JOHN H. LIENHARD Professor of Mechanical Engineering, The Breakup of Superheated Liquid Jets University of Kentucky, Lexington, Ky. Mem. ASME Observed average breakup lengths are presented for free orifice jets of superheated JAMES B. DAY water and liquid nitrogen, and subcooled water. Dimensionless, semiempirical ex­ Aerospace Engineer, USAF, pressions are developed for both flashing, and aerodynamic and/ or capillary, breakup, Wright-Patterson AFB, Ohio; and verified with data. The distribution function for breakup length is predicted for Formerly, Graduate Student, the superheated case with the help of Boltzmann statistics. University of Kentucky, Lexington, Ky.

Introduction turbance in the jet. For most cases of practical importance Re » vWe and equation (1) reduces to THE BREAKUP of liquid jets has been under fairly 1 Lb D continuous scrutiny since Rayleigh [l] first explained the mechan­ (la) ism of capill ary instability in 1878. These inquiries received con­ D vWe ~ In 2o ~iderable impetus forty years ago from attempts to improve diesel injection systems (see, e.g., [2] ), and during the 1950's, by work The term In (D/2o) depends upon the initial disturbances, and with rocket injection systems (see, e.g., [3] and [4] ). Summaries these in turn are unknown. However, for most cases of jet efflux of work done on breakup as a result of capill ary and aerodynamic this term proves to be about 12 ± 1. This corresponds with a instability are given by Huang [5] and by Grant and .Middleman variability of a factor of e± 1 in the initial disturbance. [6] . When the velocity of efflux is high, aerodynamic forces override :\lore recently, intere~t has turned toward another kind of jet capillary forces and the breakup length begins to decrease with breakup: the explosive flashing that results from the thermo­ increasing velocity. The jet now breaks up by the growth of mechanical instability of a jet of highly superheated liquid. sinuous antisymmetric waves instead of the symmetric varicose Brown and York [7] and Lienhard [8] described the spray form­ waves that distinguish capillary breakup. Miesse [12] found ing capabilities of such jets, and Lienhard and Stephenson [9] that he could correlate data for many fluids in diesel injector gave a restrictive correlation of breakup lengths as a function of nozzles operating in this range, using superheat. Flashing results in a very fine spray that is potentially useful in a wide variety of aerosol forming processes. (2) The aim of the present study is that of showing how to predict the breakup length of a given superheated jet, and its variability, This expression is restrictive in a variety of ways. It applies to under fairly general circumstances. This will require that we jets in which there is considerable turbulence, and experience determine whether or not capill ary or aerodynamic instability shows (see, e.g., [6]) that Lb decreases less rapidly with Re as will give rise to breakup before superheat does, in any situation. turbulence increases. It can even begin to rise again with Re, Therefore, we shall begin by considering what has been done at very large Re, when the jet is turbulent. Equation (2) also toward predicting the breakup of a "cold" jet. applies only for velocities above the transition point from capillary to aerodynamic breakup. Finally, it is limited to a single sur­ Jet Breakup in the Absence of Superheat rounding air density. Dumbrowski and Hooper [13] have shown In 1909, Niels Bohr [10] extended Rayleigh's analysis to include that decreasing the air pressure around a water bell' stabilizes it, viscous effects, in a prize winning paper on the evaluation of sur­ and vice versa. face tension, and Weber [11] went on to ol.Jtain the breakup If we consider that Lb depends upon the velocity, V, the liquid length, Lb, for a viscous jet in 1931. His expression was viscosity, µ, the densities of the liquid and of the surrounding air, p1 and Pa, the surface tension, rr, and D; then the Buckingham 3 pi-theorem shows that four dimensionless groups are needed to ____!:_,,_ - In !!.__ ( 1 (1) DVWe- 2o + VWe)Re characterize the process. Thus the general form of equation (2) would be where Dis the diameter of the jet, and V\Te and Re are the Weber Lb and Reynolds numbers. The symbol, o, denotes the initial dis- D- = F(We, Re,p./ p1 ) (3)

1 Numbers in brackets designate References at end of paper. In the present study, we shall assume that Lb is always ap­ Contributed by the Fluids Engineering Division and presented at proximately proportional to vWe, as both equations (la) and (2) the ,vinter Annual Meeting, Los Angeles, Calif., November 16- 20, 1969, of THE AMERICAN SocIETY OF MECHANICAL ENGINEERS. 2 A "water bell" is the spreading liquid sheet leaving the point of ?-Ianuscript received at ASME Headquarters, July 31, 1969. Paper collision of two opposing coaxial jets. Its aerodynamic behavior No. 69-WA/FE-19. was shown by Huang [5] to be strongly analogous to that of a jet.

