H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 96
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The result is sho\vn in figure 11.24.that is, that it fall to the "bottom" of the tank,(4) The maximum destabilizing g loadingt i spherical design should be used.Then, \\-it11can be sustained when the interface is fltitany slightly destabilizing orientation of the(a =90').body force, the liquid will niigrilte to the bottomWhat experimental d a t ~are t~vailableseemof the tank. On the other hand, if it is desiredto confirm the instability theory. Satterleethat the liquid should remain "above" the gas,and Reynolds (ref. 11.24) performed extensivethen one would try to design the vessel so as toexperiments in a slightly tapered tube, andthese datti are in very good agreement \\-itti tichieve a flat meniscus. This might be acfigure 11,22 over the range of 0 ~ < a < 9 0 ~ . complished by placing thin rings around the\valls, as shown in figure 11.%(u).
TheSimilar data of Masica et d. (ref. 11.25) alsomeniscr~swould become tittached to one of thesesupport these predictio~is. Jakob (ref. 1 1.26)rings, arid a11 optinlally stable flat interfacediscusses the wall-bound bubble ctise in histhereby obtained. If the g-loading is more thannucleate boiling chapter, and shoivs t i varietythe critical value for stability, the tank can beof data \vhich are in very good agreement withpartitioned to bretik up the meniscus intofigure 11.23 over the range -90°<a<00.smallerstable menisci, as indicated in figureThe case of very small negtitive radius ofLIQUIDPROPELLANT BEHAVIOR AT LOW AND ZERO GExample 1For a=,50°+* 0.667,read o n Fig.
11.22. 40°Example 2409procedure, in which the interface shape \\-asfirst specified and the container shape thencalculated. I n order to maintain the assumedmeniscus shape, a nonunifo~mpressure must beimpressed on the outside of the meniscus.Anliker and P i state that the shape of thecontainer bottom is very important in determining the stability criteria. The present workwould suggest that i t is only the containershape a t the contact point which is important,and i t is believed that the stability limits givenby the Anlier-Pi scheme for realistic meniscusshapes would be in full agreement with thecurves of figure 11.23.Flat Annular MeniscusSuppose8 = 135'a=4O0RwIR - - 1Fora-40'.RRw sin+ - - 1.41 .(pg~'l~),ritFIGURE11.25.-Two9.-.-- 1 x 0.7071.41The stability of a flat meniscus in an annulartank has been analyzed by these same methods.The eigenvalue problem for a flat meniscus isrelatively simple, and the solution is given inThe results of this stabilityreference 11.analysis are presented here in figure 11.27.read on Fig.
11.22=3.3examples of arisymrnetric meniscusstability calculations.3FIGURE11.27.-Stability(aFIGURE11.26.-Twoi(b)methods for improving stability.11.26(b). Screens can be used effectively forthis purpose.A recent study by Anliker and P i (ref.11.29) involved calculation of interface instability by the dynamic method described insectmion11.4. However, they used an inverseof a flat annular meniscus.Note that the critical Bond number for thecase of T i / T o = 6 , corresponding io a ttiil \\-irein the center of a tube, is 3.39, the same valueas obtained without the wire.
This suggests8 I n ref. 11.1, the ordinate of the illustration corresponding to fig. 11.27 is incorrectly labeled. Theright-hand side of the first equation on p. 85 ofref. 11.1 should be multiplied by R. Eqs. (11.61) and(11.64) have been corrected for typographical errorsthat were present in ref. 11.1.410THE DYNAMIC BEHAVIOR OF LIQUIDSFIGURE11.28.-Shape and stability of a rotating meniscus in a cylindrical tank.tnat the presence of a small standpipe in thecenter of a large tank would have little effecton the critical Bond number.It should be noted that the two-dime~isionulchannel, with a two-dimensional disturbance,does not form the limiting case as ri/ro-jl.This is because of the azimuthal v ~ r i a t i o npermitted in the annular tank." However, thenumerical calculations are checked by theobservation that a disturbance with no radialvariation which wiggles once and meets itselfin going around the circumference is equivalentto the two-dimensional problem, with a replaced by one-fourth the circumference, ors r / 2 .
