I
W. B. NICHOLS, L. T. CARMICHAEL, and B. H. SAGE California Institute of Technology, Pasadena, Calif.
Supersaturation in Hydrocarbon Systems n-Pentane in the Liquid Phase Bubble formation experiments with n-pentane confirm (earlierfindings that supersaturation in hydrocarbon liquids must be investigated statistically --a matter of technical interest in petroleum production and refining
A
KNOWLEDGE of the probability of forming bubbles in a supersaturated hydrocarbon liquid is of technical interest in problems associated with the production and refining of petroleum. Some experimental investigations of attainable tensions in liquids have been made. Vincent (46, 47) studied the maximum tension in a mineral oil while Gardescu (74) found it was possible to maintain hydrocarbon mixtures at pressures significantly below bubble point for short periods of time. Kennedy and Olson (20) investigated the bubble formation in liquid mixtures of kerosine and methane and found that a supersaturation of as much as 770 pounds per square inch could be maintained for short intervals. The influence of solid surfaces upon the duration of a given degree of supersaturation in liquids was investigated by Marboe and Weyl (26) whereas the effect of viscous flow upon such phenomena was reported by Jha (79). T h e behavior*of supersaturated solutions of electrolytes was studied by Akhumov and Rozen ( 7 ) while the stability of salt solutions was investigated by Tovbin and Krasnova (45). The attainable tension in water was measured by a number of investigators (3, 4, 9, 30, 42, 43) who obtained measurements varying from 750 to over 3000 pounds per square inch. More recently, cavitation in aqueous solution has been studied in some detail (32, 33, 49). T h e thermodynamics of such
systems was copsidered by Gibbs (75) and extended by Goranson (76). Related phenomena are found in gases when they are maintained at pressures in excess of vapor pressures. The spontaneous formation of nuclei in gases was studied by Sander and Damkohler (40). However, in wind tunnels, the extent of such supersaturation of air or nitrogen is small (70, 25). The process of the gas evolution from liquids was studied by Burrows and Preece (5) while the factors influencing nucleation in boiling heat transfer were studied by Sabersky and Gates (37). These investigators have established this factthe time for a particular solution to remain in a strained state must be evaluated statistically. The theory of liquids (7, 8, 22, 23) indicates that significant fluctuations exist, in the state variables at a point as a function of time. Therefore, wide variations are to be expected in the time during which a system remains in the strained condition even though the macroscopic state is held invariant. The time that n-pentane was maintained in a strained condition was measured at three temperatures. The results confirm earlier (78) findings that supersaturation in hydrocarbon liquids must be investigated statistically. Statistics
The statistical theory of liquids is usually based upon the Lennard-Jones
and Devonshire equations of state (23) which were improved by Kirkwood (22) and have been considered rather intensively by Dahler and Hirschfelder (7, 8). The results of Hirschfelder follow the work of Furth (72, 73). All of these theories of the molecular properties of liquids emphasize the fluctuations in the properties of liquids at a point with time. Therefore, the formation of bubbles in a strained liquid is expected to be randomly distributed in time. Hunt ( 7 8 ) made a simple evaluation of the probability of a bubble being formed in a given time interval. For present purposes, the times of formation of bubbles are distributed a t random. Such a situation is referred to as a discontinuous stationary stochastic process (2, 24, 35). Under these conditions the Poisson distribution has been shown to apply by a well-established proof (71, 27). With such a distribution the probability of n bubbles forming in the time interval from 0 to e is given by:
In Equation 1, (ko)" is the expected number of bubbles formed within the time interval. The probability of no bubbles forming in the interval from 0 to e may be established from Q(0, O)=
e-20
(2)
The mean value of times for the first VOL. 454 NO. 7
JULY 1957'
1 1 65
ID1
A
7
Figure 1. This is the arrangement of equipment consisting o f u pressure vessel, stainless steel thimble, three-stage centrifugal pump, and heater
The integral in Equation 13 was evaluated graphically. The relation of the mean time 3, for the first bubble to form with pressure is given by Equation 11. In accordance with the central lirnil theorem (28) for a population with a finite variance uz and mean p, the distribution of the sample mean approaches the normal distribution with variance 2,'n and mean p as the sample size 7e increases, As a consequencr of this theorem, the 9570 confidence intcrval for the mean rate of bubble formation of n points is asymptotically -+1.960/l/L times the measured rate Thus for a sample of 100 points. the 95% confidence interval is 119.67, of the measuied rate. This discussion leaves much to bc desired from the standpoint of classical statistics but appears to yield a simple means of ana1)sis of tha rxperimental data obtained. Materials
bubble to form may be evaluated from the following equation ( 7 7 ) :
The reciprocal of the rate of bubble formation may be equated to the mean time for the first bubble to form k
ab
5
(4)
Probability of the first bubble forming ' d0 is given by between 0 and 0
+
P(0,
+ do) =
&e-;hO
dt?
