H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 100
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Nussel, Derdul,and Petrash (ref. 11.51) report a visual studyof draining at zero g which clearly illuminatesthe problems. I n draining to a center hole,one invariably finds a "dip" in the interface,produced by the lower central pressure resulting from the inward acceleration. Body forcesact to smooth this dip out at 1 go, but underzero g only the relatively weak surface tensionforces remain to prevent the liquid from beingremoved in a central core. (See, for example,the analysis of Bhuta and Koval (ref. 11.59).)A schematic representat,ion of draining a t theicienticai rate a t low- a116 Zg2 g is shott-ii infigure 11.40. I t is evident that a considerableamount of liquid may be retained at low g ifadequate precautions are not taken.
If thedraining rate is sufficiently slow, the n~eniscuswill retain essentially its initial shape and themaximum amount of liquid will be withdrawn.An estimate for this maximum draining rateis quite vital.I t is, of course, entirely possible t,o formulatea mathemat,ical model of the problem; ant =-t,High g-FIGURE11.40.-Interface configuration during draining atlow and high g.424THE DYNAMIC BEHAVIOR OF LIQUIDSwhere V is the velocity in the drain, D is thetank diameter, and d the drain diameter, weobtain the maximum drain-rate criteria~e..,=(---)DV2pmax=09Thus, if We< We,,,, the meniscus should retainits general shape during draining.
Nussel et al.(ref. 11.51) experimented with a system havingd/D=O.l (for which Wem,,=105), a t Webernumbers of approximately 1.2X104 and 2.7X105. In the first case, the meniscus didindeed retain its shape arid in the second itdid not. Hence, the estimate of equation(11.109) has a t least some experimental support.When the tank Weber number exceeds thecrit.ica1 value estimated from equation (11.log),the diameter of the meniscus can be estimatedunder the assumption that the Weber numberbased on meniscus diameter is equal to thecritical value. This is equivalent to assuming that the meniscus diameter gets as largeas possible in the period of tirne which i t has toadjust to changes in its position. Hence, weestimateEquation (11.110a) can be used to estimateDJD. Then, the amount. of retained liquid isestimated asAt very low discharge rates, where the meniscus spans the tank, the ~ o l u m e of liquidbeneath the meniscus when it reaches the drainwill inevitably be trapped.
I n the simple1model presented above, this fraction is - DIH.8More accurate calculations can be made usingthe curves of figure 11.17.Further experiments using baffles a t theoutlet and a t the gas inlet indicated that suchdevices could be used to allow substantialincrease in the drain-flow rate of liquid fromthe tank with little increase in the trappedfraction. The use of a plane baffle over thetank drain a t high-flow- rates resulted in somereduction of the trapped quantity ut fixed-flowrate, probably because the central vortex iseliminated. Use of a baffle a t the gas inletgreatly reduced the trapped residuals a t agiven flow.I t is evident from this work that draining oftanks in low-g conditions will likely result inhigh trapped residuals, and therefore somemechanical means of flattening the meniscusinside the tank is highly desirable.When the Bond number is sufficiently great,a n entirely different type of retention mechanismwill become important.
Suppose a pump issucking liquid from a tank, as shown in figure11.41. The pumping demand must be lessthan the critical flow rate for the drain, whichHere H is the initial mean depth of the liquid.This volume is based on leaving an annulus ofliquid on the wall, and the volume beneath thespherical meniscus of diameter, Dm, when itsnose just reaches the drain. The fraction ofliquid retained is therefore estimated as.f=[l-(%)']+g1 Dm1(11.1lOb)When we apply equation (11.110b) to theexperiments of reference 11.51, with HID = 1.5,We =2.7XlOb, dlD =O.lO, we find f -0.88.For the corresponding experiment, Nussel et al.measured j= 0.76.,,,Fll.Q1.-Simp\c,nodel for estimate of drainchoking.LIQUID PROPELLANT BEHAVIOR AT L O W AND ZERO Gwill be determined approximately by thecondition a t which the flow over the lip of thedrain becomes crit.ica1 (Froude number unity).The maximum flow rate is therefore estimatedasZL',,,= pho~d& 5:wDump (11 111a)When the liquid depth a t the lip (ho) hasfallen below the level implied by equation(I 1.11la), the required flow cannot be delivered,and to all intents and purposes the draining isfinished.
At reduced g, the critical depth, h,,can be surprisingly large, and hence the chokingproblem should be thoughtfully considered.Assuming that the rate a t which the liquidlevel is falling is slow compared to the liquidvelocity over the drain, a quasi-steady estimateof h may be made, using one-dimensionaltheory.
