H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 34
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The situation is sum(5.57~)marized in figure 5.4, where a symbolic spec-'-Rearranging and letting v = the kinematic viscositv UIP1I-0iethylenetriaminePerchloral fluorideEthylene oxidek ~ e r o s e n e s pRP1, JP4IRFNAAmmonia (liquid)Oxygen [liquid)Oxygendifluoride(liquid) IPentaborane ---HydyneHydrogen (liquid)IIWFNA-EthanolFluorine (liquid)Nitric acid (&rousIChlorine trifluorideI I I IUMDHBromine pentafluorideHydrogen peroxide INitrogen tetroxide6 H y d r a z i n e tMonomethylhydrazine-EthylenediaminetIII.001.003.002.004.m .01Kinematic viscosity, v , cm'lsec.02.M)5.006.03.Q4I .ethylene: : r b-2060%s u c j n e solution0 100% Glycerol solutionCarbon disulfideEthylmercaptanSiliconesNitrogen (liquid)Benzene TurpentineMethyl iodideMethanolMercury IsopreneFreon [iiquidi d ~ a r b o ieirichioridenEthyl bromideIMethylene chloride.
Bromine, Ethyl etherEthyl iodide1CI, Methyl formate, Allyl chloride, Methyl sulfide-I1-etone,ChloroformI.I1Kinematic viscosity, v , cm'lsecFIGURE5.5.-Kinematic.05.(#I,iFIGURE5.4.-Kinematic viscosity: Typical values for fuels and oxidizers.-a-IAnilineviscosity: Typical values for various liquids..OB.1160DYNAMICBEHAVIOR OF LIQUIDStrum of propellant kinematic viscosities is displayed. (The kinematic viscosities of all fluidsvary considerably over the environmentaltemperature range possible in space exploration, and the particular points assigned eachfluid correspond to the temperatures quoted inthe appendix.) For purposes of a general viewof what types of inertial-viscous simulation arefeasible, it is important to note that the kinematic viscosity range shown in figure 5.4 isnearly covered by the kinematic viscosities ofliquid oxygen and kerosene which are at thesame time one of the most prevalent oxidizerfuel combinations.Figure 5.5 is a similar "spectrum" of kinematic viscosity for a variety of liquids whichmight serve, or have already served, in sloshingexperiments.
Some of these would present biological as well as mechanical hazards, as canbe easily noted.The above information on kinematic viscosity, combined 1t-ith equation (5.58a), allowsgeneral remarks on what types of inertialviscous simulation are possible. Figure 5.6 isa plot of equation (5.58a) for some significantvalues of v,. I n this plot, the line, v,= 1 ,indicates the required relationship betweengeometric and acceleration scale ratios if modelFIGURE5.6.-Simulation plot: Inertial-viscous rcaling,geometrically similar rigid tanks.and prototype contain the same fluid.
If thesimulation problem involves a prototype fuel ofkerosene, a comparison of figures 5.4 and 5.5indicates that the smallest ratio of v, which maybe achieved with the given mddel fluids isapproximately 0.1, and consequently thatinertial-viscous simulation of kerosene may bepossibIe anywhere above the line v r = O .
l onfiguie 5.6. If liquid oxygen or one of the othercryogenics is involved in the prototype, a comparison of figures 5.4 and 5.5 indicates that thesmallest ratio of V, which may be achieved isabout 1.0, or, in other words, simulation of thecryogenics is possible in principle only near andabove the line v , = l on figure 5.6. That thissituation is unfortunate can be inferred from thefollo~vingexanlple: Suppose that simulation ofa 9-meter-diameter rocket booster tank in a 3-gacceleration field is required, and that modelsare restricted to be in the laboratory a t 1 g.The specified acceleration ratio is (113) and consequently from figure 5.6:(1) If prot,otype fluid is kerosene, the smallestsuitable model tank would be approximately1.2 meters in diameter.e is a cryogenic, the(2) If the p r ~ t o t ~ y pfluidsmallest suitable model tank rvould be approximately 12 meters in diameter (excludingmercury as a practical model fluid).(3) If the prototype fluid is a cryogenic andthe model fluid is water, a suitable model tankdiameter would be approximately 40 meters.The results in the above example are basedon the fluid propert.ies outlined in figures 5.4and 5.5.
The problems arise because of thelack of fluids with kinematic viscosities significantly lower than that of liquid oxygen. I tmight be speculated that simulation of thecryogenic in the above example might beachieved uith a 30- or 60-centimeter-diametertank if liquid helium were the model fluid.However, this might be considered as a highlyunlikely possibility because of the unusualproperties of liquid helium other than viscosity.To summarize the general situation withrespect to inertial-viscous simulation of fuelsloshing, i t must be remarked that simulationin the st,rict sense with moderate-sized laboratory models can be accomplished in a limitednumber of cases of interest.
Recourse must be161SIMULATION AND EXPERIMENTAL TECHNIQUEShad to the extrapolation methods outlined elsewhere in this monograph.Dynamic Properties of Fluida CompressibilityTo continue the discussion of t,he simulationequations (5.58), equation (5.5813) allows asummary of what is possible if inertia-compressibility scaling is desired. I t is more convenientto alter equation (5.58b) to be a function ofsonic velocity ratio. SinceFigure 5.7 bears out this conclusion, since thesecombinations of Dr and a, are a t or below theline cr= 0.5. If compressibility-inertial scalingis required, artificial means of increasing fluidcompressibility may be feasible, and Sandorffsuggests some possibilities (fig. 5.8)..---(the sonic velocity characteristic ofthe fluid)(5.59)IThus equation (5.58b) becomes :Though the fluid property data of the appendixare incomplete with respect to sonic velocities,the values shown indicate that sonic velocitiesfor prototype liquids will probably range from500 to 2000 mlsec, with both liquid oxygenand kerosene having sonic velocities in thevicinity of 1000 m/sec.
