To determine the density of the air passing through the tube, density being a strong function of both pressure and temperature, it was necessary to establish the magnitude of the average pressure in the tube, pay,the temperature being known. Since the pressure in the large tube downstream of the capillary, p b , was atmospheric, it was assumed that air emerging from the capillary tube was also atmospheric, to a good approximation (Figure 2 ) . Considering the pressure drop across the capillary as Ap,, the total pressure, p T , upstream is the sum of the barometric pressure, P b , p1.n the pressure drop across the capillary, ApC (the U-tube manometer measured this pressure drop). Therefore, as an estimate: of the average pressure we can write p a y = (PT f pb)/2 = P b Ap,/2. Thus, with both the temperature and pressure in the tube determined, the density of the dry air passing through the capillary tube was established. Using this density and the corrected wet volume flow rate, the mass flow rate, M , through the tube was easily obtained. Accurate measurement of the capillary diameters was critical, since the tube dia.meter enters Equation 13 as the diameter to the fourth power. Circularity of cross section and uniformity of bore are also of prime importance. T h e cross sections and bores of the tubes were viewed under a microscope and found to be free of any major imperfections. T o determine the capillary diameter, the mercury technique of Benton (1919) was usNed and then compared to the manufacturer’s specifications. All the capillary tubes used in the experiment were within the tolerances cited by the manufacturer. With the foregoing parameters, p , p , M , and D , known, the pressure drop due to frictional resistance, ,uL, can be obtained using Poiseuille’s equation. However, Poiseuille’s equation applies only a t distances far from the ends of tubes where the flow streamlines are parallel. The losses at the ends of sharp tubes, where the streamlines are radial, can be accounted for by using the derived expansion and contraction loss coefficients. From Equation 13, the total pressure loss due to end effects is ( K , f K,)/2 ( M / 8 n ) . The contraction and expansion coefficients in the exprer.sion can be evaluated by Equations 8 and 12. Substituting numericaily in Equation 12, values of C, are obtained from Rouse (1962). Here C, is listed as a function of
Table 1.
p,
+
4;
= DJD1. T h e largest capillary diameter, D2, used in the experiment was 0.063 inch, while DL was always 0.875 inch. Hence, the ratio D2/D1 = 0.063 inch/0.875 inch = 0.072 and C , = 0.612. For the remaining capillary tubes, the diameter ratios had C, == 0.611. From the above the maxi-
&
mum value of is 0.0’72. Since the diameter ratio squared enters Equation 8, uz = 2.7 X 10-6. This value is so small compared to the other t’erms in the equation that it can be neglected. Therefore, Equation 8 reduces to
K e -- 1 t- 2K.w2C,2 - 2Cc
-
1
C2
Contraction and Expansion Coefficients
Tube Diameter, Inch 0.063 0.050 0.035 0.020
U
KC
K,
0.0052 0.0033 0.0016
1.072 1.072 1.072
0.0005
1.072
0 986 o 99i 0,996 0.999
When Cc = 0.61 1 and assuming parabolic flow exists in the capillary, K.w2 = 4/3, the contraction coefficient, K , = 1.072. If the uz terms are also neglected in Equation 12, the expression for the expansion coefficient, we have K , = 1 - 2K.+**a. Since parabolic flow was assumed in the capillary, K.vf2 = 4/3, and K , = 1 - 2 . 6 6 7 ~ . From this, we can calculate corresponding expansion coefficientsfor various values of u. T h e values of K , and K , found in the experiment are listed in Table I. Thus, the contraction and expansion losses a t the tube ends have been determined for laminar flow. Results
Figure 3 compares the calculated pressure drop, Apt, using Equation 13 and that calculated using Poiseuille’s equation, App, with the pressure drop measured across the capillary, Apt. As can be seen from the least squares fit lines for 90 observations, the calculated values (open symbols) for the pressure drop using Equation 13 are in almost perfect agreement with the measured values. O n the other hand, values (closed symbols) calculated using Poiseuille’s equation applied to each of the tube length and diameter condition of all 90 observations are obviously not in very good agreement. This large discrepancy arises from ignoring the effect of the losses due to contraction and expansion a t the ends of short sharp-edged capillary tubes. Acknowledgment
I t is a privilege to acknowledge the invaluable aid of J. W. Thomas, who supplied the motivation and assistance necessary to complete the investigation. literature Cited
Benton, A . F., J . Ind. Eng. Chem. 2, 623 (1919). Brillouin, M., “Lesons sur la ViscositC des Liquide et des Gas,” Vol. 1, p. 133; Vol. 2, p. 37 (1907). Moody, L. F., Trans. ASME66,671 (1944). Poiseuille, J., Compt. Rend. 11, 961, 1041 (1840); 12, 112 (1841). Rouse, H., “Elementary Mechanics of Fluids,” p. 57, Wiley, New York, 1962. DAVID RIMBERG Health and Safety Laboratory Atomic Energy Commission New York, N.Y. RECEIVED for review January 17, 1967 ACCEPTED June 12, 1967
L/’.s.