This gives-=VI=&(ll'lllb)The volume of trapped liquid can then beestimated asThe Bond number involved in the drainingof a large booster tank under a low-g conditionmay well be considerably greater than unity,in which case model experiments at 1 g onsmaller tanks can give insight into the drainingproblems. An experimental study of drainingof cylindrical tanks with flat bottoms in theBond number range 100-1000 has been reportedby Gluck, Gille, Simkin, and Zukoski (ref. 11.61).This range m_sy he cnnsii_dered high-Bondnumber, and the surface tension effects may beexpected to be minor. This mas indeed thecase, for these authors found that the point atwhich gas ingestion occurred depended only onthe Froude number in the drain. The height,h, from the drain at which ingestion occurredwas correlated quite well by425Note that at very high rates of drainingingestion occurred when the height at the wallwas 0.430.Another aspect of the flow of liquids frompropellant tanks which will have to be carefull?XI-atchedby the propulsion system designer isthe formation of vortex flo~v.
Since axial bodyforces do not enter the equations for angularmomentumi, the formation of vortex flow canbe expected to develop at the same rate underlor\--g conditions as under high-g conditions.When the vortex does occur, though, the resultwill be very much more pronounced under low-gconditions, with the vortex funnel penetrat'ingmuch thicker layers of liquid at low g,In the event more positive control over theliquid during expulsion is desired, one must usesome sort of flexible containment device, suchas a bladder. Such techniques are discussedat length elsewhere (ref.
11.62), and \\ill notbe treated here.Problems and Methods of Vapor Expulsion at Low gThe operation of a rocket propulsion systemoften requires that gas be allowed to escapefrom a tank without expelling liquid a t the sametime. For instance, in storable propellanttanks used with pressure-fed engines, i t may benecessary to effect a pressure reduction in oneof the tanks prior to restarting the engine.The use of cryogenic propellants is accompaniedby the urgent need periodically to relieve pressure buildup due to heating. The venting ofpressurization gas and propellant vaporspresentsno difliculty a t all under high-g conditions whenthe location of the liquid in the tanks is \\-elldefined. Under low-g operating conditions,however, the liquids may be located in a varietyof places in the tank, depending on the ievei oiand the direction of body forces acting on theliquids and, in part, on the fight history.
Someeffort is therefore necessary to assure that whenthe tank vents are opened, only gas \\ill beexpelled, and not precious liquid propellants.If the flight history is well known, it may bepossible to use the passive methods describedearlier to hold the liquid away from the vent.When this is not possible, some active means forremoving the liquid from the vent must be426THE DYNAMIC BEHAVIOR OF LIQUIDSLiquidcalculations really accurately are not presentlyavailable, and the designer must rely oneducated guesses and model experiments.We can, however, estimate the minimumtime and maximum impact velocity simply byassuming the liquid falls freely. This givesL~v,== JIGas- DrainFIGURE11.42.-Ventclearing by a temporary thrust.employed. Consider the propellant tank offigure 11.42.
The vent at the top of the tankis temporarily covered by liquid which flowedthere under the action of extraneous force encountered by the vehicle during a prior phaseof the mission. Application of a thrust in thedirection of F will cause the liquid in the tankto move toward the bottom end of the tank,clearing the vent. For proper design, we needto know both the minimum force required tobring this change about and the time requiredfor the liquid transfer.The acceleration produced by F must besufficient to destabilize the meniscus, and therequired magnitudes can be estimated, usingthe analyses and results of section 11.3.
Gooddesign would normally use several times thetheoretical minimum acceleration, which maybe a small thrust, especially for a very largesystem.Care must be taken not to use too muchacceleration, however, lest the liquid causedamage in impacting against the bottom of thetank. I t is quite important, therefore, to havea rather good estimate of the rundown timeand the velocity of the liquid mass on impact.Unfortunately, methods for making theseo=~(11.112a)~L(11.112b)With drag, the actual impact velocity may beconsiderably less. In studies of the motionof bubbles in tubes, which closely resemblesthis phenomenon, velocities of the order ofone-third that given by equation (11.112b) arefound (ref. 11.1 1).
Thus, a conservative designprocedure would be to use a reorientationacceleration something in excess of the criticalvalue determined from the stability analysis,and design the structure to take the impactvelocity of equation (11.112b). Then, allow5 to 10 times the reorientation time fromequation (11.112a) for the liquid to clear thevent.The flow behavior during reorientation hasbeen studied experimentally by Hollister andSatterlee (ref.
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