles chosen for the present work were the grooved-core Spraydry nozzles manufactured by the Spraying System Co. The nozzle consists of two essential parts, the orifice insert and the core body. For a given fluid and pressure, the degree of atomization and the flow rate are controlled by the combination of the orifice insert and the core body. For the present investigation, we have selected size orifice inserts and six core bodies, the dimensions of which are listed in Table I. Experimental Material. The following aqueous solutions nolvmers were used in this investieation: 0.11 wt 9% ~;bopol-934 (B.F. Goodrich), 0.15 i t % Carbopol-93i :. F. Goodrich), 4.5 wt % CMC-7L (Hercules), water, d glycerin (76 wt yo).The last two materials were studied make comparisons with previous investigations. The rheological determinations of these polymer solutions
Figure 3.
Sieve shaker
were made through the use of a Weissenberg Rheogionometer (Model R-16). These determinations yielded results in the form of rheological diagrams of apparent viscosity vs. shear rate for a shear rate up to 1000 see-'. From these data, apparent viscosities a t higher shear rates can he estimated by extranolation. A tvnical set of rheolocical data is shown in igical dat were also u e t e n n e u . H summary UI b m s e piuparua~is LLSWLI in Table 11. Experimental Procedures. Each test run began with the filling of the sample fluid into the pressure vessel. The system was pressurized, nitrogen gas was injected to surround the spray nozzle, and liquid nitrogen was allowed to flow into the collecting chamber to form a liquid film on the wall surface. The solenoid switch was then turned on and the sample liquid began to spray into the cut-off drawer. After the messiire o~ e a m e had assumed a steadv readine, the cut-off drawer was pulled so that the sample liquid could be sprayed into the cold chamber for a predetermined time . The drawer . > > , , was tnen vumea DacK qnlcK1y to cut on m e spray. Particles ~~y~
~~~~~~~~~~
spraycu luuu b u r n m n v n uuuu uuuuu,=u auu n-1- w a n m d down to the sieves by the flowing liquid nitrogen. One minute of flushing time was found to be enough to clean all the spray particles from the wall surface. The froaen particles in the cold chamber were carried by the liquid nitrogen through a guiding trough into the lower chamber and were collected in a set of nine sieves. A high, removable weir was used with the top sieve to prevent fine particles from flowing outside the sieve. Some very fine particles were wet-screened by liquid nitrogen all the way down to the bottom of the lower chamber. The bottom of the Chamber was lined with a large sheet of aluminum foil so th a t the fine particles could he recovered by careful decantat ion. After sufficient flushing of the cold chamber with liquid riitrogen, the lower chamber was removed from it and the i n ~ r a r n"""I" r0-0 m & lJz l v Sir.ru ~ u ' " I . I
"
y.".."yt n
t .. x oncfawarl 1
Y.."
&OXTO Y."."
.".
onn+o;nor fnr 1 . 1 "I y"l.y
*
/
Table II. P hysical Prope'Hies of Fluids at 75°F S"rf.Xe
,
Vi!icority, Fluid
CP
Carbopol-934, O.llyosoln 4.0G Carbopol-gPA 1 6 4 . cnln ,-, n"._"," 6 50 25.0* CMC-7L, 4 .5Y0 s o h 0.89 Water Glycerine, i'6% soln 31.0 3.70I Aniline 1.84: Nitrobensel le 0.88 Cyclohexane 7.83 N-octyl alciJhol 0.901 Carbon tetrachloride 8.29I 1,1,2,2-Tetrabromoethane Irl annaront vi.,-n.itv "i a _ t" - = 2 a EstimatL- 7111.1..1 YYIL.
Table 1. Orifice lnserfs and Core Bodies Orifice insert Orifice NO. diam,sm
67 58 54 46
+ha
I
...
