ML19284A738
| ML19284A738 | |
| Person / Time | |
|---|---|
| Site: | Susquehanna  |
| Issue date: | 12/31/1978 |
| From: | Weisshaupl PENNSYLVANIA POWER & LIGHT CO. |
| To: | |
| Shared Package | |
| ML17138A531 | List: |
| References | |
| 2241, NUDOCS 7903150313 | |
| Download: ML19284A738 (24) | |
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-i FORMAT 10.~4 AND OSCILLATIO.lS OF A SPHERICAL GAS BUBBLE #"E' o .s UiiDER \lATER v a C.
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PROPRIETARY IN FORMATION This document has been made NON-PROPRIETARY by the deletion of that information which was classified as PROPRIETARY by KRAFTWERK UNION AG (KWU).
The PROPRIETARY information deletions are so noted throughout the report where indicated by a) Use of the term KRAFTWERK UNION AG PROPRIETARY INFORMATION.
b) Use of blocked out areas by cross hatch bands in the report text and figures / tables, e.g.
i) . . . . " with a mass flow density of MT Kg/n2 s , , , a ;
ii) b MT mm iii) ...." should be kept below /M.atm."
7/A ,
t 8/17/78
AEG TELEFUNKEN Ffm., December 1972 E3/E2/SA Dr. Wei/po NUCLEAR REACTORS Report No. 2241 FORMATION AND OSCILLATIONS OF A y SPHERICAL GAS BUBBLE UNDER WATER o
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2H uO ao c4 NS E 23 Prepared: /s/ (Dr. Weisshhupl, E3/E2/SA) uN
'd a Checked: /s/ (Dr. Koch, E3/E2) tx
$$ Classified: /s/ (Dr. Slegers, E3/E2/SAB)
Class II l-1
Distribution list:
E3 - Sekretariat E3/V E3/Vi E3/V2 E3/V3 ,
E3/V4 E3/V3 E3/V4/KYJC E3/E E3/Et E3/E1/GK E3/E2 E3/E3 E3/Ei-LP 2x E3/E2/sA 4x E3/R E3/R1 E3/R2 E3/R3 E3/R4 E3/RS E3/R2-KL E3/Sibliothek Library HE/E-F PT-F E3/E2/SR E3/E2/MM E3/V4-KW E3/V4-KKB E3/V4-KKr
TABLE OF CONTENTS Summary
- 1. Introduction 'N.
- i' '
- 2. Oscillation of a spherical gas bubble 2.1 Equation of motion 2.2 Oscillation of the bubble 2.2.1 Variation of the radius and bubble pressure 2.2.2 Dependence of the frequency on the bubble's radius 2.3 Pressure distribution in the vicinity of the pulsating gas bubble
- 3. Formation of the gas bubble Figures References 1-3
Summary __
In the following report we examine the oscillation process of a gas bubble under water which is subjected to elevated pressure at the beginning of the process. We find a characteristic periodic variation of the pressure. The frequency of this oscillation is inversely proportional to the initial radius of the bubble.
The investigation of the pressure distribution in the vicinity of the pulsating gas bubble demonstrates that the overpressure amplitudes have an inversely proportional dependence on the distance and a directly proportional dependence on the initial value of the radius.
A greatly simplified model is set up for the formation of a spherical bubble by outflow of air from a pipe submerged in water.
It turns out that approximately N times the initial volume is generally reached in a very short period of time (less than M ms) .
1-4
- 1. Introduction If a gas bubble is under elevated pressure under water, then the gas bubble tries to come into equilibrium with the surrounding pressure:
the surrounding water is accelerated by the elevated pressure and thus the volume increase of the bubble reduces the pressure.
Due to the water's inertia effect, the bubbla expands beyond the volume corresponding to the equilibrium pressure, whereby an underpressure is produced in the bubble and the water masses are accelerated in the opposite direction. If there is no damping, the bubble is again compressed to the original pres-sure, expands again, is compressed again, etc.
The exact solution of this problem is extremely complicated and not possible at all in closed form. By suitable measures, however, the problem can be simplified substantially and solved with com-paratively little computational ef fort. We shall ther efore pre-suppose spherical symmetry for the following analysis. That means that we neglect the gravitational force and consider only the muments of inertia of the water. We shall further assume that the bubble does not rise to the water's surface during the oscil-lation process.
(This assumption follows automatically from the neglect of gravitation.
Also, the ascent toward the water's sur-face during one oscillation process is negligibly small.)
- 2. Oscillation of a spherical gas bubble 22 1,gguagigg_gf,gggigg Assuming spherical symmetry, the velocity field of the water 1-5
surrrounding the bubble is irrotational and the velocity potential reads /1,2/:
(d) f([) (1)
./L where R is the bubble's radius and r is the distance of the considered field pcint in the water from the center of the bubble.
R (t) is the velocity of the bubble's surface. The equation of motion reads:
~
g + {1 [7 2
- ' AP r P'N~}%
l (2) where p is the density of the water, p(r,t) is the pressure in the water at a distance r at time t, and p,is the pressure in the water at an infinitely large distance from the oscillating bubble (thus corresponds to the static pressure).