From the foregoing it follows that the standard deviation is the reciprocal of the rate of bubble formation as indicated in Equation 7, n', always being positive,
Integration of Equation 5 folloized by rearrangement results in the follo\ving relation between the cumulative distribution aiid the time:
e ) ] = -Go
From Equation 10 it would be expected that the natural logarithm of the mean time for the formation of the first bubble would be a linear function of the reciprocal of the square of the supersaturation pressure, as indicated in the equation,
'The n-pentane used in this investigation iras obtained as pure grade from the Phillips Petroleum Co. and was rcported 1.0 contain not more than 0.01 mole fraction of material other than npentane. It was used for these prelirninary measurements w-ithout purification except for deaeration by prolonged refluxing a t reduced pressure. 'The deaerated sample showed a specific weight at atmospheric pressure of 38.775 pounds per cubic foot a t 77' F. as coinpared to a value of 38.791 pounds per cubic foot a t a temperature of 77' F. reported by Rossini (36). An index of refraction at V 0F. of 1.35475 was obtained for the D-lines of sodium as compared to a value of 1.35472 reported by Rossini for an air-saturated sample. T h e vapor pressure a t 160" F. was 42.4 pounds per square inch and a t 278.94" 17. it \vas 183.6 pounds per square inch, which agreed satisfactorily Tvith accepied values (38). Methods and Equipment
Equation 11 has been used to correct some of the experimental data to a fixed supersaturation pressure, (Pb - Po): by assuming that
(8)
I n accordance with Equation 8, a plot of In [l -- P(0, e) ] as a function of time should yield a straight line if the formation of a bubble is randomly distributed in time.
1 166
In the derivation of Equation 3 the variation of interfacial tension with radius (44) was neglected. It might be expected that the rate of bubble formation might follow the general theory of hrrhenius. Under this hypothesis the number of bubbles formed per unit rime in the volume would be approximated by
(3)
The associated variance may be described by ( 1 7 )
In[1 - P ( 0 .
The work associated .ivith the formation of a stable spherical bubble may be approximated by the expression (75),
The following approximate equation was used to correct the small deviations of the experimental data from conditions of constant strain:
INDUSTRIAL A N D ENGINEERING CHEMISTRY
In principle, the method employed in 'this investigation involved maintaining one relatively small portion of the system a t a somewhat higher temperature than the remainder thus localizing the region in Mhich supersaturation occurred. This arrangement permitted the configuration of the system subjected to strain to be relatively simple, without interfaces, packing glands, and acute angles which might have a significant influence upon the time-strain relationships of the system. Figure 1 shows schematically the
HY DR 0 C A R BON S Y STEMS S U PE R SIA T U R A T I O N
e
equipment employed, which consists of the pressure vessel, A , provided with a stainless steel thimble, B. The threestage centrifugal pump, C, is utilized to circulate the fluid from the entrance of the pump upward through the tube, D,located at the center of the thimble, B. Mercury is introduced or withdrawn from the vessel at E. A multilead copper-constantan thermocouple is used to determine the temperature of thimble, B, relative to the bath, K (Figure 2), and the leads are shown schematically at F and F’ (Figure 1). An electric heater, G, maintained the temperature of the thimble at desired values up to 40’ F. above that of the pressure vessel,
