H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 35
Текст из файла (страница 35)
Because the basis of this type of energy is molecular, the magnitude of the surface energy forany particular interface will depend only onthe electrochemical composition and t,hermalstate of the matter on either side of the interface and the area of the interface involved.I n particular, when chemical compositions arefixed, the surface energy varies directly withinterfacial area. Thus when a liquid complet.elyfills a rigid container, the interfacial area isconstant, no change in surface energy can beeffected by dynamic motions in the fluid, andneglect of surface effects in fluid dynamic~ ~ l c r d ~ lisi n~ntirelygjustified.
Tt happens thatthe energies involved in changing the area ofa free liquid surface are small relative to theeffects of gravity, viscosity, and so forth, intanks of greater than .30-cm diameter in acceleration-fields of 1 g and greater, and interfacial effects are more often ignored thanaccounted for.T h e surface energylunit area between a liquidand a gas is customarily called surface tension, a.e(5.65)where@=the "contact angle1' between the liq~iidgas interface and the liquid-solid interface a t the intersection of the threeinterfaces.This equation, in effect, replaces the mostdifficdt surface energy iinknowns by a quantity8 which is measurable and not dimensional.The dynamic validity of this relation may bein question, but the present state of the artdictates the assumption that it is correct.I n the case of a fuel tank filled with a liquidand a gas, the energy which must be supplied t omake a change in the areas of the three interfaces iswhereAA,,=thetotal change in the area of t,he(mz) i ~ t e r f ~ c e .1f ecluation (5.65) is used to eliminateUSGAE=~ s L ( A A s Q + ~ s L )+a(AALff+AAsG cos 9)(5.67)Since the total area of the tank walls is somefixed constant, an increase in the area of thesolid-liquid interfnce must be accompanied by164THE DYNAMIC BEHAVIOR OF LIQUIDSan equivalent decrease in the nrea of the solidgas interface.
ThusAAsc+~ S L = O(5.68)andEquation (5.69) indicates the several parameters important. in simulation of interfacialeffects under the assumption that equation(5.65) is dynamically valid, and the similarityanalysis can proceed as before:Parameter1Change in surface energy- - Change in interfacial area:Liquid-gas - - - - - - - - - - - - Solid-liquid - - - - - - - - - - - - (Contact angle- - - - - - - - - - - - - 1Surface tension (liquid-gas) -SymbolDimensionM Y T-2aEYY------------AA,,AASL0MF2a1Employing the repeating variables as beforeThe first T-termof equations (5.70) is consistentwith the previously noted inertial scaling offorces. Maintenance of the relative magnitudes of A E and the total energy involved insloshing in model and prototype is basicallywhat is desired and, if equation (5.69) is valid,will be assured if o and &ILc are properlyscaled. Consequently T~~is a dependent variable, in a sense.
Both ?r16 and xr6 are restatements of *, for any area. Thus, geometricsimilarity of the interfaces is implied. The lasttwo terms, n4, and s48, are the only new onesintroduced in this section.The first of these two says that the contactangle must be the same in model and prototype.For simulation purposes, this criterion amountsto the proper selection of model tank surfacetreatment to achieve the proper contact angle.Since most prototype fluids M-ouldbe expectedto be "~vetting" (8+0°) when in contact withthe enclosing tank, "~vetting" model fluids arenecessary, and most of the model fluids quotedin the appendix are probably wetting withrespect to metals. A wetting prototype fluid\till probably rule out a liquid metal modelfluid such as mercury unless some practicalsurface treatment can be devised to makemercury "wet" the model tank.
I n any event,the simulation of contact angle in the model isa suhject which has not been extensively treatedin the lit,erature and subsidiary experimentationin any given case to see what contact angles canbe achieved would be necessary if this parameteris thought of importance. It may not benecessary to achieve exact correspondence ofcontact angles in model and prototype forwetting or nonlvetting fluids (e=oO, or 180°),since 8 appears in equation (5.69) as cos 8, andthus small errors in achieving model contact.angles \{-ill not greatly affect the ratio betweeninterfacial energy changes in model andprototype.The last *-term of equations (5.70), u 4 can~be shown to be basically the same result as wasfound previously (refs. 5.10 through 5.13).Equation (5.71) denotes a replacement operation carried out on *,,.
The result is the \veilknon-n "Weber number" (also written in thesquared form, see ch. 11)The reciprocal of 77,~ multiplied by gla, a1-ariant of s,,,and by r2/D2,a variant of r l ,results in the well-known Bond number (seech. 11)whereg= gravitational accelerationr=radius of meniscus in an equivalentcapillary tubeThe Bond number compares the relative magnitudes of gravitational and capillary forces,while the Weber number compe.res capillaryforces to so-called "inertial forces."SIMULATIONEXPERIMENTAL TECHNIQUESFor purposes of showing what simulationpossibilities are available in a simulation plot,i t can be noted that the surface tension, u, andfluid density, p, always appear as a ratio, andthus a "kinematic surface tension," 4, may bedefined+=; uSubstituting innotationT.Y)(5.73)and converting to ratioConsulting the appendix for typical values of+, it can be found that prototype liquids a tnormal operating temperatures have values of4 ranging from += 10 to 100 (cm3/sec2).