Journal of Basic Engineering SEPTEMBER 1 970 / 515 ndicate it to be, and we shall work with only one value of p./ p1. The function, F, was ignored in Dergarabedian's original formu­ This will be p 0 / p1 = 0.0012, which corresponds with any liquid lation, but added later by Forster and Zuber [l.'l] and Plesset whose specific gravity is close to unity, discharging into a stan­ and Zwick [16) to account for the role of heat conduction in caus­ dard atmosphere. We shall therefore attempt to correlate data ing the bubble to grow. for sharp-edged orifices using These latter studies showed that after the bubble grows a n order of magnitude beyond its unstable equilibrium radius, the Lb = F(He) (4) inertia terms, rF + (3/2)f2, cease to be important. The asymp­ DvWe totic solution of the remaining equation applies through almost the over the entire range of efflux conditions. entire growth of the bubble. This solution was given in [1,i] in Once data have been obtained to form this correlation, only a terms of the specific heal, cp, the latent heat, h10, the superheat, part of the breakup problem will have been completed. This !:iT, the saturated liquid and vapor densities, p1 and Pu, and the breakup length will only apply if the jet does not first break up as thermal diffusivity, a, as a result of flashing. Our second objective will then be to predict cv!:iT) ( P1 ) _ 1- the flashing breakup length and its variability. R = ( -- - v1rat (H) h1 u Pu Jet Breakup Under the Influence of Superheat Photographic evidence indicates that a jet shatters when a bubble The Delay Time. We shall now redevelop some ideas from refer­ grows to about R = D. Therefore, we can approximate l ,12 a, ence [9) in such a way as to provide n.ecessary background and 2 (U ) facilitate the subsequent development. The delay time, td, be­ l,12 = :~ c:~uT)' (;J tween the efflux of a jet and ib breakup, will be used here instead of the breakup length because most of the prior work has been The longer component of t" is usually td1-the idle or dwell done in terms of time. The change is unimportant since td = time. 3 To characterize this, let us consider the solution of equa­ Lb / V. The delay time is composed of two components: an idle tion (,)) for small r. References [15) and [16] show that the func­ lime, td,, during which an unstable bubble nucleus in the fluid tion, F, can be neglected in this range, and reference [14] show. "dwells" before it begins rapid growth, and a time, td,, for rapid that equation (;>), with the initial conditions, r(0) = 1 + E and growth of the bubble up to the size at which it will fracture the r(O) = 0, admits the solution jet. r ( l 2 l )- '/, The calculation of both these components of the delay time T = - + - - - dr (W) 3 will make use of Dergarabedian', [14] bubble growth equation: Ji+, 3r 3 r 3 r - 1 This result is plotted in Fig. 1 for an initial perturbation, E, 2 ,.;: + - f - -- + /?(physical properties, time) = 0 (.i ) equal to 0.01. Here we see that, depending upon t-he magnitude 2 r of E, the bubble might grow very slowly indeed for a long time, where r is a dimensionless form of the bubble radius, R, before it picks up speed. r= R/ Ro (6) Reference [9] provided ihe basis upon which we wish to build a correlation for td,- There it was argued that the discharging and the independent time variable, T, is liquid might contain "weak spots" that could be triggered into 3 T = t [ (p, - Pamb) ] lj, (7) 3 This is especially true in water. For this reason, td, was simply 2 4p1a neglected in reference [9 ]. ----Nomenclature------

A cro0 , sectional area of jet, 1\ = 1rD'/ 4 time, or a 11 y ra11dom variable in the context of equation cP speeific heat at constant pressure (24) D diameter of jet- fully contracted l, characteri.,ti<" dwell time of a single bubble, tc < l,11 F any u11 specified fun ct ion l,1 delay time, I,, = t,,, + l,12 Fo Fourier number, Fo = at/ D' t,,, dwell time required for a bubble to commence rapid f(x) distribution fun<"tion of x growth after jet efflux 2 Ml potential barrier to bubble 11ucleation, !:i(J ¾1raU0 t,12 time required for a bubble which has beg1111 rapid g, "degeneracy" of a random event growth lo shatter the jet h1" latent heat of vaporization 1,11 average cl well ti me Jo, J, Bessel fu11ctions of the first kind of order zero and one t, a particular value of the random variable, t 1,. breakup length of a jet, Lb - Vt" t,,,,13 the root mean /)th moment of the di;tribution of evenh llf n the roots of Jo in t, defined by equat ion (26 ). 111 unspecified exponent, equatio 11 (27) We Weber number, We p1 V'D/ a X total number of breakup events y radial coordinate, y = 0 on jet centerline A, number of breakup events between t, _, and l, a thermal diffusivity Pr Prandtl number, Pr= µ / p1a 1J number specifying a moment of a distribution. 8ee PamU ambient pressure equation (26) p,, vapor pressure at 'J' = 'l'o o initial disturbance in a jet interface !:ip P,, - Pamb, the exte11 t of superheat expres,;ed in terms of E = initial displacement of r from unity pressure µ liquid viscosity H bubble radius Pa density of air surrounding jet