80 74
Y"
screening. The sieves had been wrapped with a sheet of asbestos cloth to prevent moisture condensation. The container was clamped to the shaker and screened for 30 min
0.0343 0.0571 0.0812 0.1066 0.1396 0.2060
Core body
Flow area, NO.
cm2
10 16 20 17 21 27
0.00232 0.00496 0.00800 0.00993 0.01808 0.03100
172 Ind. Eng. Chem. Process Dei. Develop., Vol. 1 1, No. 2, 1972
_.1
~
~
I
Density,
tension,
g/cma
dynesfcm
0.997 0.997 1.013 0.997 1.200 1.017 1.198 0.774 0.820 1.585 2,950
x
106 sec-1
72.0 72.0 72.0 72.0 67.5 42.3 43.3 24.5 27.0 26.2 49.0
during which time 2 liters of liquid nitrogen were poured intermittently over the container to maintain low temperatures. When the screening was complet'ed, the sieves were transferred into the weighing box, which contained blocks of Dry Ice. They were allowed 1 hr to come up to the temperature in the box before the asbestos cloth was taken off aiid the sieves weighed. The box was always saturated with water vapor and the loss of the sample weight due to evaporation was found t'o be negligible, as checked by repeated weighing of the same sample over several hours. Material recovery was found to range from about 80 to ll0Yc. The loss of material was chiefly due to some particles being left on the inside surface of the chamber, while the gain of material was due to errors in the spray time recorded by the timer. When large nozzles were iii operatioii, t,lie suitable spray time was of the order of 1/2 see. I n bliese cases, the spray times determined by the electric timer, which was triggered by the iiianually operated drawer, were relatively less accurate. Res uIt s
Folloriiig the esperimeiital procedure described above, a total number of 113 experimental runs was obt,aiiied, of which 15 were made oil water, 6 011 767, glycerilie solii, 17 on 0.11% Carbopol-934, 45 011 0.15% Cahopol-934, and 29 011 4.57,CMC-7L. Thirteen nozzles of various sizes were used with the orifice diameter ranging from 343 to 2066 p , Pressure was varied from 125 to 1700 psi and liquid flow rate from 2.65 to 68 g/sec. Average recovery of the sprayed particles was 90.4%. Correlation of Mean Drop Size. Based on tlie result of the sieve analyses, the drop size distribut,ioii was found to be in good agreement' with t,he square-root norinal distribution-Le,,
An indicat~ioiiof the validity of the above espression is provided by the fact that plots of cumulat'ive percentage against square-root of diameter on probability paper result iii straight lines, as shown in Figure 5 for selected data which are typical. The average value (%l'z or e ) can be determined from such plots. The entire set of data can be found in reference (Wang, 1969). If one assumes that the rheological characterist'ics of the polymer solution in the atomizing process can be represented in terms of the apparent viscosity corresponding t o some characteristic shear rate, it is simple to show through di, that, the mean drop size can be correlated with the physical variables according to the following espression : I -
where A is the ratio of the orifice area to the total flow area of grooved slots. To determine the form of the relationship in Equation 4 using the experimental data, it is necessary to define the characteristic shear rate for the atomizing process. The hydrodynamic phenomena relating to the disintegration of a liquid sheet are extremely complicated. However, viewed in the most simplified way, the liquid disintegration by the pressure nozzle can be considered to take place right a t the
Figure
5. Drop size distribution of CMC-7L, nozzle 58-21