Themodel fluids shown have an identical range ofvalues of + for the fluid temperatures shown.Consequently, an outside range of possiblevariation in + isIf liquid oxygen is the prototype fluid and themodel fluids are restricted to the temperaturesshown in the appendixIf the prototype fluid is kerosene and the modelfluids are restricted as before,Equation (5.74) is plotted in figure 5.10 for theforegoing ranges of 4,.
I t is apparent thatscaling of surface tension with models signscantly smaller than the prototype and acceleration ratios less than 1 is not a very likely proposition. Fortunately, the range a,<l is nottoo important for surface tension scaling, sincethe laboratory model a t 1-g acceleration impliesquite high body forces in the prototype relativeto surface tension forces.The possible simulation ranges on figure 5.10for a,>l show the possibility of simulation of.(FIGURE 5.10.-Simulationplot: Inertial-surface tensionsimulation, rigid tank (model fluid temperaturesrestricted to normal ambient).relatively low gravity phenomena with smallmodels in a 1-g acceleration field (about whichmore is said in ch.
11).Surface tension, as with all model properties,is variable with temperature (u+O as the fluidtemperature approaches the critical temperature), and i t is instructive to estimate the orderof magnitude of model fluid temperature necessary to result in +, less than (0.1). Empirically, the variation of u with temperature is asequation (5.75)whereT,=new temperatureT,-referencebiiipeiiit'=GT,=critical temperatureThusConsidering model fluids which are liquid a tatmospheric pressure and room temperature166THE DYNAMIC BEHAVIOR OF LIQUIDS(T2=20' C) , the maximum range of variationin the density ratio factor in equation (5.76) isapproximatelyCritical temperatures of the aforementionedmodel fluids may range from 180' C to 1540' C.If it is assumed that model fluids at ambienttemperatures may be chosen to result in[+,IT2=0.5, then the value of 4, a t a newtemperature TImay be crudely approximatedbyIt can be seen that the order of magnitude ofTI must be 150' C and more just to change 4,from 0.5 to 0.1.
As a consequence, the possibility of achieving 4, less than 0.1 exists, but atthe expense of quite extreme model fluidtemperatures and pressures.For the simulation of low p a r i t m yin thelaboratory (a,>> I), it can be seen from figure5.10 that extremely small geometric scale ratiosare required for the simulation range shown.Unf ortunately, even the liquid metals tend tohave kinematic surface tensions not too muchgreater than that of water ( ~ 9 cm3/sec20nearthe freezing point) because of their higherdensity; thus, hopes of achieving 4,>10 cannotbe placed too high. If simulation of lo-' genvironment is required, for instance, a geometric scale ratio be tween 0.003 and 0.03 wouldbe required in the laboratory at 1 g (for a 200centimeter-diameter prototype tank, modeltanks of from 0.6 to 6 centimeters might berequired).
However, if the "laboratory" acceleration environment can be reduced toby use of drop towers, and so forth, the requiredgeometric scale ratios would range between 0.3and 0.03, which would appear to be a practicnlrange.5.5SIMULTANEOUS SATISFACTIONSCALING CRITERIAOFALLBecause of its importance, inertial scaling isincluded in all the foregoing discussions pertaining to scaling various dynamic properties offluids and materials.
The approximate simulation ranges, D, versus a,, for the followingsimulations have been discussed:(1) Inertial-viscous simulation(2) Inertial-compressibility simulation(3) Inertial-cavitation simulation(4) Inertial-tank elasticity simulation(5) Inertial-surface tension simulationSince in any given real problem more than oneof the above combinations may be involved, thepossibility of simulation where any two to fiveof the above simulation criteria are involved isan important consideration. Unfortunately,from the point of view of clarity there are 26combinations of the above simulations (taken2, 3, 4, and 5 at a time), and only a few of thesecan be considered here.Since many of the most challenging practicalfluid dynamic problems of the present andimmediate future involve lou- gravity behavior,some illustrations will follow involving thepossibilities of adding to the iner tial-surf acetension-scaling criteria, the criteria resultingfrom the consideration of other fluid properties.A simple way of approximating simulationranges is to superimpose the simulation plotspreviously exhibited.
This procedure is anoptimistic one, since fluid properties are notindependent of one another, and i t may well bethat the fluid properties dictating some extremebound in one type of simulation will not dictatea corresponding bound in another type.Nevertheless, if figures 5.6 and 5.10 are superimposed, and net simulation ranges are drawn,there results figure 5.11, pertaining to simultaneous simulation of inertial, viscous, andsurface tension effects. While the extremerange and the simulation range s h o ~ ~forn akerosene prototype do not differ from thecorresponding ranges for surface knsion scalingalone, the simulation range for liquid oxygen istruncated a t the high acceleration ratio end.Dalzell and Garza (ref.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