He p1VD/ µ p1 density of saturated liquid, p1 = density of jet at any Ro equilibrium bubble nudeu-; radiu-;, Ho temperature r R / Ro p" density of saturated vapor r, perturbation radius, r, - r - 1 a surface tension bet ween a saturated liquid and its vapor T temperature T dimensionle.;s time. See equation (7) Tc.1. temperature 011 the ax i,; of a jet Tc T fort = t, To temperature of superheated jet at efflux dimensionle.;.-; dwell lime. See equation (16)

!:iT the superheat, !:i'J' == 7'0 - T,., if; dimensionless superheat. See equation (17)

516 / SEPTEMBER 1 970 Transactions of the AS ME Fo• .0205 :~v,~~10 --- - - .91 r Solution of Equation (I) - >--' .,_o' 0 OJ• . 8 - ..::.__ !_ __ ··7 ~ 2,0r- ·-- -2 0: - ___,_____ L " t sg First Order Approximation to the ~ .6 r Solution of Equation (I) - / i'j 1-- ] 0 r:1+£ coshT '/. 0: ; ~ c ! 4~---'----'-- -'--~-----'-- 0 1.5 ~ .0 C12 03 04 06 ·.;; 2 C Founer Number, Fo= at/0 E / 2 0 [ // {r=(l+,)+n/2 Fig. 2 Variation of jet centerline temperature with Fourier number I / ,- (accurate for r' r , -rh=O)= I+, 1/, ,- close to I+,) , ~,___ ./- _ -I 0 2 3 4 5 6 7 (16) Dimensionless Time, T

Fig. 1 Early growth of a vapor bubble when r(r = 0) = 1.01 and a. dimensionless snperheat, or dimensionless jet diameter,

(17) unstable nuclei by whatever "noise" might exist in the environ­ Using these expressions to eliminate td, and tip from equation ment. The most effective source of noise is, in turn, that gener­ ated by other flashing bubbles. The energy required to trigger (l.''i), and noting that A = f D', we obtain a nucleus in the liquid is Frenkel's [17] "potential barrier" to nucleation, tiG. The potential barrier is the free energy of a iJ>y; = const (18) bubble with respect to the surrounding liquid, and Frenkel fo und Once the constant in equation (18) has been determined ex­ 2 tiG = ½1rer Ro ( 11) perimentally, t he expression should predict the mean breakup length for any superheated li quid. where Ro is the radius of an unstable equilihrinm nnclens. It is One tacit assumption has been made throughout these con­ given by the force balance on a bubble as siderations, namely, that i:;;; will be sufficiently short that cooling 2er of the jet wi ll be unimportant. Actuall y, the interface of the jet Ro= (12) will assume the saturation temperature, T sa,, corresponding with Pv - Pamb Pamb as soon as it is formed. The validity of the assumption can where Pv is the vapor pressu1·e corresponding with the tempera­ be checked by solving t he heat conduction equation. If the tme of the superheated liquid, and Pamb is the ambient pressure. Peclet number, VD/ 2a, is high (10 or more) axial conduction But, once a nncleus is formed, it must survive existing back­ shonld be negligible and t he problem becomes gronnd noise for a characteristic time, t" before it has sufficient size for its grnwth to " run away." From equation (7), T(D/ 2, t) = T,at T(y, 0) = To - T ,,., 4p er' ] '/, (19) t = T (€) __ / ____ (13) c c [ (Pv - Pamb)" oT I . o Oy y - 0 Fig. l indicates that for an initial distmbance, f = 0.01, r , would he about 2 or 3. where y is the radius of the jet and To is the temperatme of the Withont reproducing the formulation of the probabili ty argu­ emerging superheated liquid. The solution (see, e,g., [18]) is ment in reference [!J], we can indicate how it went: The probabil­ ity that there is a nncleus that will survive until rapid growth T-T,at ;., J,(Mn2y/ D ) ( 'F) = 2 L.., ----- exp -4Mn o (20) begins is proportional to jet area, A, inverne 6G, and inverse t,. To - T,at n=l MnJ,(Mn) The delay time in turn shonld be invernely proportional to the probability of sm,ival. Thus we can write for t he average, i:;;;, where the Mn's are roots of the Bessel function of the first kind of of l,11: zeroth order, J 0, and the Fourier number, Fo, is