A. The system is brought to equilibrium at some predetermined temperature and is maintained at a pressure well above the bubble-point pressure for a specified time, usually several hours. During this period the pump, C, is operated only intermittently. The temperature of the thimble, B, is then raised a predetermined amount in order to bring the bubble-point pressure of the fluid within the thimble, B, above that of the main body of %quid. The pressure is then carefully reduced to a value just above the bubble-point pressure of the main body of liquid by the withdrawal of mercury. The situation existing under these circumstances is shown in Figure 3, in which the equilibrium vapor pressure of n-pentane is a function of temperature. The pressure within the vessel is indicated at P corresponding to the equilibrium temperature, T,. The vessel, A , is maintained at T Aand thus the state, H, corresponds to the conditions within the pressure Lessel, A . The temperature at B (Figure 1) is shown at state, J. The equilibrium bubble-point pressure at TBis above the pressure in the vessel thus yielding a supersaturated or strained state at J. The magnitude of the supersaturation pressure is given by Pb
-P
(14)
From Figure 3, the equipment described herein is only suitable for use with systems in which the equilibrium bubble-point pressure increases with an increase in temperature. Figure 2 shows the general arrangement of the associated equipment. The pressure vessel, A , (Figure 1) is shown within the agitated silicone bath, K (Figure 2). The thimble, B, is located above the three-stage centrifugal pump, C, and the tube, D,serves to introduce the fluid from the pump into the thimble. The bath, K, provided with an impeller, L , is driven by gears through the packing gland, M . Rotation of the impeller
U
’ Lo
associated Figure 2. This shows details of the equipment . . with the general arrangement in Figure 1
circulates the silicone Auid within the bath upward around the outside shell and downward around’ the vessel, A . A mercury-oil interface is provided in N so that the pressure can be measured by the balance, P (39). The quantity of mercury within the pressure vessel, A , is controlled by the chamber, R, which is connected to it with stainless steel tubing approximately 0.09 inch in inside diameter (Figure 2). A bell, S, is provided within the chamber, R, to supply oil to the compensated shaft-cylinder combination, T (6) which seals the shaft driving the impeller of the pump, C. Gages and valves at U permit the introduction and withdrawal of mercury and the determination of the approximate pressure in the vessels, A and R. The temperature of the agitated silicone bath, K, is controlled through a modulated circuit (34). Temperatures are related to the international platinum scale by a strain-free platinum resistance thermometer (27). For the ready determination of pressure as a function of time, a strain gage type of transducer, V, aided in control of operations, whereas the mechanical injector, W , maintained isobaric conditions. The system is brought to physical equilibrium at a pressure markedly higher than P (Figure 3) and the thimble
heated to an appropriate value, T B . The pressure is then gradually decreased to a chosen value, P,yielding a known degree of supersaturation. The supersaturation of the state is determined from the temperature of the thimble, B, and the pressure is determined by the balance, P (Figure 2 ) . These data, together with a knowledge of the vapor pressure of the compound as a function of tem-
TEMPERATURE
OF
Figure 3. Pressure-temperature diagram for n-lpentane VOL. 49, !NO.7
JULY 1957
1 167
-G
-X -Y
,C
Figure 4. This sectional view shows the details of the pressure vessel, thimble, and centrifugal pump constructed of chrome-nickel or other types stainless steel
perature, permit the supersaturation pressure to be calculated. The transducer, V , is employed to follow the small random variations in pressure as a function of time. The dependent variable is the time at wfhich the strained state re-
Table I.
Z+
Figure 5. This shows the details of the shaft-cylinder combination
turns to equilibrium by the formation of a bubble. Details of the pressure vessel, A , and thimble, B, are given in Figure 4. The vessel was constructed of a chrome-nickel steel. It was advantageous to use a
different type of stainless steel for the closure to avoid galling between closely mating parts immersed in silicone. A three-stage centrifugal pump, C, constructed of chrome-nickel steel was used to attain equilibrium. The location of
Experimental Results for n-Pentane in lsochoric Equipmeni Bubble
No. ot
ldentification
Exptl. Pt .
Supersatn. Pressure, Lb./Sq. Inch
Bubble Formation Time, See. Uncorrected Corrected
Std. Dev.,
Lb./Sq. Inch
Formation
Kate, Corr.