nozzle orifice, provided that t,he operating pressure is sufficiently high. If this were the case, the appropriate shear rate can be taken as the ratio of V,,/'h, n-here V,, is the axial velocit,y of t,he liquid a t the orifice based 011 t'he liquid film thickness h. This latter quantity, h , can be estimated from information on the ratio of tlie air core diameter of tlie spray cone to the orifice diameter. 011this basis it was estimated that, over the range of olieratiiig coiitlitioiis iised in t,his work, V,,lh varied from 0.59 X 10; sec-' to i . 5 X lo5 see-' (\Tang, 1969). For conveliieiice, an arbitrary value of 2.0 x IO5 sec-l-i.e., the midpoilit o n a logaritlimic scale-was chosen to be the characteristic shear rate for the atomization process correspoiiding to t,lie experimental coiiditions of this investigat'ioii. This characterist,ic shear rate is well above the limit of the Weissenberg Rheogoniometer, a i i d to estiniate t'lie appropriate apparelit yiscosity it was necessary to estrapolate the rheological data over t,wo cycles in shear rate. This was tioiie by drawing extensions to t,he apparelit viscosity-shear rate curve as shown in Figure 4. The plots are oii log-log paper, and the esteiisioiis were dran-11 by following the trend of t,he dat,a,rather than by drawing straight, lilies. This extrapolation procedure naturally introduces uiicertainty. However, as show1 later, tlie dependence on the alq)xrent, viscosity was found to be rather weak (see Equatioii 10). Consequently, the error caused by the estrapolatioii noultl be relatively uiiimport,aiit. To obtain a more general correlation, the meaii drop dianieters obtained from the 113 runs were combined with tlie 97 runs obtained by Nelson arid Stevens (1961) for organic fluids. The incorporation of these data iiicreases markedly the range of Reynolds and Keber numbers coilsidered. The water data obtained in 13 ruiis by Selson arid Stevens were not included because they were not in agreement with the bulk of their orgaiiic liquid data, ailti because they were not consistent wit,h the water data obtaiiied hi this work. Through a careful search for the best straight-line fit on a log-log plot, we were able to correlate t,he mean drop diameters for the entire set of 210 ruiis using the following simple equation :
I'
=
-0.60 X
+ 1.40
(Sa)
where y
=
log10
();
Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 2, 1 9 7 2
173
t
v
oT1115
t
WOR
- nozzles is related to tlie thickncs5 OF the liquid shcct. For o u r swirl-type pre,?sui.e iiozzles, the liquid 5lieet thickness is the film thickness a t the nozzle orifice since the ceiitral portion of the orifice is occupied bj. the air core. Fihii thickness is det,errniiied by flow behavior inside the stj-irl chamber. Taylor (1950) suggested that the fluid flow inside the siyirl charnber could be represented by a free vortex aiid that the trailsport, of the fluid toward the orifice was by way of a bouiidarj- layer along the n-all surface of the En-irl chamher. Based on tliia model, lie performed ail aiialysis for the lainiiiar bouiidary layer gron-tli of n Sen-toiiiaii fluid l q - solving the bouiidary-layer equations of niotioii usiiig the monieiitum 174 Ind. Eng. Chem. Process Des. Develop., Vol. 1 1, No. 2 , 1972
=
loglo [ (5) ( h ~ R ' R , ) - 0 , 4 3 6 ( N ~ ~ - e ) o . 2 ' s ( A-0'339] )
(6c)
where 6 = boundary layer thickness, 1.1, aiid n: = volumet'ric nieaii drop diam, p . The correlation is shown iii Figure 7 . The average deviation of tlie esperiiiieiital points from the equation line was 11.87%, aiid is com])arable to that' of the correlation based on dinieiisioiial aiialysis (Equation 5 or Figure 6). Standard Deviation Correlation. The standard deviat'ioii measured from the straight line of cuniulat,ive weight percentage on a probability paper has the dinieiisioiis of (p)l'*. Therefore, it, is more convenient to use s/dc - in t'he correlation. Since the experimeiital values of s / d D z covered only a-~sinal1 range, it \\-auld be bet'ter t o stretch the rat'io of s / d D 2 to get n inore nieaiiiiigful correlation. This could be accomplished by multiplying s!dE by anot'her dimensionless group, say t8heWeber iiumber, as was done by Kelson (1958) and Selsoii arid Stevens (1961) L - d e r t'hese circumstances the correlation for staiidard deviat,ions could be represented by the following equation: I
I.