(14) Fo=at/ D' (21) or, after suhstitntion of equations (11), (12 ), and (13), Fig. 2 shows the relation between the centerline temperature, T, 1., and Fo, computed from equation (20). One point on this [;;; ~ 1/ A(pv - Pamb)'/, (15) eurve is of great interest to us: Equation (J.i) shows that 6p is proportional to t-'h, but the Clau~ius-Clapeyron equation shows This dimensional result was verified with data obtained in that, to a first approximation, tip is proportional to T - T,at• flashing water jets. The data exhibited wide variability and Therefore, if T - T,., decreases more rapidly than t-'h or emphasized that equation (Lj) gives the average of what is, in Fo-'h, the delay time will be increasing faster than time is pass­ actuality, a broad distribution of td,, We shall therefore show: ing. This point is reached at Fo = 0.0205 (see Fig. 2). first, how to generali ze equation (10) into an ex pression that will Thus cooling will protect the jet fr-0m flashing if the fo llowing work for liquids other than water; and second, how to predict the criterion is met: distribution of td, about this average. "(;, ~ 0.020,j D'/a (22) Dwell Time Correlation. The preceding discussion shows that r;;; should depend upon p1, D, er, and (Pv - Pamb), These comprise Nondimensionali zation of this criterion with the help of equations five variables in three dimensions-time, length, and force. The ( 16) and (17) gives Buckingham Pi-Theorem shows that two dimensionless groups are needed to fully characterize the phenomenon. We shall _ Pr Re ,;, 0.020.J .../We (23) choose these to give a dimensionless dwell time, : ~ y;

Journal of Basic Engineering SEPTEMBER 1970 I 517 ------~----~------•-,

125psi Prmure ~ ...-----sofl!y Valve Safety GOQt - -f/ Valve Pressure Thermometer Control - -- Ga,ge Vatve ··--ri , Bleed 1 Valve 1--7 I t 11 _,.- Filler yll I I I,- PluQ I ' 'I 5 5 5 3· dio. Quieting Section \ " 0-Rino r----~1==;~ Support I" ThermOCOJp le Arm Leads ' (Five Places) ~/\ ,02-·-~ 450 Insulation

Orifice

Metering Orifice y

Fig. 3 Schematic diagr~m of water loop VIZZ I e)zzz 1 ; 1 Pl ug ' , ' r;s=,' Pos, hon,nq ! 1 B Brocket J:l- orifice where Pr is Prandtl number, µa/ p1. Equation (23) is a conservative criterion because considerable - -Orific e cooling has occurred before it is reached. It has been shown by Plug Day [19] that the superheat energy has been reduced by about 32.percent, even though 7' - 7',at has dropped only 9 percent, at Fig. 4 Schematic diagram of blowdown tank this point. However, since the breakup length is a random vari­ able, equation (23) gives a probable limit and not an absolute li mit. Thus cooling "hedges" against any such random occur­ The substitution of r = 1 + r,, r, « 1, in equatio11 (ii ) with F rences of flashing as might occur in apparent violation of equation = 0, gives (23). The Variability of the Dwell Time. It was shown by Lienhard and 1\ - r1 = O; r 1(0) = E, f 1(0) = 0 (28 )

Meyer [20] that the generalized gamma distribution function, 2 after the elimination of terms on the order of r1 or less. The solu­ which contains most of the common distribution functions as tion of equation (28) is r, = E cosh r, so special cases, can be obtained by the methods of statistical mechanics. The function is r = 1 + E cosh r ; r - 1 « 1 (29) For very small r, this can be approximated as 1,,,,13.f(t) (3 ( m )m//3] ( t )m-1 [ I'(m/ (3) (3 t,,,.13 r = (1 + E) + Er2/ 2; r - 1 « 1

T 1 (30) 1 « X exp [- _111,_ ( - )/3] (24) (3 l,m /3 These approximations are included in Fig. 1 to show how they compare with equation (10). where the constants are explainable in terms of the con straints Equation (30) shows that as time passes an initial perturba­ on the distribution. These are: (a) conservation of the events tion, E, will slowly grow to a new effective perturbation E(l + or elements that are being distributed- the dwell times in this r 2/ 2). This suggests that the likelihood of survival of a nucleus 2 case. If there are N, of event t,, N, of event t2, and N; of event t,, will generally increase as r , so we shall assume t.hen the total number of events, N, is given as

