H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 94
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These trends are infact observed experimentally.Let us nest ask if R drop prefers t o be floatingfree of the surface or if it has lower potent,ialenergy in a "mall-bound" state. A bit ofgeometrical calculation (fig. 11.12) lends to arelationship between the capillary areas forwall-bound and free drops (or bubbles) ofidentical volume on a plane ~ u r f n c e . ~The capillary area of each I\-ill be proportionalto D2, and the total capillary area is thereforei ( 2 - 3 cos @+codB)]113(drops)113Since V is fixed, we conclude that the capillaryarea is of the formThe capillary area mill therefore be least whenthe drops are merged into one.
The generalproof of this is a classic problem in the calculusof variations, where one finds that the shape ofsmallest surface area for given volume is asingle sphere. We conclude therefore that thedrops will tend to coalesce to form a singlelarge sphere.By similar arguments, i t is easily demonstrated that small free bubbles will tend to join,~ d = [ ~ ( 2 + 3 c o s 8 - c o s 3 ~ ) ] (bubbles)Ac treeThese ratio* are less than 1 for all 8. Con~ ~ q l l ~ n t we~ l y conclude:that the wall-boundconfigurations are more stable. If a freefloating drop (or bubble) collides with a wallnot too violently, it \ i l l stick. I n zero g, dropsor bubbles prefer to be wall bound.Similar geometrical calculations have beenmade for the case of liquid in a spherical tank,and the results are summarized in figure5 F o r bubbles A ,= Al+ cos OA N W , where -4 .VH- is thenonwetted area.398THE DYNAMIC BEHAVIOR OF LIQUIDSNon -wettinq systemsWettins systemsV00.20.40.6Liquid volume fraction*Contact angle81.0High g(b)Liquid volume fractionFIGURE11.13.-Comparison of capillary energies of possible configurations in a spherical tank.11.13(a).
The ratio of the capillary areas forthe wall-bound and free-floating configurationsare plotted versus the liquid volume percentagefor several wetting and nonwetting coiltactangles. The conclusio~~is as stated previously,that in every case both the liquid and gasbubbles are wall bound in the preferred configuration. The ratios of the high-g capillaryareas (assuming a flat interface) to the wall-bound zero-g capillary areas are shown- infigure 11.13(b). Note that the capillary areawould indeed decrease as the fluid passes froman initial high-g state to a zero-g state afterthe removal of the body force field.
An exception is observed for the particular situatioxlwhere the liquid meets the wall a t the contactangle, in which case no motion would ensuefollowing removal of the body force.399LIQUID PROPELLANT BEHAVIOR AT LOW .4ND ZERO GThese predictions have been qualitativel~observed in free-fall experimerrts. However.with wetting liquids in large quantity, the ullagebubble sometimes becomes embedded. Thisis attributed to inertial effects and it is feltthat after an extended period of very small g,the bubble would drift toward the wall andeventually become \\-all bound.In the foregoing discussions, n-e tacitly assumed that the interface was a spherical sector.In a zero-g hydrostatic system, the pressureson either side of the interface will be uniform,and equation (11.13) then requires that thesum of the reciprocals of the principal radiiUoa2Q40.6asRatlo tlquld -wtume-to-container -volumeof curvature be constant.
Thus, an axisymmetric meniscus in zero g will take the shapeof a spherical sector, provided it extends tor=O; an annular me~liscusdoes not meet thisrequirement. If the Bond number is not toogrent, it is a fair approximation that the meniscus is spherical, and this idealization con beused in estimating the confiylnation of manycapillary systems. Clodfelter (ref. 11.17) usedthe spherical sector model to determine theposition of the meniscus ill a spherical tankunder zero g, and design graphs adapted fromhis report are sho\vn in figure 11.14.LOoa2a40.6asRatio liquid-volume to - container volume-FIGURE11.14.-Meniscus configuration in a sp'herical tank at zero g.-Lo400THE DYNAMIC BEHAVIOR OF LIQUIDSParallel Meniscus Systems in Zero gParallel meniscus systems are of considerableinterest in a variety of space systems. Theirpreferred zero-g configurations may be determined using the minimum capillilry area ideas,and sometimes alternatively by force balanceconsiderations.
For example, consider the twomenisci formed by two concentric tubes, asshown in figure 11.15(a). This parallel meniscus system has applicatio~i11nder certain conditions in large booster fllel tanks as a propellantpositioning device, for, if the radii are properlychosen, the liquid will flow into the inner tubens the vehicle enters zero g. We can make itsimple analysis for the preferred configuratio~~.Neglecting the thickness of the inner tube, andassuming perfect \vetting, the following pressuredifferences can bc dcr*ired directly from thedifferentinl equations for the shapes of the innerttnd outer menisci, or from simple force balances.Now, if Pa is less than Pa, t,he liquid will flouto t.he inner tube. Henceliquid moves up the annulusliquid remains in any positionIIliquid Inoves up the center tube)(11.56)TJius, if tlie i~lllerrtldius is snlaller than halfof the outer radius, the liquid will flow to thecenter.
This prediction was substantiated inthe experiments of Petrash and Otto (ref.11.19).It must be erriphusized that the tendenciespreviously indicllted may be overridden bydynamic effect>. For example, if the liquidwere sutltlenly dru\vn up through the inner tube,following the renloval of the body force, itsmomentun] may carry it out of the inner tubeinto the top end of the tank; this possibilityshould be guarded against in any design utilizing this sort of ~ t a n d p i p e .