=
0.071 5' - 0.211 X
+ 1.062
(7a)
1.2g6)
(7c)
where
s = log,,(Sne - 0 4 7 1 S K e
This correlation is shon-ii iii Figure 8. The average deviation froin the equation line was 10.84%. I. 1lie experimental results iiidicat,ed a relation between 5 aiid s. Ai plot of s vs. .?. is showii in Figure 9. Standard deviations \\.ere approximately correlated with the mean drop sizes by tlie following equation: s = 0.076(f)0,72
(8)
where s = square-root standard deviat,ion, ( p ) * I 2 and 2 = volumetric memi drop diam, p , A similar equation for pneumatic atomizers was report,ed by Gretziiiger aiid Marshall (1961). Discussion of Results
Comparison of Experimental Results with Previous Works on Grooved-Core Swirl Nozzles. To compare experimental results with previous invest,igat,ions, tmhe correlation for ineaii drop size (Equations 5) can be ren-rit ten as :
50
Y F 4
":
9
0
THIS
WORK
NELSON'S ORGANIC D IO'
2
5
2
IO'
5
IO'
2
I
-1
* N
6
m 5
a
49
.;
30
-
I
(10)s 20
Figure 8. Correlgtion of standard deviation with physical variables
0
I
or
I 200
E
WORK
I
ORGANIC
DATA
I
I
303
400
(microns:
~ ~ - 0 . ~ p - o . o4 .3 ~~ ~ ~ - o . ~ z ~ o . ~ i a ~ - o . i ~
(la)
P
We note first t h a t the experimental results are in agreement with those obtained by Nelson and Stevens (1961) for organic fluids. I n the correlations proposed by Nelson and Stevens (1961), the average axial velocity, based on liquid film thickness, and the spray cone angle are chosen as correlation parameters while the total pressure, P , and the factor, A , are used in the present work. The quantities P and A are direct operating variables while the average axial velocity and film thickness require more involved measurements, or the use of supplementary correlations for their estimation. The mean deviation for the present mean drop size correlation (Equations 5) is 12.85% as compared to 8.25% obtained by Nelson and Stevens. On the other hand, an inspection on Nelson's plot of mean drop size for organic fluids indicates t h a t the bulk of the data can be represented by the following equation:
Y
=
-0.54 X
+ 1.2
(IW
The values for Y calculated from Equations 11 and Equations 5 for a given value of X are within 9% of each other. Therefore, it can be stated t h a t the two different sets of the dynamic and nozzle size parameters lead to essentially the same numerical equation. A comparison between the correlations for standard deviations reveals t h a t they are within 12% of each other. The mean deviation of the standard deviation correlation (Equations 7) is 10.84y0 as compared to 13.0y0 for. the Nelson-Stevens' work. To compare the experimental results of Turner and Afoulton (1953) with the present work, the relationship between the mass flow and pressure must be established first. For the type of swirl nozzle used in this work, the data for water was found to obey the following relationship (Wang, 1969) :
CQ = 0.22 and
I
100
5 = 25.1
THIS
NELSON'S
IO
A-0.455
Figure 9. Correlation of standard deviation with mean drop size
If these expressions are assumed t o be valid for the nozzle used by Turner and AIoulton, their Correlation for mean drop size can be written as: Q
=
30.8 D
- 0 . 3 6 ~ - 0 . 1 1 W o ' 16 P -o.ii,n.7i~o.m
(14)