L N,=N (25 ) Thus the dislribulion function for td, should be equation (24) n=l with (3 = 1 and m = 3, or (b) A (3th moment of the distribution is known. If (3 = 1, then (31) t, .. , is a simple mean, l. If (3 = 2, then t,,n, is the root-mean­ square moment, t,.,., etc. Thus In the course of this discussion we have offered five predictive expressions, all of which require experimental verification or com­ L N,t,/3 = ll,,n13) 13N (26) pletion. These are: the correlation equation (4), equation (9 ) n=l which anticipated that a jet will shatter when R ::,,, D, equation (18) which requires an experimental constant, the criterion (23), (c) The degeneracy, g,, or number of ways in which the event and equation (31 ). In the next section we shall report data can occur at the ith level, is of the form which will serve these ends and which will overlap the aerody­ namic and fl ashing breakup regimes. Yi r-.., t/n-1 (27)

where mis a constant. Neither m nor (3 need be integral but both must be positive. Experiments Equation (24) is the basis upon which it is possible to predict Two kinds of apparatus have been developed for this study. the variability of the dwell time, tdi- The appropriate moment of One is the hot water loop shown in Fig. 3, the other is the liquid the distribution is the simple mean given by equation (18); thus nitrogen blowdown apparatus shown in Fig. 4. These apparatus f3 = 1. The specification of m requires that we first consider the and our experiments are fully described in reference [19] and very early growth of a bubble. We can find this by solving for a we shall only briefly describe the experiments here. small perturbation, r,(r), around the equilibrium radius, r = 1. The hot water loop was used to deliver both cold and super-

518 / SEPTEMBER 1970 Transactions of the AS ME r

t2in,dio Orifice ~ in. dia Orifice 1,0 I T·Tsot= 93°F T-T50t= 90°F t;;"1=. 788 msec \J1 =.816 present data I reference[9]

o~7 '.0 ',I:., _J ?6 .c 3 O' 5 i; zlz _J C. 4 ~ m~ "' 3 o Orifice dio.= l/32in. 3 ~"' C • Orifice dio.= l/16in. 0 ·;;; • Orifice dio,= 3/32in. C E 'v Orifice d,a.= I/Sin. 0 2 I Variability of Doto

.,"' Equation (3 2) c .5 0 -.,~ I E 0 •92~------,:3.--~4---/,5---,5!,---!7!,--!.8-J;9,-+.I0~---136 0 Reynolds Number, Re X 10-4 imensionless Dwell Time, tdl / fct1 Fig, 5 Variation of dimensionless cold-water breakup length with Reyn- Fig. 6 Dwell time histograms for water. The symbols key to Fig, 7. olds number

1 3 1 L heated water to / ,., ' / , 6, / ,,, and / 8-in-dia sharp-edged orifices. ~ = 2.75 X 10 10 Re-2 (32) The breakup leJ,gths of cold jets, and their variability, were DvWe measured with the help of a strobe light, and the character of m the range of aerodynamic breakup. There is a transition breakup was investigated wit.h still photographs. region in the neighborhood of Re = 48,000 (for Pa l p 0.0012); The superheated jets exhibited far greater variability in Lb, 1 = and below He = 35,000 equation (la) is sat.isfied. In the present and far greater violence in breakup. They were photographed case in (D / 2o) J 1..5 ± .5 which is typical of such data. with a Hycam motion picture camera, from a distance, at about = Equation (32) reveals a far stronger decrease of Lb with Re 8000 frames/sec. The jet velocity, V, was computed on t.he basis than did previous data for inject.ors and turbulent tube discharge. of the upst.ream gage pressure and a velocit.y coefficient. Care The Dwell Time for Superheated Jets. A total of eleven motion pic­ was taken to insure that the flow was substantially subcooled ture records with useful dwell time data are available to us. Six upstream during the superheated jet experiments. movies of water jets were made and interpret.ed by Stephenson The liquid nitrogen blowdown tests could only be done with [23], and reported in [9]. One was made by Lienhard using superheated liquid since t.here was no way to bring the nitrogen Stephenson's water loop which resembled our own and is pre­ below its saturation t.emperature in the pressure vessel. Both sented here as part of the present

Journal of Basic Engineering SEPTEMBER 1970 I 519 1,4r----.------~------~------5 Experimental Doto for Orifices

} H,gh data ,esul1ing from standing b,eokup 8 :~!:~ ~:~ \~: ~~ }Lienhard 8 Stephenson [ 9 J • Water 1/16 in dio 1.2 Orifice Superheat Average ~ Waler 1/8 in die dio. 6 T °F delay Nitrogen 1/16 in dio Do ir,. f;:;i msec 0 • Infinite Superheat L im1t 0 3/32 75 2.47 ~ 3/32 83 2.23 Note: Superheot,b.T °F, is given f:., 3/32 90 ,82 inside symbols I ==: :!: 40¾ of best fit l 7 5/32 62 1.21 t: .a 3/32 83 ,81 0 •• 1/16 93 1,65 D ·c 1/16 87 1,26 ."0 .6 • 1/16 89 1,78 8 0 • -~ Open symbols ore data from reference[9] E ~ • Closed symbols ore present dolo o 8 .• 0 IOF':::.L--!c-2-_L__-14---'---6!---'---'-8-_j__lOL__j__-J,12--'---·__J14'-_J C ,4 E o Inverse Dimensionless Superheat, 1/'f'=(a- //J.pD)x K)5 • Equation (31) Fig . 8 Variation of dwell time with inverse superheat