Factor,
P
Bubbles/ (Sec.) (CU. Ft.)
160' F. 9218 9220 9226 9228 9230 9232 9234 9236 9238 9242 9246 9252
5 14 8 4
7 6 6 13 13 16 4 14
4.520 16.322 144506
0.4272 0.5255 0.6189 1.8726 0.5436 2.5914 0.8679 0.2498 0.6739 1.0337 5.6901 0.7235
1.548' 1.230 1.262 2.002 1.261 1.555 1.314 1.611
0.6243 0.3554 0.6628 0.7292 0.6068 0.4716 0.2924 0.6884
2,767' 4.040 4.542 6.034 5.709 6.034 7 354 6.766 %
7.553
4425 16117 8630 4730 9315 4155 6420 17040 23300 29820 1710 7185
4472 16319 8888 5898 9493 4710 6597 17070 23286 29814 1687 7193
1.0106 1.0125 1.0299 1.2469 1.0191 1.1336 1.0276 1.0018 0 * 9994 0.9998 0.9865 1.0011
0.026 0.007 0.013 0.019 0.012 0.024 0.017 0.007 0.005 0.004 0.068 0.014
861 1000 4260 3019 2420 3157 1340 1014
855 1098 4288 3563 2750 3157 1481 994
0.9930 1,0980 1.0066 1.1802 1.1364 1.0000 1.1052 0.9803
0.133 0.104 0.027 0.032 0.042 0.036 0.077 0.115
280' F. 9268 9270 9274 9276 9278 9280 9282 9284 a
Average supersaturation pressure.
1 168
INDUSTRIAL AND ENGINEERING CHEMISTRY
H Y D R O C A R B O N SYSTEMS S U P E R S A T U R A T I O N
250
2
m A
$200
$4
a n
w
!150
rf
2 2 W
W
7
a 100
2 0
J
!! I-
b
a
50
3
2-2 B n u)
2500
5000 TIME
7500 SEC.
10,000
Figure 6. Experimental results with isochoric equipment
2
2500
SO00 TIME
7500 SEC
10,000
Figure 7. Supersaturation pressure for n-pentane as a function/of time
TIME
SEC.
Figure 8. Experimental results with heated thimble equipment lcq000
the thermocouple leads (Figure 1) is shown a t F and F'. An electric heater, G, maintained thimble, B, a t the desired temperature. The details of the shaft-cylinder combination are shown in Figure 5. A clearance of approximately 2 X 10-6 inches was provided between the shaft, X,and the cylinder, Y . The cylinder was arranged to yield some pressure compensation by elastic deformation which reduced the clearance a t the higher pressures. The oil introduced a t 2 in a lantern near the middle of the cylinder eliminated leakage and resulted in negligible contamination of a particular sample over a period of several months residence in the equipment. For a number of preliminary measurements a spherical, isochoric vessel was employed. The primary equipment consisted of a spherical stainless steel shell provided with a flexible diaphragm of the aneroid type through which the pressure within the vessel was measured as a function of time. Construction of this vessel and the associated temperature control eqyipment were described in some detail (47). The vessel was filled with n-pentane liquid a t a temperature of approximately 40' F. When heated, the pressure rose to about 1000 pounds per square inch above vapor pressure where it was held under nearly isobaric, isothermal conditions for several hours. The temperature was then gradually lowered until the pressure within the vessel was an appropriate amount below the vapor pressure of n-pentane and the system was then maintained under isothermal conditions until a bubble was formed. The associated rapid rise in pressure to the vapor pressure a t the temperature in question indicated the bubble formation. T h e careful control of temperature to time during adjustment to the prescribed value was not
sufficiently precise to avoid marked variations in the pressure with fespect to time during the period of strain. For this reason the equipment was not suitable for extensive investigation of the phenomena of supersaturation in hydrocarbon liquids.
I/
50,000
20,000
l0,OOO
9
i
a-
5000
m"
'ta
Experimental Results
2000
A typical set of experimtntal results obtained with the isochoric equipment for n-pentane (Figure 6) corresponded to the pressures measured with the balance, P, (Figure 2) as a function of time. I n addition a nearly continuous record of pressure was obtained by the strain gage transducer, V. From the data (Figure
7
1000
500 6
I
I
I LB PER
Loo
I
SO
IN
Figure 9. Effect of strain duration of supersaturation
upon
Table 11.
