H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 85
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Pressures corresponding to the 25" configuration are generally lower than those corresponding to the50" configuration.(7) For a given ccnfiguration, the variationin pressure over the head is not great and inmany cases appears to be nearly constant.(8) For a given configuration, the pressureat a point on the head is approximately proportional to acceleration and can be expressedin kg/cm2 by multiplying the acceleration in gby a constant which varies from 0.22 to 0.42,depending upon the particular configuration.(9) Pressures for the quarter-full conditionsare from 5 to 10 percent higher than corresponding pressures for the half-full conditions.An exception to this is the smooth-wall tankwith spherical head in which case the pressuresare practically identical for both full conditions.(10) Pressures on the conical head are approximately 60 percent higher than the corresponding pressures on the spherical head.(11) The addition of ring frames causes asignificant decrease in head pressures, the decrease being on the order of 25 percent.(12) The average time of prototype pressurebuildup is of the order of 0.3 second and theaverage time of duration is of the order of 1.6seconds.(13) The maximum pressure in a prototypewith conical head, at 50" inclination, quarterfull, decelerated at the rate of 0.6 g, is estimated to be 0.25 kg/cm2, with peak developedin 0.13 second and lasting for 1.3 seconds.iarer Experiments a i Souihwesi Research Iniiiiii;e(ref.
10.6)The objective of this program was to assesspossible scale effects due to fluid viscosity andsurface tension, since it was found that extension of the almost perfect prototype scalingof reference 10.3 to larger prototypes orfluids less viscous than kerosene was not possible. Toward this end, a 28-centimeterdiam-366.ITHE DYNAMIC BEHAVIOR OF LIQUIDSeter tank with an ellipsoidal head was fittedon the same apparatus used in reference 10.3and a series of vertical firings was carried outwith three different test liquids.
The upperbulkhead in this tank was instrumented toindicate total force rather than pressures.Insofar as data of direct use in design areconcerned, these experiments were of littlevalue. By virtue of equation (10.32)) and thefixed stroke of the apparatus, i t was necessaryto fill the tank to 78 percent full in order tomake the impact forces begin to build up soonenough during the stroke. I t was found, evenso, that no definite maximum on total force wasreached prior to the end of the accelerationstroke. The initial portions of the forcerecords were correlated, however, with thefollo~vingresult,s:(1) The beginning of the force pulse forvertical firings is only very slightly later thanpredicted by equation (10.31).(2) The magnitudes of the initial portionsof the force time histories are nearly proportional to relative acceleration, E.(3) .In indication was found that viscousscale effects would complicate the extrapolation of data.Experiments atNASALangley (ref.
10.5)The experimental apparatus used in theseexperiments has been described in chapter 5(see figs. 5.29 and 5.30). Basically, t.he tank isattached to a drop weight by a cable passingthrough a system of pulleys. Upon release ofthe weight, the tank is accelerated upward to adesired velocity (depending upon the releaseheight of the weight), whereupon the dropweight is arrested. The tank continues upward and is decelerated by gravity and anelastic cable (which remains slack until thedrop weight is stopped). The decelerationtime history of the tank produced by thisappara tlis is shown schematically in figure 10.8.The period of time denoted by T,in figure 10.8is the experiment duration. For the tank anddrop weights used, the absolute decelerationof the tank during the experiment could beaccurately described byg=- p-n0g sin(=&)-Time historiesI( n,g )RelativeaccelerationTS 1-I!Tank accelerationFXCURElO.U.-LForce and acceleration characteristics oftest facility (ref.
10.5).where time is reckoned from to in figure 10.8.Since a particle in the tank would be in freefall after the drop weight is stopped, therelative acceleration between tank and fluid isP=nOg sin(=i)The authors (ref. 10.5) chose to correlatetheir results with the relative acceleration,n,g, shown in figure 10.8, which is the acceleration a t the time of maximum measured impactforce. Total durations of the experiment,T,;times of initial contact, t,; and maximumrelative acceleration data, no, are not avnilableand, therefore, it is not possible to constructt,he general capabilities of the apparatus as inthe previous discussion; in fact, much of thebehavior of this apparatus depends on the sizeof the tank.I n these experiments (ref.
10.5), a 22cen timeter-diameter cylindrical tank havinghemispherical ends was utilized (fig. 10.9).One hemispherical bnlkhead was attached tothe remainder of the tank through a balancesystem so that total force could be measured,and a pressure cell was also fitted in the centerof this bulkhead.
Geometric variations in thistank included Z-ring baffles and "screen"baffles as indicated in figure 10.9. The tankwas filled 28 percent by volume in all tests.With this particular tank, relative acceleration amplitudes, n d , up to about 3.5 g wereachieved and data were displayed with asso-367LIQUID IMPACT ON TANK BULKHEADS,$vmeForceTank detailsBaffle detailsQuiescentFIGURE10.9.-Model tank geometries (ref.
