H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 101
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11.63). As the acceleration isapplied, liquid begins to flow down the wallin a relatively thin sheet of liquid. If theBond number, pgG/a, is of the order 10 orless, a large central bubble forms along theaxis of the cylinder; but if the Bond numberis of order 150 or greater, the formation of t.hewall sheet is accompanied by the delayed formation of a central plateau of liquid whichgrows down the axis of the cylinder, narrowingas it falls. The time trajectory of the wallsheets in these cases is given approximately by(for d/ro<1.2)d/,0=0.6gr+0.41713.5<Bo<10d/ro=0.65r2(11.113a)B0c15O (11.1 13b)where=tJg/(2ro)(11.113~)and d is the fall distance.In the lower Bo case, the large central bubbleslows down as it approaches the top of theLIQUID PROPELLANT BEHAVIOR A T LOW AND ZERO GOne of the methods presently being consideredas a means of locating liquid propellants inrocket tanks involves the use of electricallyinduced body forces in dielectric materials.The term which has been applied to this is"dielectrophoresis." A theoretical explanationfor this can be obtained using electric fieldtheory, that is, Landau and Lifschitz (ref.11.67).
This theory has been specialized to thestudy of the behavior of fluid surfaces inelectric and magnetic fields by Melcher (ref.11.68). The theory as applied to a dielectricliquid with uniform dielectric constant andwith no free charge indicates that a surface l3force is exerted directed away from the liquida t the liquid-gas interface (where a change inthe dielectric constant occurs). Consider, forexample, the force on a cylindrical body ofliquid partially filling the annular space betweena pair of concentric cylindrical electrodes. Ifthe liquid is concentric with the electrodes, thetheory tells us that there is no net force exertedon the liquid.
If, however, the liquid nlnss isperturbed to a nonconcentric position, thesurface force eserted on the liquid farthestfrom the central electrode I\-ill decrease andthat on the liquid surface closer in \ i l l increase.I n effect, there is a net force acting to reducethe displacement of the liquid. Hence, theliquidbe stabilized around the centralelectrode where the electric field intensity isgreatest for this electrode geometry.The theory explaining dielectrophoresis indicates that the surface force acting on the liquidis proportional to E2,E being the electric fieldintensity. This means that the polarity of theelectrode system does not affect the directionin which the net restoring force acts.
I n fact,alternating voltages can be used. The surfaceforce will be periodic but \\-ill always act in thesame direction.A A ; f f i ~ l i l + ~-----n~r i c o c in p r n c t i r ~with the useof direct currents. No liquid is a perfectdielectric. A finite current will be conductedin even the most nonconductive material.Such a current eventually results in the accurnulation of an electric charge of one polarityThis charge0.
the free surface of the liquid.la For pgr:/u>5.The rise velocity is considerablyslower at lower Bond numbers. and is zero a t the criticalBond number. See, for instance, ref. 11.64.13 Somc reports have erroneously held that the electricforce is n body force.tank; consequently, a slight projection of thevent pipe into the tank will help insure thatno liquid is removed.Hollister and Satterlee (ref. 11.63) have madea simple shallow-water analysis, which seemsto predict the velocity of the bubble in theregion near the top of a hemispherical tankrather well. They suggest use of the velocityof rise of bubbles in tubes, which is approsimately (ref.
11.11) l2;until the bubble nose has risen to 0.23ro of thetop of the tank. At this point they patch tothe shallow-water analysis, which indicatedthathv=2 (11.113e)To6where h is the distance of the bubble nose fromthe end of the tank. This procedu~reprovides areasonable estimate for the time required forthe bubble to reach the projected vent pipe.An interesting approach to noncondensablegas bubbles is being explored by Funk andWelch (ref. 11.65). They consider a systemwhich suddenly passes from a high-,a state tozero g, and look for the statistical distributionof bubble sizes that exists when the bubbles arefirst formed.
While this work is still in apreliminary stage, it does indicate that certainaspects of statistical mechanics are applicableto bubble distributions; in fact, a hfaxwelliandistribution of bubble sizes seems to be found.The use of nonwetting screens for passiveseparation of small amounts of liquid from avapor stream has been successfully employedby Smith, Cima, and Li (ref. 11.66). Usingwater-air mixtures, containing of the order of0.8 percent by mass of water, separation e5ciencies of the order of 95 percent were achievedin small-scale bench tests.i5427Dielectrophoresis*aU*-Y"-"J428THE DYNAMIC BEHAVIOR OF LIQUIDSdistributed on the free surface is, of course,mutually repulsive. I t s effect is just theopposite of that of the surface tension of theliquid.