A THIRD FIACTOR IN BOILING NUCLEATION HE hypothesis is based on earlier photographic studies of T b o i l i n g (Westwater et al., 1961). T h e virtually instantaneous appearance of a bubble after a pause in bubble formation a t a site implies a short period of extremely rapid growth. Photographic studies alscl show that the growth of the bubble is very much less rapid after it has broken away from the surface.
Thus, there is initially a very short period of very rapid growth, followed by a sharp drop to normal growth while the bubble is attached to the surface, followed by yet another sharp drop in growth rate as the bubble detaches from the surface. T h e simplest possible pattern for this system is shown by the solid line in Figure 1 which illustrates the constant growth rate durVOL. 6
NO. 4
NOVEMBER 1 9 6 7
603
Bubbles come only from specific points in a hot surface, and these points initially contain vapor only intermittently. To explain this, two conditions have been considered: one is a location particularly favorable to forming or holding a vapor and/or dissolved gas nucleus; the other is a sufficient degree of superheat, either to form the nucleus, or to allow the repeated budding of a residual nucleus. These two factors alone do not adequately explain the experimental results, but call for a third factor. This is the presence of pressure fluctuatius, generated b y the bubbles themselves, which tend sometimes to suppress and sometimes to favor nucleation at either adjacent side. A hypothesis is presented suggesting when this third factor should come into play, and experimental results testing the hypothesis are analyzed. These experimental results tend to confirm details of the hypothesis that cannot be readily explained b y the current theories involving only the two factors noted earlier.
Experimental and Analytical Procedure
dV ---dB
N Figure 1 . hypothesis
e
B
Bubble growth rate vs. time, illustrating the
ing bubble attachment. T h e dotted line suggests the prevalence of overshoot and oscillations in such physical systems. As groivth of a bubble implies an equivalent displacement of liquid, an increased growth rate causes acceleration of liquid and, thus. the initiation of a pressure wave. This pressure \vave suppresses nucleation by raising the total pressure on nucleation sites. A decreased groivth rate and the resultant liquid deceleration imply a rarefaction pressure wave Lvhich could be of considerable assistance in nucleation given a favorable combination of local surface conditions and local degree of superheat. Alternatively, pressure waves themselves are knoivn to create rarefaction waves as, for example, those occurring on their arrival at an interface with a much more compressible medium. Bubble formation from such effects have been photographed (Benjamin, 1966) in cavitation studies. .4ccording to this hypothesis, it would be expected that the nucleation of a second adjacent bubble would be suppressed a t the instant of the primary nucleation. but favored immediately afterivard. Budding of a residual bubble should also be favored a t the instant of bubble break-off. T h e possible oscillations suggested by the dotted line should also affect nucleation if they are strong enough. Oscillations during bubble growth have been observed by the authors in high speed films. T h e heat transfer process itself tends to amplify minor oscillations because increasing the heat transfer rate increases the bubble growth rate, and the resulting acceleration of the liquid forces the bubble against the heater wall, thereby aiding heat flow into the base of the bubble. Decreasing the heat transfer rate produces the opposite effect. From this, it could be expected that the influence of oscillations would be greatest at high heat fluxes. 604
l&EC FUNDAMENTALS
T h e hypothesis has been tested by examining films of boiling taken a t 4000 frames per second to determine whether there is evidence of a nonrandom relationship which is consistent \vith suppression of nucleation of a second bubble a t the instant of nucleation of the first, and favorable to the nucleation of a second bubble immediately following that of the first. The boiling of distilled water was photographed on alumina- (5 microns) finished stainless steel strips (0.01 0 inch thick), which \cere electrically heated (Young and Hummel, 1964). T h e evidence indicates a strong preference for nucleation of a second bubble one frame time after the first. This predominant feature of the experimental results is not adequately explained by the concept of thermal cycling. Further examination of the film suggests that nucleation after break-off is favored and that there is a hint of bubble oscillation affecting the nucleation of bubbles. ,4brief examination of the literature will be made to support the hypothesis. Figure 2, Lvhich shotvs 12 successive frames of film taken a t 6000 frames per second, illustrates the hypothesis and the method of analysis. Nucleation of a bubble in the second frame is followed one frame later by nucleation of a second bubble in the third frame, and this is again follo\\ed by nucleation of a third bubble in the fifth frame, all in separate locations. hToadditional nucleation events take place in the follo\\ing frame. For the purposes of analysis. these relationships will be identified by a frame count between a nucleation event and the nearest possible preceding cause of the type being considered. For example, in the frames shovn, the count for the third bubble would be 2 relati! e to nucleation ( S S 2 ) of the preceding bubble. The count for bubble 3 could have been considered 3 relative to the first nucleation, but the count starts relative to the immediately preceding nucleation. If two bubbles appear in the same frame, then the frame count for one would be 0 (.Y.YO) and the frame count for the other uould extend back\\ ard to the preceding nucleation event. The nucleation events in each complete film are tabulated in Table I according to this frame count. This method of count-
Table I.