Then, using equation (1 1.70), Ire wouldpredict pgr2/a= I, which agrees with figure 11.27.Shape and Stability of a Rotation AxisymmetricMeniscusSeebold and Reynolds (refs. 11.30 und 11.31)hare made similar calculations for a rotatingmeniscus in a cylindrical tank, nssl~mingthat9 It is important to appreciate that the two-dimellsional chnnnel idealization, while simplifying theanulysiw, does not let\ci to meaningful results in thestnbility problem; for this reason the results of Concus'pioneering study (refs. 11.31 trnd 11.22), which set themethocl for the stability nnnlysis, have not beeninclndecl herein.the liquid is in a state of solid body rotation.Some of their results are given in figure 11.28.I t is interesting to note that a rotating meniscusnlrty be unstable even a t zero g, enabling theliquid to move entirely to the outer perimeter.This has important applications in liquid-vaporseparation.For the special case of zero contact angle andzero Bond number, Seebold and Reynolds(ref.
11.30) obtained a simplified stabilitycriteria for the rotating meniscus in an annulartank. Their results may be written asHere r , and ro are the inner and outer radii ofthe tank, w is the angular frequency (radtsec),and Q2 is the rotational Weber number. IfQ2 exceeds this critical value, or if the meniscuscontacts the bottom of the tank, the liquidm o w s rapidly to the outer wall.This theory indicates that even very slowrotations will produce full liquid separation inlarge tanks. For example, in a 3-meterdiameter tank, a rotation of the order of tw-orevolutions per hour is sufficient to cause sepnration, once the st,at.eof solid body rotation hasbeen achieved. Seebold and Reynolds present nLIQUID PROPELLANT BEHAVIOR AT LOW AND ZERO Gnumber of other data pertinent to the design ofrotating tanks for low g.
I n particular, the reference includes a more complete description of acalculation method for axisymmetric menisci,and a detailed discussion and analysis of thestability problem.The exrperiments of Seebold and Reynoldsindicate that figure 11.28 provides a conservative estimate of the critical rotation parameter.Separation was usually obtained at somewhatlower Borid and RTebernumbers, primarily dueto "dynamic overshoot" in the reorientationperiod.Blackshear and Eide (ref. 11.32) have examined the meniscus in a tank whirling aboutan axis perpendicular to the tank axis.
Thismotion produces an effective body force gradient, but the shape and stability can be determined essentially by the value of the bodyforce at the meniscus surface. Hence thestability criteria, estimated using a criticalequivalent Bond number of 0.84 (for fullywetting liquids), ~vouldbeFIGURE11.29.-Supporting pressure.This length nius t not be exceeded if the meniscusis to be supported, and such considerationsshould be nlude if the liquid is in a nearlys a t ~ ~ r a t esta1.e.dIf the liquid is saturated, noinverted meniscus can be supported.
Thisfact must be considered in the design of cryogenic licluicl storage systems.Stability in Capillary-Supported SystemsHere L is the distance from the whirl axis tothe meniscus, and w is the whirling angularfrequency.Condition for Sufficient Supporting PressureThroughout the previous three sections wetacitly assumed that the pressure forces inthe gas are s a c i e n t to support the meniscusin an inverted position. This must be examined in each application.
The pressure atthe uppermost point in the fluid will be lessthan that at the lowermost point by an amount(see fig. 11.29)Pz- PI=pgh(11.73)where h is the dist,ance between these twopoints. The pressure at PI must not be lowerthan the saturation pressure of the liquid atthe particular temperature involved, or elsevaporization is likely to occur at the top of thetank.
The maximum liquid height which canbe supported is therforeThe menisci discussed in previous sectionsare supported in an inverted position by pressure forces. In contrast, consider the capillarysupported systems of figure 11.30. This systemcan also be analyzed by the minimum potentialenergy method. However, instabilities generally occur a t very low values of the Bondnumber, based on the radius of the supportingmenisci, and hence the simpler force-balancemethdds, utilizing the idealization that themeniscus is spherical, are usually quite adequate.
For example, equating the total pressure variation along the axis of figure 11.30(a)to zero, one finds2a-=pgh(11.75)r1Since r,>r, a stable system is possible only ifSimilar analyses yield the results sho~\-nforfigure 11.30(b).In a series of experiments on a system suchas figure 11.30(c)-(e), Hollister (ref. 11.33)THE DYNAMIC BEHAVIOR O F LIQUIDShil r * ~The inverse problem of a gas column in aliquid under zero g was studied b y Rayleigh(ref. 11.35).
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