~An analysis of theminimum-capillary-energy type, which is relevant to this problem, involves the determinationof the container geometry in which n liquid offixed volume can be contained with the leastcapillary energy. Consider a family of cylindrical tanks with hemispherical ends, as shownin figure 11.15(b).
Assuming full wetting, theliquid volume isV= *r2h(11.57a)The capillary area isCombining, one finds that the least potentialenergy for fised V is obtained with the largestradius. Hence, should the liquid squirt up tothe top end of the tank of figure 11.15(a), itla)FIGURE11.15.-Menisci in typicaltanks.6 This has beer1 den~onstrateclin small-scale expcrilnents carried out by Siege1 (ref. 11.1'3).40 1LIQUID PROPELLANT BEHAVIOR AT LOW AND ZERO Gmay well stay there in the absence of anyrestoring body forces.The foregoing example serves to illustrate thedangers of relying on the predictions of thistype of zero-g analysis, and the need for carefulconsideration of dynamic eflects, either byanalysis or by properly scaled model experiments.
Both of these topics mill be discussedin subsequent sections.We have seen how the spherical segmentmodel can be used in zero-g meniscusconfiguration analyses.T o make accuratedesign calculations a t nonzero Bond numbers, one must use a more accurate meniscus-shape equation. I n many systems, themeniscus is axisymmetric, and n very laryeclass of configurations can be h:indled whenthe characteristics of such s meniscus areknoit.n. me shall no,v consider a generaltreatment for axisymmetric menisci, the resulk of which can be usect in designproblems.Shape of Axisymmetric MenisciConsider the infinitesimal ;~nnularring C I I ~from the meniscus (fig.
11.16). ii force bulance in the vertical direction provides thedifferential equation for the static interfaceu-:s(r=r dr- (Pg-P,)d.s)ds(11.58)The pressure is assumed to have a hydrostaticdis tGbutionP,=Pz,- pgh(11.59)where PI is the pressure in the liquid justbeneath the interface, and PI, is the liquidpressure a t the bottom of the meniscus.Introducing the dimensionless quantitieswhererc=u/(PK-Pz0)arid combining equations (1 1.58) and (1 1.59),one finds7pThe gas density is neglected. To consider it, replaceby p z -pgin the analysis.FIGURE11.16.-Forcebalance for axisymmetric menisci.$ ( R %)=I?d fi (I+ BH)(11.61)-1second differential equation is obtained fromthe geometric conditionwhich.
\\-hen tliferentiateclgivestint1dH d-+2 H dR d2R--0d S d S 2 cis d S 2-normalized,(11.63)Equations (11.61) and (11.63) form IL pairwhiclr may be solved numerically. The problem may be treated us 2\11 initial value problem,in which we prescribeR1(0) =1ICalculations of this type have been carriedout a t fixed B, yielding values of a along theway. These results were then cross-plottedfor selected values of ar to obtain the curves offigure 11.17. I n these figures, V represents thevolume of revolution bounded by the meniscusand the plane of the contact line. Thesecurves may be used in design calculationsinvolving any axisymmetric meniscus, whichmust extend to r=O.T o illustrate the usefulness of these curves,suppose we wish to calculate the general configuration of a meniscus in s spherical tank as402THE DYNAMIC BEHAVIOR OF LIQUIDSF~XJRE11.17.-Configurationparameters for axisymmetric menisci.ct function of the percent volume loading, todevelop a curve similar to those in figure 11.14.I t is a simple mtttter to read tdhe requiredvalues frorn figure 11.17, and the calculatiorlmethod is outlined in tl~ble11.1.
Figure 11.18shows the results of this calculation for a purticular case of rnodernte Bond number. Kotethttt the depth of the merliscr~s is less thanpredicted by the simple zero-g (circular segment) model of figure 11.14. This points u pthe fact that the zero-g analysis, ttlough inerror, may often provide u sin~plecormervutiveestimate of the rneniscus extension. Whenmore accurute results tire desired, the curvesof figure 11.17 can be used.Approximate calculations for configurntio~lsnot covered by the curves discussed above canbe made using some assumed shape for themeniscus. For example, Satterlee nnd Chin(ref. 11.20) have found that nn ellipsoidalapproximation gave quite adequate results foraxisymmetric menisci a t low Bond numbers.Stability of Axi~~rnrnetricMenisciI n a zero-g environment, a meniscus will holdits position and shape as the container is slo\vlyinverted.
At sufficiently low g, the inversionmay ulso be made and the liquid left "hnnging"a t the top of the container. There are important situations in which one would like toLIQUID PROPELLANT BEHAVIOR AT L O W AND ZERO Gdesign the container such as to maintain theinverted meniscus, and other instances where itis important that the liquid not remain in theinverted position. In these engineering problems, knowledge of the stability of the meniscus,with respect to a destabilizing body force field,is of considerable utility, and me now will takeup this important question.T h e concept of interface stability can beintroduced by drawing on common experience.403Consider a mercury-in-glass thermometer.Held in any position a t normal g, the interfaceis st'able, and the mercury does not run downthe tube.
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