10.5).ciated accelerations a t time of maximum force,n,g, as low as 0.5 g. The sample force andpressure time histories shown in reference 10.5imply that the total duration of the decelerationpulse, T,, ranged between 1+ and 3 times thetime necessary for the first portions of thefluid to contact the dome.The effect of a first-mode lateral sloshexisting a t the beginning of deceleration wasinvestigated, and this was found to affectsignif?cant,ly the mode of fluid motion.
Whenthe initial fluid surface was quiescent, thefluid behaved in the same may as in reference10.3. Depending on the portion of the lateralsloshing cycle a t which the deceleration started,the fluid tends to t,ravel up one side of the tank,around the bulkhead, and down as illustratedin figure 10.10. This is virtually the samebeh8vior as shown in the experiments ofreference 10.3 when the apparatus was inclined.Maximum pressure data obtained are reproduced here as figure 10.11. Though no appreciab!e alteration iil i o t ~ idome impact forcewas found by the addition of Z-ring baffles,the "screen" baffles made a visible reduction(fig. 10.12).A scale effect study was also made by varyingthe temperature of the fluid in the tank to altervapor pressure and viscosity, and by addingn detergent to reduce surface tension.ModalFIGURE10.10.-Representation of liquid motions (ref.10.5).Acceleration,gasFIGURE10.11.-Pressure versus acceleration in smoothwall tank (ref.
10.5).10Force,kgn0 Quiescentn-01.020Acceleration,3.040gasFIGURE10.12.-Force versus acceleration with and withoutscreen baffle (ref. 10.5).368THE DYNAMICBEHAVIOR OF LIQUIDSThis program led to the following tentativeconclusions :(1) The liquid exhibits one of two flowpatterns depending upon the condition of thesurface prior to arrest: i.e., (a) if the liquidsurface is undisturbed or quiescent, it travelsin a series of streamers; (b) if the surface isoscillating in its fundamental antisymmetricmode, a portion of the liquid travels up one sideof the tank, around the dome, and down theopposite side.(2) The impact force appears to be dependentupon the relative acceleration of the tank a t thetime of impact.(3) For a given acceleration level, there appears to be no significant difference in themagnitude of the modal and quiescent impactforce.(4) The pressure in the center of the dome isabout twice as high as the value obtained bydividing the average force by the projectedarea.(5) The force level was not significantlyaltered b y the inclusion of ring baffles; however,a reduction in force of approximately 30 percentwas observed with the inclusion of the ?<-inchscreen baffles.(6) For the range covered in this investigation, no dependency of the force or pressure onthe vapor pressure, surface tension, or viscositywas observed.(3) Fluid vapor pressure, p,(4) Mass density, pThe nature of the problem is such as to make arelative deceleration of importance.
The various similitude analyses result in scaling parameters analogous t.o the Euler, Reynolds, FVeber,and cavitation numbers; a set of t.hese sufficientfor present purposes may be written asPressure coefficient = p / p /7igd,Viscosity parameter =iig&lGSurface tension parameter =pf7igd;/aCavitation index= (pa- p,) /p,iigd,(10.33)I n effect, sc.nlinp of pressures by the first relation is correct if the rem:~iningthree paranietersare satisfied or are not important. Satisfactionof tliese is not possible for every conceivablecase.
Figure 10.13 indicates the ranges ofviscosity parameter and surface tension parameter which have been attained in references10.3, 10.5, and 10.6, compared with an outsideprotot!-pt> rnnge. Tlle prototype ringe S P U I ~ Sconditions from a 2-meter-diameter tank containing kerosene to tt 9-meter-diameter tankRef. [ l a 51Ref. [ l a 61' nq d 31Range of possible prototype values410.4 SUMMARY OF LIQUID IMPACT STUDIESFrom the point of view of practical design,the results obtained in the studies reviewed inthe previous sections do not permit everyquestion to be disposed of with great confidence;however, a review of some of the divergentconclusions arrived a t and a gross comparisonof results will be attempted in the paragraphs tofollow.Fluid Scale EffectsThe previously cited references have considered the following fluid properties to be ofpossible importance :(1) Viscosity, p(2) Surface tension,1lo*10'I1II10.losloa10'IloalomSurface tension parameter,Ref.
[ l a 51Ref.IE6]Ref. [ l G lRange of possible prototype values101°II10"10'II110"10'10"1lo1'110"10"ngd;Viscosity parameter,y,fFIGURE 10.13.-ApproximateuIloaPf "d;ranges of viscosity andsurface tension parameters.~-A,-.:i\I-,- ..~*-*----- .. ... .LIQUID IMPACT O N TANK BULKHEADScontaining liquid 0-xygen and subject t o a 2-gdeceleration. While reference 10.6 indicatesthe possibility of a viscous scale effect, this isbased on analysis of initial portions of the totaldome force records and thus can serve only asa warning, not as a source of quantitative designdata. On the other hand, no scale effects werefound in the st.udy of reference 10.5; however,this conclusion is a qualified one, and it may beseen by examining the evidence presented insubstantiation in reference 10.5 that, indeed,no systematic differences in impact force wereobtained which were substantial enough to overcome the normal scatter of data.
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