1.0 4.(' 10,.---,-----,--,--,rr-..rr---~-~~--,--,~~~- Dimensionless Dwell Time, td ;~ 1 1 8 • Waler l/16"dia 0 Water 1/32'' dio Fig. 7 Comparison of predicted and observed distributions of dwell 6 A Water l/8"dio times D Woter 5/32"dio r ~I~ 0 Nitrogen 1/16 11 dio u8 "'0 4 ~ q created by the preceding bubble. i':>tand i11g breakup i.-; very near 1 ~ 0 the mean ld1 when it occurs. Equation (31) lies in the middle of the histogram points. The points scatter broadly because there are only 100 to 300 events on - dwell tim e a single 100 ft reel of film exposed at 8000 frames per sec, in t he cases reported. Presumably the T;;; data would scatter less if plenty of event::; were available at any condition. Eqnat,ion (31) i.; therefore borne out in our results. The eleven dwell times have been nondimen::;ionalir,ed in ac­ cordance with equations (16) and (17), and plotted in Fig. 8 011 Fig. 9 Variation of dimensionless dwell time and dimensionless cooling lime with superheat ~- ~ 1 cf> versus y,- 1 coordinates. 01nce '±' must approach 7,ero as - ap- 1/1 proaches zero, the origin i::; also a Iegit.imate point. The least­ squares-fil, straight. line through these points is given by in the fignre into equation (32) yields Lb = 37.6 in.-mnch longer than breakup actually required in this ca::;e. What then actually cf>i/, = 2.12 X 10 13 (lSa) caused the jet to break up? Possibly moisture conden::;ed onto The conelation coefficient for these data is 0.762 which indicates the lip of the orifice, roughening it. We believe that it, is more acceptable correlation. plausible that some bubble growth was taking place, even though The data are repeated on In cf> ver::;us In i/, coordinates and the no individual bubbles can be clearly identified. The physical prote<'tive cooling limit as given by equation (23) is also included. properties of nitrogen are stwh that equation (8) predicts a much Here we see that the cooling time equal::; the dwell time at a time slower bubble growth rate than for waler. Therefore, bubbles that exceeds the observed dwell time in every in::;lance. Thus the failed to perforate the nitrogen jets as they did the water jets. cooling criteri on is not violated in any instance. Eqnatiun (23) The wide uncertainty on the one nitrngen data point in Fig. 8 explain, why B1'ow11 and York suggested that breakup would not stems from the fact that the first appearance of bubbles had to occm before a certain fai rl y high superheat was obtained, and be identified during this kind of "shredding" breakup- a situa­ wh;1· Lienhard and Stephenson found no breaknp in water jets tion that was unavoidable with our apparal tL~. The average belo\\' :1.bout 260 F. In these cases, 'l' - T ,., was low enough flashing breakup length was only about 1.6 in thi,; case. Fig that l,1, appro:1.ehed this criterion. JO(d) shows a liy11id nitrogen jet al a higher superheat than in Some Qualitative Results of the Photographic Observations. Fig. I 0 Fig. lO(c). Here there are obvious examples of flashing which show, fom typical photographs from the present study and one acts lo augment breakup. Fig. I0(e) is a pictme of the flashing from refcrenre [9]. I II one case a rectangular 1-in. marker is of a highly superheated water jet in the complete absence of either visible jnst below the jet. These photographs illustrate several capillary or aerodynamic breakup (from reference [9] ). The of the phe11omena \\·e have been discussing. delay time here is shorter, by virtue of both higher superheat and Fig. JO (a) shows a condition of capillary breaknp i11 a water larger cross-sectional area, than in Fig. lO(b ). Several simulta- jet. Fig. l 0(/J) :,;hows an example oi a very symmetrical bubble in 11eous flashing events are evident in Fig. lO(d) while only one the process of flashing in flashing in a waler jet. This is abonl as occurs in Fig. 10(/J ). largr as bnbbles ever grew during flashing and it is aboul fom times the diameter of the jet. With a11y asymmetry in the loca­ tio11 of the bubble, bmsting would occur at a smaller diameter Summary than this. The typical bubble would grnw to about twice the If flashing does nut occur fir::;l, in a jet of superheated liquid jet diameter, or to R = D. Thus the assumption that bursting leaving a sharp-edged orifice, the jet will either break up as a result occnrs at was a reasonable one to use in equation (9). U = D of aerodynamic instability, in which c:1.se Lb will be Fig. JO (r·) shows what appears to be aerodynamic breaknp in 10 a liquid nitrogen jet. However, substitution of the parameters L0 = 2.7,5 X 10 D -VWe/ Re'; Re > 48,00U (32a)