Number of sites 1
- M/’V
T ,
Extrapolated .YAYl Actual .l’.Vl Actual S h ’ O
Summary of Numerical Results Film ,Yumber 7 2 3 4 10 8 10 11
0.947 2 31 8 6 22 0
0.951 1 28 6 0
15 1
0.944
1.92 10 2 16 0
5 4
0.962 0.978 1.15 0.81
3 6 10 3
1 4
5 0
Other Results Consistent with the Hypothesis FRAMES
W
& W
m
0
3688
FRAMES
0 5 10 1520 25
15
BUBBLES FRAMES SITES
2
. ..
10
5
3562 IO
w
0 0
0
5
3266
IO 15 2 0 2 5
0 5 I O 15 2 0 25
BUBBLES 8 3 FRAMES 3 8 3 8 SITES
0U08LES 137 3620
FRAMES SITES
lM l u .. . ..
. . . . .. .. *
0
5 IO 15 20 25
5 IO 15 20 2 5
I..
0
“
5 IO 15 2 0 25
N-N FRAME COUNTS
Figure 3.
Number of events
Each roll refers to a separate 1 0 0 - f o o t roll of 16-mm. fllm taken a t 4000 frames per second. The number of frames of fllm which a r e useable, the number o f bubbles observed, and the number of sites from which these bubbles come a r e noted. The flux i s in E.t.u./hour foot2
A’,2’1 is given in Table I along with 6, the root-mean-square deviation of the data for the last 21 frame counts from the random curve. This analysis shows that for film 2, the value for n’n‘l is 6 . 3 ~from the curve. The probability that such a deviation is part of a normal population is 1 in 10’0. Nucleation of a second bubble is indeed favored one frame time after a bubble nucleation. Individually, the other films are not quite so convincing, but collectively they lend increased support to the validity of the hypothesis. T h e events in one film counted as iYN1 would involve various nucleation sites, sometimes a t opposite extremes of the field of view (0.5 inch wide X 1 inch deep). Some events which were given a higher count probably stood in an A’s1 relationship to some nucleation events outside the field of view. Consistent linking of nucleation a t two different sites having a closely spaced time cannot be explained by thermal cycle alone, even if we grant that the beginning. or low point, of the thermal cycle for these two sites can be related by the fact that departure of one bubble frequently sweeps another bubble from the surface. To terminate the cycle within one frame of identical times by nucleation would also require essentially equal and reproducible waiting periods for the two sites, while in fact Han and Griffith (1965) have noted on Table I V of a recent paper: “The waiting period, T , changes from 17 (T,,,,) to 130 (T,,,,) . . . ” Additional complication occurs because it is not always the same two sites that are related to one another in our photographs. Finally, both current theory and the experimental observations of this study indicate that the nucleation sites involved are by no means equivalent in probability of bubble emission. In fact, instead of having essentially equivalent waiting periods, the waiting periods should vary widely. Table I shows that the evidence for the suppression of nucleation of a second bubble a t the instant of nucleation of a prior one is equally compelling. In the few recorded cases of iL’A’0, one nucleus is always considerably larger than the other suggesting that the two occurred a t the beginning and end of the time of one frame. I t is fortunate for analysis that the frames-per-second ratio was not less. There is also indication of a preference for nucleation near break-off peaking a t B’Vl. 606
I&EC
FUNDAMENTALS
Other expected manifestations of this interaction have been observed in our films and in the literature. For example, in a film with many bubbles and many potential nucleation sites, one might expect combinations of long periods without nucleation v i t h periods during Lvhich several nucleations follow in rapid succession. This pattern clearly occurs with a greater than random frequency. Marto and Rohsenow (1965) and Rohsenow (1 966) obtained unstable boiling of sodium where, for periods of approximately one minute, the boiling superheat would alternate between a value less than 50’ F. to about 150” F. Sound periods corresponded to periods of lower surface superheat. The greatest sound intensity occurred when the temperature first dropped. Westwater (1959) and Foust (1965) reported a violent vapor eruption in boiling methanol which triggered a “shock wave” of minor bursts set off by the major one. Colver and Balzhiser (1964) photographed such a chain of bubble triggering. Observers of boiling water (Young and Hummel, 1964) and boiling potassium (Colver and Balzhiser, 1964) have reported temperature fluctuations of seconds, and thus much longer than a bubble lifetime. A considerable literature has developed on cavitation-i.e., the formation of vapor by violent mechanical disturbances and by ultrasonics. I n conclusion. a strong tendency for nucleation to occur one frame time after a preceding nucleation has been observed on electrically heated, polished stainless steel strips. This phenomenon cannot be adequately explained in terms of thermal cycling. .4n explanation in terms of a sonic interaction of a bubble with its surroundings is supported by photographic data and by evidence in the literature.
Acknowledgmenl
The authors thank R . K. Young and G. E . Williams for permission and aid in photographing samples on their apparatus and Young and L. Gevaert for helpful discussions. T h e indulgence of the University is also acknowledged with gratitude for allowing the senior author to be supported for the summer from an exhausted account. The kindness of the Aero-Physics Department in lending us their movie equipment is also acknowledged.
Literature Cited
Benjamin, T. B., 6th Naval Hydrodynamics Symposium, IYashington, D. C., Sept. 28-Oct. 4, 1966. Colver, C. P., Balzhiser, I