Experimental Results for n-Pentane in Heated Thimble Equipment Supersatn. Bubble Formation No. of Pressure, Std. Dev., Rate, Exptl. Lb./Sq. Lb./Sq. Time, bubbles/ (see.) Identification Pt. Inch Inch sec. (cu. ft.) 160' F. 19078 19078 19082 19082 19086 19086 19090 19090 19096 19102 19104 19104 19107 19107 19107 19114 19118 19120 19120 19130 19130 19135 19135 a
2 28 7 14 7 23 2 17 3 21 3 64 3 2 3 5 1 2 2 2 2 2 2
18.29a 13.21 18.99 13.74 19.23 13:47 23.05 8.91 23.14 9.02 22.78 8.96 26.56 27.04 13.51 31.34 31.65 36.27 37.42 35.10 36.75 35.57 34.92
0.0000 0.5653 0.2286 0.4930 0.2673 0.3376 0.0000 0.4050 0.0000 0.1844 0.8520 0.4405 0.1155 0.7778 0.2754 1.0663 0.0000 0.2121 0.7071 0.2828 0.0707 0.1442 0.7071
242 29265 10345 14854 12023 23735 437 20062 1457 41250 4805 126900 1418 1899 3414 5454 682 246 480 540 823 442 624
10.054 0.083 0.235 0.164 0.202 0.103 5.568 0.121 1.670 0.059 0.506 0.019 1.716 1.281 0.713 0.446 3.568 9.891 5.069 4.506 2.956 5.505 3.899
Average supersaturation pressure.
VOL. 49, NiO. 7
JULY 1957
1169
5 '
LB. PER
bringing the supersaturation to a predetermined value as the temperature was gradually reduced. As a result, the supersaturation was changing with time for a significant part of the total period. Such behavior further complicated the statistical analysis of the results. The results of 20 n-pentane measurements with the isochoric equipment are recorded in Table I for temperatures near 160" and 280" F. The average values o f supersaturation pressure and time of strain have been indicated. T h e standard deviation of the variation in pressure with time was included as well as the uncorrected time of strain. The details o f the calculation of the average pressure and the corrected time of strain for isobaric conditions of strain were discussed in an earlier section. A total volume of 0.00876 cubic foot and a surface area of 0.22245 square foot were involved. Figure 8 shows results for n-pentane obtained with the heated thimble equipment. The sample is maintained under isothermal conditions throughout the period of strain. Only a total of 0.000411 cubic foot of sample with a surface area of 0.0567 square foot within the thimble is subjected to the strain and the conditions are nearly isobaric. For this reason the results can be obtained for a particular set of conditions directly from a series of experimental measurements (Figure 8). Experimental results obtained with the heated thimble equipmert are siimmarized in T a h l e 11. By using the information of Table IT. adjusted by Equation 11 for small deviations from isobaric conditions, Figure 9 shows the natural logarithm of the time of strain as a function of the supersaturation pressure. The applicability of Equation l l is shovm by the natural logarithm of the time of strain as a function of the reciprocal of the square of the supersaturation pressure (Figure IO). As predicted by Equation 7, there is a marked deviation from the straight line fitted to the data by standard statistical methods ( 2 9 ) . The data of Figures 9 and 10 are based upon a temperature of 160" F. and were ob-
SP. IN.
l00,000 50,000
20,000
10,000
v
W
5000
m" 2000
I
I
7
I
L
0.0025
00050
(p,*)-' Figure 1 0.
00075
IN? PER
00100
0.0125
LBZ
Applicability of Arrhenius equation to supersaturation
h - 0.5 - 1.0 T=7
%, - 1.5 U
4
y
-2.0
-2.5
-3.0
2500
5000 TIME
Figure 1 1 . Probability of a bubble's not being f o r m e d
7500 SEC.
Table 111.