Comparing Equation 14 with Equation 10, it can be seen t h a t the constant coefficient' and t'he exponents for p and p are close, while the exponents for D9,u , and A are quite different. The discrepancies are mostly due t o the narrow ranges covered by the physical variables in Equation 14. The differeoce in experipieiit'al conditions, however, could be an important factor in causing the discrepancies. No comparison was made between the correlations in this work with those of Tate and Marshall (1953) since their study was restricted to water only and the correlation was obtained from a relatively small amount of experimental data. Darnell's correlation was found to give higher values for the mean drop size. I t should be pointed out t h a t Darnell's data were based on the samples taken a t a distance of 10 in. below the orifice. He observed that samples taken a t the 10-in. distance were coarser than those at' the Lin. distance. The cause of the difference in mean drop size at the two distances below the orifice is most likely due to the coalescence of the initially formed drops, I n the freezing method, the coalescence is effect,ivelp suppressed, and, therefore, the mean drop size should be smaller than those observed by Darnell. The present correlation was also found to be in agreement with the results cf the theoretical analysis on droplet forniation from atomizat'ion processes. Both ;\layer (1963) and Fraser et al. (1962) concluded that t'he mean droll diameter is proportional t o u1'3 or ArTl-el'a. Other theoret,ical considerations made by Holdroyd (1933) for liquid jets and by Keller and Holodiier (1954) for liquid sheets under unbalanced pressures also yielded the same dependence of mean drop size on a properly defined Weber number. I n this respect, the correlation represented by Equation 9 is in good agreement with the theoretical results. Justification on the Use of Apparent Viscosity Based on Extrapolation. Perhaps the main feat,ure of the present correlations which is open to question is the extrapolat,ion procedure used t o estimate apparent viscosity. Furthermore, the validity of characterizing the nonNewtonian behavior of fluids in atomization process using the simple concept of apparent viscosity, also requires substantiation. The fact t h a t a successful correlation for mean drop size was obtained can be used a t least as an ad hoc justification t o the Ind. Eng. Chem. Process Des. Develop., Vol. 1 1, No. 2, 1972
175
-.,
less tightly stretched after several times of L"..y.Il. mean drop diameters were not as sensitive to the sieve accuracies as standard deviations. To check on the efficiency of the sieve shaker, several pictures were taken of the spray drops in a few sieves. Enlarged photographs (Figure 11) clearly revealed t h a t the spray drops retained on a given sieve were quite uniform in size and spherical in shane. and t h a t no drops smaller than the sieve openings were observed in the sieve. Shattering or C oalescence of Spray Drops. The shatter ing or secondary atomization of the very initial drops b Y +. *..-+--T Tx "f + h n uIIuIII:I air friction is very. ,l:z-..T+ u11115ull uu I I U L I Y I U I . 111 l l r l l ylly able theories of t h e disintegration mechanism of a liquid drop, it appears t h a t secondary atomization is not an important factor in drop siEe distribution for pressure nozzles, since the initially formed drops are decelerated instead of being accelerated by the ambient gas. However, possible shattering of the frozen drops resulting from collision with the wall surface of the collecting chamber should he ascertained. Enlarged photographs showed that there were no f ractured particles. Coalescence of the spray drops can be nminimized hy using a high flow rate of liquid nitrogen. _I
..:-...
OD
' 1 ' 1 1
5rlo'
I
,
10,
Re
I
I
I I I I I ,
!?pi
IO'
Figure 10. Correlation of 818, for polymer solution
".."