Where surface tension causes liquidmasses to form into shapes with minimumcapillary area, the accumulated charge willcause the liquid mass to be unstable and tobreak up wvith the attendant formation of moresurface area rather than less. If, instead,alternating currents are used, the accurni~lution of a net charge on the free surface of theliquid can be kept to lot\--enough levels that theinstabi1it.y just described is eliminated.
Thiseffect has been described recently by J. b1.Reynolds (ref. 11.69).Depending on the electrode geometry considered, there tire u number of regimes ofliquid behaviol \vhicli rnay be described. Consider again, for example, the concentric cylindrical geometr: used previously:(1) When no potential is applied between theelectrodes, the liquid will tend to break up intoa series of discrete spherical drops much like astring of beads. This instability is caused bythe action of surface tension and can be explained by the tendency towvard shapes wvithminimum capillary area. (See sec. 11.3.)(2) The application of a constant potentialbetween the electrodes will result first in thestabilization of the liquid into a cylindricalmass centered around the inner electrode.However, when the surface charge has built upsufficiently, the liquid cylinder will againbecome unstable and will break up.(3) Application of a n alternating potentialbetween the electrodes will result in the formation of a stable cylinder of liquid about theinner electrode.(4) If the frequency of the current is toohigh, higher order instabilities are observed.Further, as a practical matter, the reactivecurrent, which flows because of the fact thatthe electrode system acts as a huge capacitor,becomes excessive.(5) High voltages are required in applicationto rocket propellant tanks to produce reasonablebody forces.
However, potential gradientsmust be kept below breakdown quantities.Otherwise arcing will occur which could bedisastrous, particularly in oxidizer tanks. Thisrequires that voltages be limited or thatstaging of the electrodes be employed.This thumbnail sketch of the phenomenonof dielectrophoresis indicates how i t works.A number of applications have been consideredfor this technique of liquid positioning.
First,liquids can be held in the drain end of a tankready for withdrawval to a using system suchus a rocket engine. This, of course, requiresthe proper choice of electrode geometry. Second, the tank vent can be located in the vaporspace away from the liquid. This also requiresproper choice of electrode geometry. Third,bubbles formed as a result of nucleate boiling,as in cryogenic rocket tanks, can be forced o l ~ tof the liquid muss nnd combined with the gasesin the ullage space. A fourth applicationenvisions reducing possible trapped residualpropellant quantities occurring wvith the rapiddraining of a propellant tank under reducedgravitational conditions.Cryogenics appear to be the best applicationfor dielectrophoresis because they generallyexhibit high dielectric constant properties andconsequently high specific electrical resistivity.This results in modest and quite feasiblepower requirements.
Storable liquids ordinarily have much lower dielectric cons tan tsand specific resistivity properties. The highpotential required, coupled with relatively highcurrent flows, render the use of dielectrophoresis infeasible for room temperature liquids.Application of a force of a different typethan the normally experienced unidirectionalgravitational body force actually removes thelow-g character of the behavior of tankedliquids.
A new set of liquid equilibrium shapeswill occur in response to the shapes of this newforce, and the behavior of the liquid may nolonger be dominated by capillary forces.11.6 HEAT TRANSFER TO CONTAINED LIQUIDSAT LOWgIntroductionOne of the major reasons for studying themechanics of liquids a t low g is to determinethe nature of convection heat transfer underLIQUID PROPELLANT BERAVIOR AT L O W AND ZERO Gthese conditions.
Knowledge of importantheat transfer modes is important in controllingliquid temperatures and resultant vapor pressures. I t may be expected that natural convection, the prime mechanism for heat transferto or from a propellant, will be much reducedat low g. Consequently, diffusion of bothenergy and mass will be more important thanunder normal conditions. One must be particularly careful in analysis to assess the importance of each energy transport mechanismbefore discarding any. I n this section, we shalldescribe some of the work in the study ofenergy transport under low-g conditions.
Weshall consider, in order, storable liquids, crgogenic liquids, and low-g boiling heat transfer.429Temperature nonunifor~nity in the liquidresults in differences in vapor pressure. Thesedifferences in vapor concentration set up adiffusive flow of the propellant vapor in thegas phase from regions of higher concentrationto regions of lower concentration.
Propellantwill be evaporated from the free surface inregions of higher temperature and condensedon cooler portions of the surface. This mechanism is quite important, particularly at zero g.At sufficiently high g these diffusive processeswill be enhanced by convective motions, particularly if the hot portion of the system is"below" the liquid in the (effective) g field.The dimensionless parameter, which is important in determining the onset and strength ofconvective processes, is the Rayleigh numberTemperature Control in Storable Propellant TanksThere are several reasons for temperaturecontrol for storable propellants.
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