10,000
Comparison of Predicted and Experimental Standard Deviations Mean Time
G), the supersaturation pressure or strain was computed in Figure 7. I n this instance an average supersaturation pressure of 4.54 pounds per square inch was maintained for 8630 seconds or approximately 2.4 hours before a bubble was formed. Figures 6 and 7 indicate that the technique involving an isochoric vessel leaves much to be desired in maintaining a constant value of supersaturation pressure for an extended period. Also, some difficulty was experienced in
1 170
?io. of
Points
INDUSTRIAL AND ENGINEERING CHEMISTRY
Temp., F.
Supersatii.
t o First
Pressure,
Bubble,
Lb./Sq. Inch
See.
Bubble Formation Rate, Buhhles/(Sec.) Ft.)
Std. Dev.n, Set.
Eq. 5
Eq. 8
8521 1317
0.010
0.050
0.011 0.049
2499 10354 57692
1.117 0.139 0.038
1.019 0.13% 0.057
((211.
Isochoric 12 8
160 280
5.79 1.41
16 4 3
160
25.82 13.48 8.94
11278 2262
Heated Thimble 2177 17457 64485
Deviation from mean time. Fixed supersaturation pressure, Pb - Po.
H Y D R O C A R B O N SYSTEMS SUPERSATURATION
.
tained from the thimble equipment. T h e data from the isochoric equipment were obtained over a significant range of strains while those for the heated thimble apparatus were restricted to two to three narrow ranges of supersaturation pressures. The results of the corrections for nonisobaric conditions are given in Table I. These corrections, which were made by iterative application of the methods of analysis, are only approximations that permit the experimental results to be compared under the conditions of constant strain. Utilizing the corrected data, Figure 11 shows the probability of a bubble not being formed under one typical condition of strain as a function of time. TO calculate the ordinate of this figure, the probability of a bubble forming up to a given time was approximated by the ratio of a number of trials with supersaturation times less than that of, but including, the trial in question to the total number of trials. This approach to the evaluation of probability leaves much to be desired (77, 3 7 ) but affords a simple means of presenting the nature of supersaturation phenomena. T h e confidence limits for such an analysis are poor with the small number of investigations reported here for each set of conditions. A straight line was fitted to the data by the method of least squares (Figure 11). According to Equation 8, the line should pass through the origin with a slope equal to the negative reciprocal of the mean time. T h e intercepts and slopes were compared with Student’s t test (50) and no significant difference was found. T h e standard deviation of the times from the mean time was computed and given in Table 111. The agreement is good for the number of points involved. A chisquared test (50) of the frequency of occurrence of points within various intervals does not indicate any significant discrepancy between the data and the theoretical distribution, but it does point up the need for a large amount of data under constant conditions. These results indicate that the proposed distribution (48) is followed by these data and that the formation of bubbles is random with respect to time as was predicted from the theory of liquids (78). From Table 111, the probability of a bubble forming as a function oftime for a given degree of strain is shown in Figures 12 and 13. The poor confidence limits described earlier apply to the inforniation of these figures. For this reason wide disagreements from such prediction are encountered for a particular case. The data obtained with the isochoric equipment involved nearly 21 times the volume of n-pentane and nearly 4 times the surface area employed in the investigation with the heated thimble equip-
0
d
u
Q-
W
-I
m
m 3 m LL
0
TIME
SEC.
Figure 12. Probability of bubble being formed in heated thimble equipment
5000
l0,OOO
15,000
20,000
25,000
TIME SEC.
Figure 13. Probability o f a bubble’s being formed in isochoric equipment VOL. 49, NO. 7
JULY 1957
1171
ment. Each set of measurements was corrected to a unit volume basis (Tables I and 11). After making this correction there is little to choose in a statistical sense between the experimental data from the two pieces of equipment. T h e data from the two types of equipment are in rough agreement when compared on a unit volume basis. T h e data are insufficient in number to establish the effect of surface on bubble formation. These preliminary results indicate that a large number of experimental trials are required for determining the statistical behavior of supersaturated hydrocarbon liquids. Furthermore, the data of Figure 13 indicate a marked shift in the probability function as the temperature increases. T h e probability of a bubble being formed a t a given degree of supersaturation was much greater for a particular time of strain a t 280’ F. than was the case a t 160’ F. Further experimental work is necessary to establish whether or not these trends indicated by this limited study are reflected in the general behavior of supersaturated hydrocarbon liquids.
p
=
y
= interfacial tension, lb./ft. = time, sec.