I
',>..~.I:...:, 'Cv,,en q u l u -:A nluugc,,
..-.-
^..., & k c
II ,l"W
:- A--..""""A
A"",. ":"^ IZ U G r l C L I D r U , IIITaLI U'vp U l o r
and standard deviation could be larger than those obtained a t a lower pressure. This clearly shows t h a t if the heat trs fer rate is not high enough, coalescence of the drops definii takes place. This is in agreement with Darnell's observai t h a t the spray drops collected a t 10 in. below t h e orifice larger than those collected a t 5 in. below. Effect of Ambient Pressure. It has been obsery in many studies on fan-spray nozzles t h a t t h e mean drop size is inversely proportional to the '/6 power of the ambient gas pressure (Fraser et al., 1962). When ambient pressure is 2 atm, mean drop size would be 12% less. Since the collecting system used in the present work is virtually a closed one, it is necessary to release pressure build-up by the vaporization of liquid nitrogen prior to the introduction of the sprays into the cold chamber. The pressure is released through a vent hole near t h e bottom of the lower chamber so that the pressure inside the collecting chambers is atmospheric when the spray is injected. The effect of ambient pressure on mean drop size has been discussed extensively Ir)y Dombrowski and Hooper (1962). ~~
Appendix. Boundary Layer Analysis of Power Fluid in Swirl Flow (Figure 12) The flow of an incompressible fluid through a swirl-type lressure nozzle is an interesting but complicated problem. &any authors including Doumas and Laster (1953), Tan* awa and Kobayashi (1955), and Taylor (1950) have atempted to describe the fluid behavior in the nozzles as an nviscid flow, neglecting the viscous and the boundary-layer tffect. The sufficiency of the inviscid model can be tested by :omparing with experimental results of spray cone angle, lischarge coefficient, and liquid film thickness. At high Reynilds numbers, agreement between theoretical and experinental results is fair. Tanasawa and Kobayashi (1955) have dso considered the importance of viscous dissipation due t,o the radial gradient of the tangential velocity. Taylor (1950) suggested t h a t the boundary layer of the h i d on the wall surface of the swirl chamber could be an mportant factor in determining the nozzle performances, specially a t relatively low Reynolds numbers. H e proposed ;hat the main body of the fluid in the swirl chamber could be :onsidered as a free vortex, while the boundary layer was
responsible for the transport of the fluid toward the orifice. The result of his theoretical analysis indicated that, even with a slightly viscous fluid such as water, a large portion of the fluid arrived a t the orifice by way of the boundary layer. Extensions and applications of his analysis to other problems have been niade by Binnie and Harris (1950), Cooke (1952), La1 (1967), Sinbel (1959), and Weber (1956). Hodgkinson (1950) has made use of Taylor's analysis t o determine the design features of a swirl nozzle and to predict the mean drop size. H e showed t h a t the boundary layer thickness, 6, was related to the pressure, P , by Figure 12. Boundary-layer thickness in swirl chamber
-
Since = C ~ ( ? r D z ' / 4 ) ( 1 / 2P / p ) and assuming that f a z we get from Equation A-1 the following result:
6,
301
I
Using Equation 12, we obtain
5
(NR,)-1/3(A)-0.15Z
DZ
(-4-3)
Comparing Equation A-3 with Equation 9, we see t,hat the agreement is reasonable as far as the functional forms of the equatioiis are concerned. However, n o direct comparison of t'he boundary-layer thickness with experimental mean drop size has been made by previous authors. We have intended to make a quantitative comparison between the calculated boundary-layer thickness and the experimental mean drop size for both Sentonian and 11011Newtonian fluids. For this purpose, we have extended Taylor's analysis to cover those simple nonNc~~-tonian fluids whose relations between shear stress and shear rate can be represented by the power law model. Referring to t'he coordinate system shown in Figure 13 and using the formula of Taylor, we can st'ate that our probleni is t o seek a solut'iori for Equations A-4, A-5, and A-6. This can be summarized as: Boundary-Layer Equations
t Figure 13. Coordinate system for boundary-layer analysis
(iii)
zc7
(iv) u ,
=
0 and
=
0 at r
D
= I outside . boundary
r
=
8111 0
layer
R,
Shear Stress-Shear Rate Relation .e =
-nz
I_ n - 1.
11 -
, 2
(A:A)
2
(1-7)
A
where 7 = viscous stress tensor, A = rate of strain teiisor, A : A = second invariant of A . Lysing an order-of-xnagllitude consideration based on the equat,ioii of continuity, Equation -\-7 can he reduced t o the simple form :
(A-5) Integrating the equations of motion over the boundary-layer thickness and eliniiiiating 218 by the equation of continuity, m-e obtain
Equation of Continuity
Boundary Conditions (i) v,
=
v$
=
0 at e
=
CY
where a = half apex angle of the conical swirl chamber, 6 = boundary-layer thickness, and D = circulation constant of the free vortex outside the boundary layer. Assumptions Involved