If we insert Eq. (1) into Eq. (2) we get:
l' f Yf_ jo(AN'&
r 4 A L A4 2.2__gggillagigg_gi_thg_bybblg 2 . 2:.l_YeEietien_e f _the_Ea digg_ggd_bghblg_gggg gggg For r = R we obtain from Eq. (3) the equation of motion for the bubble's surface:
")
T . 'R + } <R.z = pt"R,M j%
S 1-6
The velocity of the bubble's surface is generally small compared to the speed of sound in the bubble's gas. The pressure at the surface p(R,t) is therefore simultaneously the pressure in the bubble's .nterior.
Since the oscillation process proceeds rapidly enough, we can assume an adiabatic change of state. Using the adiabatic equation of state
. V = V, (5) we can calculate p(R,t) from the initial state (p g,Vg,T g) and obtain with Vg = 4rR /3:
b 34 f k dI = 8- (6) where p g = p (R,0) .
With Eq. (6) we now write the equation of motion (4) in the final form g' 1. f -. b. , t: - b (7)
A This nonlinear differential equation of second order can be solved by a numerical method and yields the time variation of the bubble's radius. The as.ociated pressure variation in the bubble can be determined from Eq. (6).
That pressure variation is illustrated in Figure 1 for an initial radius Rg =L%4 m, an initial pressure p g
= I(((([j(absolute)
/ and 1-7
a static pressure p, =N2Q2Q6' kg/cm (absolute). We again obtain the characteristic o illation behavior of a compressed gas bubble as described previously in Report AEG - E3- 2208 /3/. The frequency of the o'scillation is f = k%%% Figure 2 shows a parameter study Starting from the reference case (R = k% Nim, p g = E%%%%%1 g
(absolut e) , p, = kN003kg/cm 2 (absolute)) one parameter was varied in each case while holding the other parameters fixed in order to demonstrate the influence of that quantity on the frequency, maximum radius and minimum pressure. Especially conspicuous is the strong dependence of the frequency on the bubble radius, which obeys a 1/R 1aw: QM%%%%%%%%%%%%%%%%%%%%%%%
N%%%%%%%%%%%%%%%%%%%%%%%h%%%%%%%%%%' Thus, the smaller the bubble 's radi us , the higher is the frequency of the oscillation. On the other hand, the initial pressure p g to which the bubble is subjected has only a relatively slight effect on the frequency.
2 2 2_gggggdgggg_g{_tbg_jpgcggggy_gg_tbg_yghblgig_gadigg The 1/R depender.ce of the oscillation f requency can also be seen directly from the differential equation (7) for the motion of the bubble's surface. For that purpose, we first replace R(t)/Rg by x (t) and write the equation for *he variable xy(t) associated with a particular initial radius R g(1) :
d %ld 'I [o (8) 48 L cU<[M 1= x'N 4 _ /b clU + A 5, kT, - $ $
We now seek the solution x2(t) belonging to the initial value Ro(2) ,
1-8
where Rg (2) =a -
Rg (1)
(9) is to be valid.
We write the differential equation for x2 (t) , express R by means of the above relation in terms of Rg ( } and bring the quantity a onto the left side of the equation, bringing it into the differential quotient. We obtain:
x*k).d 1(0 J NW{$\2 2
s - Po x2-3w(z) i %
d(US 1 ol% -(RlT, .] f (* )
We now replace the expression t/a by t' and thereby obtain:
A g' (gt ). d >e"l$ +en Ekk[Nb 2.
hlgj $
fo 3w
,y (11) c (),' ' 1 ck' 2 ~
(Qff $
f We now compare the equation thus obtained with Eq. (8) for xy(t) and see that the two differential equations are identical if in Eq. (8) we repl 7e t by t' in a purely formal manner:
(12) d'A[E) ,1 d% W 4 Po ,y'#(g)~ S K,S, h g , z. *I g,, (qm)2. f 9 c, The two equations now have the same differential operators, which are applied first to xt(t') and then to x2(at'). In other words, the formal replacement t t' in the solution xy(t) leads to the solution of the differentia 1 equation for x2It)
- 1-9
X.,(q d ) = Yf [
~
, (13 )
With t' = t/a, we then obtain b
X2. ( k C 24 \ (14)
Q I 0:
p (A x2.( M' = xa p ( 7.1 6)e (15)
This means: I# x 1(t) is the solution associated with Rg ( ) and x
2(t) is the solution associated with Rg ( , then one is obtained from the other by multiplying the time axis of one solution by the ratio of the initial values of the bubble's radius.
For the oscillation frequency we therefore obtain the relation:
g o(di (16)
[_ 4(U Thus, the oscillation frequency is inversely proportional to the radius.