B
B
correction factor
mean time for first bubble to form, sec. 6, = time of strain to first bubble, sec. 0: = time of strain to first bubble adjusted to fixed supersaturation pressure (P6- PO),sec. 8h = mean time to first bubble for specific conditions, sec. (&)e = corrected time to first bubble for nonisobaric conditions of strain, sec. Bm = mean time corresponding to a given value of supersaturation pressure, sec. p = mean Q = standard deviation =
Superscripts
*
= time average of quantity during run
Literature Cited ( 1 ) Akhumov, E. I., Rozen, B. Ya., Zhur. Fiz. Khim. (USSR) 27, 1760 (1953). ( 2 ) Arley, N., Buch, K. R., “Introduc-
tion to the Theory of Probability and Statistics,” Wiley, New York,
Acknowledgment
This work was a contribution from Research Project 37 of the American Petroleum Institute a t the California Institute of Technology. H. H. Reamer and W. M. DeWitt contributed to the experimental program while Virginia Berry and Betty Kendall aided in the preparation of the results. W. N. Lacey reviewed the manuscript.
1950. (3) Briggs, L. J., J . A$@. Phys. 21, 721 (1950). ( 4 ) Budgett, H. M., Proc. Roy. Sac. (London) A86,25 (1912). ( 5 ) Burrows, G., Preece, F. H., Trans. Inst. Chem. Engrs. (London) 32, 99 (1954).
sity of -Wisconsin, October
(8) (9)
Nomenclature
(10)
A
(11)
b d e k In n %
P Pb
pb
PS
P
Q T W
-
a
= frequency factor in Arrhenius
rate expression specific gas constant = differential operator = base of natural logarithm = parameter in Poisson distribution = natural logarithm = statistical sample size = rate of bubble formation, bubbles/(sec.) (cu. ft.) = pressure, lb./sq. inch = bubble-point pressure, lb./’sq. inch PO = fixed supersaturation pressure, lb./sq. inch = supersaturation pressure, lb./sq. inch = probability of bubble being formed = probability of bubble not being formed = temperature degrees Rankine = work necessary to form stable spherical bubble, ft. lb. or B.t.11. 16 n y 3 = __- (lb./sq. inch)2 3 6t =
1 172
(12) (13) 114)
15, 1954. Ibid., WIS-ONR-15, January 3, 1955. Dixon, H. H., Sci. Proc. Roy. Dublin Soc. 14, 229 (1914). Faro, I., Small, T. R., Hill, F. K., J . AppZ. Phvs. 23, 40 (1952). Fry, T. C., “Probability and Its Engineering Uses,” Van Nostrand, New York, 1928. Fiirth, R., Proc. Cambridge Phil. Soc. 37, 252 (1941). Zbid., p. 276. Gardescu, I. I., Ozl Gas J . 30, No. 41, 22 (1931).
(15) Gibbs, J. W.,“Collected Works,” vol. I, Longmans, Green, New York, 1931. (16) Goranson, R. W., $‘Thermodynamic
Relations in Multi-Component Systems,” Carnegie Institution of Washington, Washington, D. C.,
1930. (17) Gumbel, E. J., Natl. Bur. Standards, Appl. M a t h . Series 33, U. S. Govt.
Print. Office, Washington, D. C. (1954). (18) Hunt, E. B., Jr., private communication, January 8,1954. (19) Jha, S. D., Kolloid-Z. 137, 162 (1954). (20) Kennedy, H. T., Olson, C. R., Trans. A m . Znst. Mining Met. Engrs. 195, 271 (1952). (21) Khinchin, A. I., ‘(Asymptotische
INDUSTRIAL AND ENGINEERING CHEMISTRY
Gesetze der Wahrscheinlichkeitsrechnung,” Chelsea Publishing Go., New York, 1948.
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RECEIVED for review July 3, 1956 ACCEPTED December 26, 1956