520 / S EPTEMBER 1970 Transactions of the AS ME r

""' • Pr Re .,.,;, "' > 0.020,, _ ;- y, (23) - vWe

Flashing will orcur in an average distance given by

(34) -- where[,; is obtaina1i>le from equation (18a), and td, is obtainable from equation (9). Actually Lb is a random variable. The extent (a) 0.051-in-dia cold water jet; V 24 fps; Re 3820; We 845 of its variability is specified by the variability of t,ll and this in turn is given by equation (31 ). In the nitrogen jets la, > la,; thus early nucleation was followed by slow bubble growth. This combination (possibly helped by other systematic complications) made breakup very hard to de­ scribe in this case.

References 1 Lord Rayleigh, "On the Instability of Jets," Proc. London Math. Soc., \' ol. 10, 1878, p. 7. (b) 0.051-in-dia superheated water jet; V = 108 fps; :i.r = 70 F; Re = 2 Schweitzer, P. H., "Mechanism of Disintegration of Liquid 170,000; We = 2:l,300 Jets," Jom. App. Phys., Vol. 8, 1937, pp. 513- 521. 3 Heidmann, ;\1. F ., Priem, R. J., and Humphrey, J. C., "A Study of Sprays Formed by Two Impinging Jets," NACA TN 3835, 1959. 4 Ingebo, R. D., "Drop-size Distributions for Impinging Jet Breakup in Airstreams Simulating the Velocity Conditions in Hocket Combustors," NACA TN 4222, 1958. 5 Huang, C. P., "Dynamics of Free Axisymmetric Liquid Sheets," College of Engineering Bulletin 306, Washington State University, Pullman, Aug. 1967. 6 Grant, R. P., and Middleman, S., "Newtonian Jet Stability," AIChE Jour. , Vol. 12, 1966, pp. 669- 678. 7 Brown, H., and York ..T. L., "Sprays Formed by Flashing (c) 0.051-in-dia superheated liquid nitrogen jet; t.T 6 F; V 29.4 Liquid Jets," AIChE Jour., Vol. 8, 1962, p. 149. fps; Re = 63,300; We = 12,000 8 Lienhard, J. H., "An Influence of Superheat Upon the Spray Configuration of Superheated Liquid Jets," JounNAL OF BASIC ENGi• NEERING, TR.,Ns. ASME, Vol. 88, Series D, 1966, pp. 685-687. 9 Lienhard, J. H ., and Stephenson, J. :vi:., "Temperature and Scale Effects Upon Cavitation and Flashing in Free and Submerged Jets," JOURNAL OF BASIC ENGINEEHTNG, TRANS. ASME, Vol. 88, Series D, 1966, pp. 525-532. · 10 Bohr, N., "Determination of the Surface-Tension of Water by Method of Jet Vibration," Phil. Trans. Roy. Soc. London, Series A, Vol. 209, 1909, p. 281. 11 Weber, C., "Zurn Zerfa!l eines Flussigkeitsstrahles," Z eit. Ano. Jfath. Mech. (ZAMM), Vol. 2, 1931 , pp. 136- 154. 12 Miesse, C. C., "Correlation of Experimental Data on the Dis­ (d) 0.051-in-dia superheated liquid nitrogen jet; t.T 10 F; V = 37,7 fps; Re= 81,900; We= 21,200 integration of Liquid Jets," Ind. and Enor. Chem., Vol. 47, 1955, pp. 1690-1701. 13 Dumbrowski, N., and Hooper, P. C., "The Effect of Ambient Density on Drop Formation in Sprays," Chem. Enar. Sci., Vol. 17, 1962, pp. 291- 305. 14 Dergarabedian, P., "The Rate of Growth of Vapor Bubbles in Superheated Water," Journal of Applied Mechanics, Vol. 20, TRANS. ASME, Vol. 75, 1953, p. 537. 15 Forster, H.K., and Zuber, N., "The Growth of a Vapor Bubble in a Superheated Liquid," Jour. Applied Physics, Vol. 25, 1954, p. 474. 16 Plesset, M. S., and Zwick, S. A., "The Growth of Vapor Bub­ bles in Superheated Liquids," Jour. Applied Physics, Vol. 25, 1954, p. (e) 0.079-in-dia superheated water jet [9]; t.T = 79 F; V = 106 fps; 493. Re = 272,000; We = 45,000 17 Frenkel, J., Kinetic Theory of Liquids, Dover Publications, New York, N. Y., 1955, p. 374. Fig. 10 Examples of jet breakup under a variety of conditions 18 Carslaw, H. S., and Jaeger, J.C., Conduction of Heat in Solids, 2nd ed., Oxford University Press, 1959, Section 7 .6. 19 Day, J. B., "Combined Effects of Superheat and Dynamical Instability on the Breakup of Liquid Jets," MS thesis, University of or it will break up as a result of c11,pillary instability 111 which Kentucky, June 1969. case 20 Lienhard, J. H., and Meyer P. L., "A Physical Basis for the Generalized Gamma Distribution," Quar. App. Math., Vol. 25, 1967, pp. 330-334. . Re< 411,000 (33) 21 Swanson, G. T., "Unsteady Flow from a Pressurized Vessel," MS thesis, Washington State University, Pullman, 1967. with relatively little variability in either case. These results are 22 Huang, C. P., "The Behavior of Liquid Sheets Formed by the restricted to Pa l p1 = 0.0012 and they will give slightly high values Collison of Jets," College of Engineering Bulletin No. 291, Washing­ in the transition range 3.5,000 < Re < 60,000. ton State University, 1965. 23 Stephenson, M. J., "A Study of Cavitating and Flashing Flashing will not occur at all if it ha:s not occurred after a dis­ Flows," Washington State University, Institute of Technology Bulle­ tance of 0.020.5 VD'/ 01., or if tin No. 290, 1965.