2.3 Pressure distribution in the vicinity of the pulsating gas
_____ bubble ____________________________________________________
The water mass surrounding the oscillating gas bubble is accelerated by the overpressure and underpressure in the bubble. At a suf-ficiently large distance from the bubble, however, the water is 1-10
no longer influenced by the oscillation and the pressure pre-vailing there is determined by the static pressure. Thus, the pressure prevailing in the vicinity of the gas bubble decreases with increasing distance from the bubble from p (R,t) to p, (o r ,
if there is an underp essure in the bubble, increases from p(R,t) to p,). From eq. (3) we can now calculate this dependence of the pressure on r when we know the time variation of R:
24J EfI '1s #(4).M6)
R (17)
F k &, Rn+f E .x Si A .@ ,1 g If we inserg the solution R = R (t) frcm Eq. (7) into the above equation, we obtain the time variation of the pressure at any distance r from the bubble's center.
Because of our assumption of spherical symmetry, the pressure in the water depends only on the distance r, and not on t'e polar and azimuthal angles. Also not taken into consideration in our formulation are the effects of the surfaces bounding the water mass, since the model presupposes an air bubble in an infinitely extended medium. For that reason, the model also fails if the bubble is located near the water's surface.
If we are interested only in how the pressure amplitude decreases with increasing distance from the bubble, then a simple analytic expression for the dependence of the pressure on the distance can be ob tained from Eq. (17): At the time t =0,T, 2T, 3r , . . . . ,
the pressure in the bubble has its highest value. Therefore, if we set t = 0 in Eq. (17) and use the initial conditions 1-11
'R (0) = 7?,
]P 0)- O v ( 8) then we first obtain from Eq. (4)
R (0) _,, N I
- [o ~h 20 __ (19) o[ o[
and finally from Eq. (17):
p (x, 0) = p, + - (p, p) .
Thus, the pressure peaks p (r,o) - p,above the static value p, decay inversely proportionally with the distance. ((pg. p ) is the pressure peak in the bubble above the static value). This relation is illustrated in Figure 3. The value of the pressure amplitude above p, has dropped to one n th at a distance corre-sponding to n times the bubble's radius. For example, for a bubble radius of % cm the overpressure amplitude at a distance of 51 m has dropped to one tenth, whereas for a radius of 1' cm that happens only at a distance ofL% m. Thus, these overpressure amplitudes decay very rapidly, with the magnitude of the bubble's radius playing a not inconsiderable role: The greater R g, the g re a. . r is the range (considered absolutely) at wDich pg -
p has dropped to one nth ,
1-12
- 3. Formation of the gas bubble _
The equation of motion of the bubble's surface, Eq. (4), can also be used_ for an approximate calculati sa of bubble formation by expulsion of air through a pipe projecting into the water. The air forced out of the pipe must first overcome the inertia forces of the surrounding water in order to be able to expand. In contrast to'the problem of the oscillation of a gas bubble in which the mass of the gas enclosed by the water does not change, we must now figure on a mass supply extending over some period of time. This state of af fairs is allowed for by assuming that the newly supplied mass just compensates the pressure reduction corresponding to the expansion of the bubble, so that a constant pressure prevails during the inflation process. Although this model is certainly only a rough approximation, it can be used to get some idea of the time variation of the inflation process.
We now write Eq. (4) in the form
~ L
%Rf 2
- 6 f) C constant (21)
By integrating this equation we obtain a relation between the propagation speed of the bubble and the instantaneous radius:
24P r-Y- = 'L.b' %
R 3]
(22)
(R, = radius at the beginning of the inflation process; corre-sponds approximately to the pipe's radius).
1-13
From Eq. (22) we obtain as the limiting speed:
['g f (f j ._ Y un 3 '
3 The expansion speed for R >> R depends only on the pressure dif-a ference Ap.
k(t) is illustrated in Figure 4 for a few values of op and R .
The increase of the bubble's radius with time is shown in Figure 5 and the increase of the bubble's volume in Figures 6 and 7.
Depending on the pressure, about L%%%%1h ms are required to go from the initial radius R a = L%%1 m to a radius k%%%%%%%%%1 (R = k%1 m) . Thus, in general, the inflation process takes Place in a very short interval of time.
1-14
Pigure 1 Pressure Rg =1Tm variation in an oscillating spherical gas bubble for pg =L% kg/cm p, =1%T kg/cm '
) k
< 1 Periode =
P 70-f 1 period Pc g, g _
l l
0,6 - 1 i static value 0,4 - 1
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _i. s t a t is c h e r y 02-3 l Wert
- l i
i i I l l l l l 0 40 80 12 0 160 200 240 1
280 320 360 t/ms Bild 1:
Druckverlauf in einer schwingenden kugelf6rmigen Gasblase fGr Ro =Mm pO = Gat poo = V/A a t
hRAFTWERK UNION AG PROPRIETARY IhTORMATION Figure. 2 through 7 1-16 through 21
References
/1/ H. Lamb Hydrodynamics Dover Publications, N.Y., 1945
/2/ H. A. Johnson Boiling and two phase flow for heat transfer engineers in: University of California lecture series May 27 - 28, 1965
/3/ Weisshsupl, Schall '
Calculation model to explain the pressure oscillations in the suppression chamber after vent clearing AEG - E3 - 2208, March 1972 1-22