H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 92
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The Bond numberis seer1 t o be simply the ratio of the Weberand Froude numbers'l'he regimes of flo\v as determined by thesethree parameters are shown in figure 11.5.In each cttse, viscous effects may or may notbe important, and this question must bedecided separately.This monograph deals primarily with problems associated with sloshing; for gravitydolninated sloshing, the natural sloshing frequency, w, is correlated by the dimensionlessgroup (ro is a characteristic system dimension)Under capillary-dominated conditions, it ismore convenient to use the kinematic surfacetension in the normalization, and hence towork with the dimensionless frequencyQ2=?&?p/u(11 .lob)Note thatWhen we extend the inviscid, irrotationalanalyses of the earlier chapters to include theeffects of capillary forces, we will find thatFIGURE 11.4.-Bondnumber for atmospheric dmg,centrifugal, and gravity gradient accelerations.Sometimes the Froude number is defined a sv/J&.LIQUID PROPELLANT BEHAVIOR A T LOW AND ZERO G391FIGURE11.6.-HydrostaticFIGURE11.5.-Hydrodynamicregimes and characteristicresponse times.regimes.G2 can be expressed as a function of the Bondnumber f o r any particular configuration; forlow Bond numberswill approacha constant,.wiiile for high Busd numbers R2 will approachthe constant values given earlier.
The importance of the Bond number as a parameterin low-g sloshing should no\\- be evident.z2Response Time EstimatesIn section 11.4, we will discuss the natl~rlilfrequency of liquid sloshing motions, fromwhich characteristic tinies for capillary-dorninated and gravity-dominated motions \\-illemerge. These characteristic times are foundto beT = m g for the gravity-dominated regime(1l.lla)T=Ifor the capillary-dominated regime(1l.llb)These expressions provide order-of-magnitudeestimates for the time required for reorientatiorlof liquid to tttke place follo\ving the transitionfrom one hydrostatic regime to another.rI hese estimates are she\\-n quantitatively fortypical systems in figure 11.6.
Note thtit.the reorientation period at leasererlLlinlaroe t:Lnk.THEcan beANDOF CAPILLARY SYSTEMSIntroductionCapillarity h:is been it subject of considerableinterest to surface chemists for several decades,and there is a vast body of related literatureconfronting the e~igineerconcerned with lo\\--:sloshing. I t is known that capillary .' effectswise as a result of rather short-range molecularinteractions, and there are various theoriesfor predicting the surface tension. For acomprehensive summary, see reference 11.7.A presentation more oriented to the temperament of an engineer is given by Bikermarl(Xf. 113 ) .Although the microscopic interpretations andanalyses are interesting and important, forengineering purposes macroscopic representations are much more suitable.
I n this section,I\-eshall review the key ideas of the macroscopicapproach to capillarity, emphasizing in particHistorically, a "capillary" tube was so small t h a t i tcould only admit a hair (capella). F o r a n interestinghistorical review, see Maxwell (ref. 11.6).392THE DYNAMIC BEHAVIOR OF LIQUIDSular those aspects which are relevant to thelow-g sloshing problem. Other treatment ofcapillary hydrostatics and hydrodynamics isgiven in references 11.9, 11.10, and 11.11.
Tothe novice to this field, we strongly recommend the educational motion picture ofTrefethen (ref. 11.12).Surface Tension and Contact AngleI t hns been found that liquids behave as ifthey \\-ere covered by n contrt~ctiblemembrttnein uniform tension. This tension acts alongthe surface and tends to make its surface assmall as possible. The force-per-unit lengthacting normal to any line drctwn in the surfaceis defined 11.; the interfacial or surface tension.Consider ti spherical bubble of gas embeddedin a liquid, as shown in figure 11.7. If wedenote the surface tension by a, then the tensileforce acting along ti great circle is Fs=axD.For static equilibrium, this must be btilancedby the pressure force, F,= (?rD2/4)( P I - -Po),where P t and Po represent the pressrlres insidennd outside o f the bubble.
The presslvedifference is therefore given byand is known as the capillary pressure.If the interface is not spherical, the capillarypressure difference across the interface at anypoint is given by (refs. 11.8 and 11.9)where rl nnd r2 are the principal radii of curvature at theit point. This relation forms thebasis for the analysis of every hydrostaticinterface in which capillary phenomena areimportant, and is often called Laplace's law.The surface tension is generally consideredto be a thermodyna~nicproperty of the interfaceand is a function mainly of temperature for agiven interface. Holyever, during and shortlyafter the formation or destruction of newsurface, the apparent surface tension may differsome\vhat from its equilibrium value. Thismodified surface tension is called the dynamicsurjace tension (ref.