Journal of Basic Engineering SEPTEMBER 1 970 / 521 DISCUSSION V. E. Schrock 6 The authors have provided a thorough and interesting study on the mechanisms of liquid jet instability and their regimes of influence. This is an important contribution that will be of great practical value in safety oriented studies of both thermal and fa~t nuclear reactorn. (a) V = 78.30 fps, Re= 107,200, We = 39,000 In fast reactor studies it will be important to know the mecha­ nism~ of droplet formation and the distribution of their sizes when an over power transient results in fuel melting and cladding failure. Although the geometry of the coolant channels and fuel elements in these systems precludes the possibility of long jets, the concepts presented by the authorn might be useful in such studies. Typically, in this application, the jet fluid would be molten U02 and the ambient fluid would be molten sodium for which the density ratio is two orders of magnitude larger than it wa.s in the authors' experiments. (The j/)t would most probably be subcooled.) A useful addition to this paper would be the (b) V = 95.34 fps, Re = 130,600, We = 57,900 authors' thoughts on the importanee of the density ratio. As a more general comment it is not dear to this reader whether liquid droplets produced by flashing or aerodynamic instability are smaller on the average. Do the au thorn' photographs reveal a difference? It would abo be interesting for comparison if the authors could include a photograph of aerodynamic breakup of a cold·water jet.

Authors' Closure We appreciate Prof. Schrock's interest in this work and his (c) V = 123.1 fps, Re= 168,900, We= 96,450 suggestion that it might be applicable in reactor safety work. He Fig. 11 Aerodynamic breokup of a 0.197 in-dia cold water jet has expressed specific interests in the effect of the density ratio and in the mechanics of droplet format,ion. The influence of the density ratio has not, to our knowledge, a jet, and to answer Prof. Schrock's req11est, we include in Fig. 11 been studied extensively. The best information that, we are a series of three pictures of the aerodynamic breakup of a large jet of cold water taken a few feet from the orifice. They show aware of is in the results of Dumbrowski and Hooper [13]. They showed that reducing the densitv of the air around a water bell that the Hayleigh waves on the surface are i1tcrea,ingly amplified below atmospheric conditions, s·uppresses aerodynamic breaku~ by aerodynamic forces as the velocity increases. We have not as yet mea.sured droplet sizes; however a few very effectively. Unfortunately the cylindrical jet differn superficial observations can be made. The droplets res11ll i ng sufficiently from a water bell as to make quantitative comparisons from simple Rayleigh breakup are on the order of the jet diame­ impossible. No doubt the de1mity ratio is a very significant ter. Those resulting from aerodynamic breakup are increa~­ :variable, and its signifiance probably becomes greater as Re(p1/ p,) rncreases. ingly fine as the velocity is increased. In our experiments the flashing of superheated jets produced much finer droplets than To amplify Oil the basic complexity of aerodynamic breakup of aerodynamic breakup did. But we cannot argue that. this would 6 Professor, Department of Nuclear Engineering, University of still be the case at jet velocities beyond the range of our experi­ California, Berkeley, Calif. ments.

Reprinted from the September 1970 Journal of Basic Engineering

522 / SEPTEMBER 1 9 7 0 Transactions of the AS ME