Almost exactly thesame range is shown in the appendix for"modeling" liquids. As a consequence, theprobable greatest ratio between model andprototype sonic velocities would be 4 and thesmallest 114. Thus, a maximum range for c,2may be:O.O6<c,2<16(5.62)FIGURE5.7.-simulationplot: Inertial-compressibilityscaling, geometrically similar rigid tanks.IFlr-Thin wall polyetylene tubingheat sealed into compartmentsIf kerosene and liquid oxygen are of interest,the probable mwimum range of c,2 would beEquation (5.61) is plotted in figure 5.7 forthe above ranges of c,, with the extreme limitsshown. Prob'ably more practical limits arerepresented by the range between cr=0.5 and 2.SandorfT (ref. 5.9) concludes, in effect, thatnone of the common model fluids is sufficientlycompressible to meet LOX-JP-4 simulationneeds for geometric-scale ratios of 118 and 1116and acceleration scale ratios of 0.2 and 2.0.ISuspend in tank cavil so thatcapsules are uniformly distributedthroughout fluid volumeFIGURE5.8.-Methods of artificially increasing compressibility (ref.
5.9).162THE DYNAMIC BEHAVIOR OF LIQUIDSof Fluids: CavitationThe last simulation problem involving fluidproperties is equation (5.58c), for cavitationeffects:Dynamic PropertiesSinceA P = Po-P,wherePo=gas pressure above the fluidP,= fluid vapor pressurethis equation may be satisfied independentlyof the fluids and scale ratios if the absolute gaspressure is assumed not important to thesimulation. The model gas pressure,may be adjusted to achieve the desired ratio,AP,, given model and prototype fluids, geometric scale ratio, acceleration scale ratio, and(PO)F the pressure of the prototype gas.
Consequently, the simulation possibilities with thisparticular simulation design philosophy are notlimited, in principle. I n practice, adjustmentof model gas pressure means experimentingwithin a rnriable pressure chamber, or designingthe model tank structure to withstand possiblymuch greater pressure differentials than arefeasible.It is instructive to consider the magnitudes ofpressure differences Po-P, which may be encountered in the prototype. For pressure-fedrocket propulsion systems, the gas pressuremay be very high, perhaps much greater than 6atmospheres.
For pump-fed propulsion systems, which appear to offer weight advantagesover the pressure-fed systems for large boosterrockets, a typical minimum gas pressure may be1.5 atmospheres. Thus a range of PO-P, fortypical booster rocket fuel tanks may be 0.5 to6 atmospheres, assuming the range of fuel vaporpressures shown in the appendix. Alteringequation (5.58~):Po- P,(5.64)Again consulting the appendis for typicalmass densities p for fuels, an expected typic,alrange for (Po- P,)/p is(the units of the above are atmosphere-cm3/gm)From the point of view of convenience inmodeling, i t is of interest to see what may bedone if (Po),,,,= 1 atmosphere and the modelfluid is near normal room ten~perature. Assuming the applicable model fiuid vapor pressures and densities given in the appendix:The foregoing ranges allow some very npproximate simulation regions to be drawn withequation (5.64), figure 5.9.
I n this figure, theregion below (AP),/p,=0.07 corresponds tosituations where the prototype booster fueltank contains a fluid of very low vapor pressureand may thus indicate a simulation range wherecavitation may be quite nnimport ant. Theupper half of tlhe region bounded by (AP),/p,=0.07 and 7 rol~ghlycorresponds to the situationwhere the p r o t o t ~ p e fluid has a very highvapor pressure, or is n cryogenic, and i t canbe seen that the model must be bigger than theprototype in this case for the range of acceleration scale ratios important t o simulationof rocket booster tanks (a,=l to 0.1). T oconclude, it may be remarked that the restrictions placed on model fluids (room temperature) and 11lode1gas pressure (atmospheric)severely limit what may be done, and, ifcaritation simulation is import ant, the modeldesigner will very likely be faced with reducing the model gas pressure apprednbly.Interfacial EffectsI n the simulation of fuel sloshing, the primary interest is in the case where a liquidfree surface exists.
Consequently, three interfaces will exist in most all the problems ofinterest: liquid gas, solid liquid, solid gas.hlolecular attractions between like and dissimilar molecules near the interface give riseto surface effects. The existence of an interfacerepresents the presence of a surface energywhich is associated with the work which must163SIMULATION AND EXPERIMENTAL TECHNIQUES0Considerable data on surface tension of variouscommon liquid-gas interfaces are available. Unfortunately, corresponding surface energieslunitarea for solid-gas interface, us,, and for the solidliquid interface, us&, is almost entirely lackinga t least for fluid dynamic modeling purposes.The problem is that these last quantities aredifficult or impossible t o measure, and in factsome nrgument may exist as t o the nature ofthe solid-gas interface in the presence of a liquid,the amount of liquid vapor adsorbed, and thechemical homogeneity of the solid surface control, or a t least considerably modify the natureof this interface.Static relations between surface energy/unitarea for the three types of interface are embodied in Young's equation.usG=usL+a coswIFIGURE5.9.-Simulationplot: Inertial-cavitation simulation, rigid tank, model fluids at room temperature andambient pressure of 1 atmosphere.be done against the mutual attraction ofmolecules on each side of the interface toeffect a separation along that interface.
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