11.13). Fortunately, thesedynamic effects may be neglected in manyengineering anttlyses, in pctrticular for the onesat hi~nd,and we shall not consider t h e m further.The srirfnce tension is a monotonically decreasing function of tempertttllre, ctnd vanishesat the critical point. Values of the surfticetension of several important liquids are givenin the appendix.When t i liquid drop comes in contnct with tisolid surface, three angles are formed in aplane perpendicular to the three-phase line inthe solid surface.
For drops on flat surfaces,the solid plinse occ~~pies180°, while both thegaseous and liquid pliitses occupy the remniriing180". The angle measured within the liquidbetween the solid and the tangent to the liquidgas interface a t the three-phase line is calledthe contact anglr (fig.
11.8(a)).T h e value of the contact angle is related tothe relative magnitudes of the niicroscopic adhesive and cohesive forces (ref. 11.8). If thecontact angle is less than 90°, the liquid is saidto "wet" the solid; if the contact angle isgreater than 90°, the liquid is said to be "nonwetting" (fig. 11.8(b)). Both wetting andnonwetting liquids will adhere to solid surfaces.Consider now a cylindrical droplet of liquidresting on the surface of another liquid, bothunder a third liquid (or gas), as shown infigure 11.8(c).
If we define the individual surface tensions of the three interfaces as before,then equilibrium of the contact line requiresthat423=U13 cos B+ u12cos 4(11.14a)u13sin B = u12sinFIGURE11.7.-Sphericalgaa bubble embedded in a liquid.4(11.14b)T h e above relations have been reasonably wellsubstantiated by independent measurements ofthe three interfacial tensions and contact angles.LIQUID PROPELLANT BEHAVIOR AT L O W AND ZERO GWetting(b)Non -wettingSolidFIGURE11.8.-Contact angles.Consider next the three cylindrical interfacesformed by a solid, a liquid, and a gas, as shownin figure 1l.S(d). The condition of horizontalequilibrium can be sat,isfied ifHow-ever, the condition of vertical equilibriumcannot be satisfied. Nevertheless, this pictureof the interfacial force "triangle" is commonlyemployed and justified on the basis of its anslogy to equation (11.14a). Equation (11.15) isoften called Young's equation.Although surface tension and contact angieare manifestations of microscopic forces, thereis a great deal of evidence which indicates thattogether they define an appropriate macroscopicrepresentation of capillary phenomena.
Forexample, the height to which liquid will rise ina capillary tube can be predicted in terms of thesurface tension and contact angle, both independently measurable, and these predictions393are confirmed experimentally. The shapes ofliquid drops on a surface can be computed interms of the contact angle and surface tension,and these computations are in agreement withmeasurements.
Henceforth, we will considerthat contact angle and surface tension do indeed allow capillary phenomena to be treatedmacroscopically.Although the surface tension is a relativelyinvariant property of the interface, the contactangle is not, and hysteresis is common. If alittle liquid is added to a drop, it is observedthat the base does not change, but that thedrop merely changes its shape, and consequentlythe contact angle changes. After a period ofgrowth. the base will suddenly expand in ajerky manner, and the contact angle will againbe reduced.
The greatest angle obtained ongrowth is termed the adnancing contact angle,and the smallest angle on removal is the receding contact angle. A similar effect is observedwhen a drop is placed on a plate which is subsequently tilted. The advancing and recedingof the contact angle, characteristic of the hysteresis process, have been measured for mercuryon a tilted glass plate. Figure 11.9 presentssome quantitative data for t-his case. Thishysteresis seems to depend somewhat upon thedrop size and orientation in the gravity field.Hysteresis is also observed with a moving interface.
The angular amount of hysteresis is notexcessive, but its existence is perhaps the mostdisconcerting factor in the contact angle-surfacetension representation of capillary phenomena.An exception is found for pure fluids on clean1800160'0:='.01ma140°120Oo0aF l o u ~11J.-ContaCt~angle variation with inclinationfor mercury on glass.394THE DYNAMIC BEHAVIOR OF LIQUIDSsurfaces, where no hysteresis is observed(ref. 11.14).Both the contact angle and the surface tensionare dependent on the condition of the interfaces.Contamination by foreign material influencesthe value of the surface tension some\\-hat, andalters the contact angle even more.
Sensitivityto sr~rfaceand environmental conditions makesreproducible measurements of the contact tlngledifficult. This partial uncertainty in contactungle is IL key factor which often makes predict ions of capillary phenomena subject touncertainty. The difficulties are the greatestwhen the contact angle is near 90°, for slightchanges in this range can change a wetting fluidvery rapidly into tt nonwetter.Capillary ThermodynamicsThe thermodynamics of a capillary systemare useful in formulating the conditions ofequilibrium and stability.
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