H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 67
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8.33 and8.34)) Buchanan et al. (ref. 8.31), Baird (ref.8.30)) and Kana and Dodge (ref. 8.35). Thetheory given here is due to Bleich (ref. 8.33), asmodified in reference 8.35.Although the mathematical details of bubblemotion tend to be complicated, it is fairly easyto describe the physical processes that occur.When the tank is vibrated up and down, theeffect on the liquid is the same as if gravity werevarying. For the moment, only the changes inthe fluid pressure caused by this varying gravityneed be considered.
During the upward acceleration, the pressure increases over its staticvalue, and, consequently, the volume of thebubble is smaller than for static conditions.The pressure gradient at this time creates a.net. positivn ~h_~_=gei~ ~ ~ J ~ J T aE ~~ d~th_aJ T , .-rs.bubble gains a certain amount of upward momentum. During the downward acceleration,the pressure change is negative, and the bubblevolume increases.
A net negative buoyancy iscreated, and the bubble gains a certain amountof downward momentum. But the downwardmomentum is larger than the upward moFIGURE8.23.-Bubble-tank-liquidmentum; that is, the bubble volume (and the5b = Wall thickness2Lx0 cos cutdynamic system.THE DYNAMIC BEHAVIOR OF LIQUIDSi--.
, ''The bubble is assumed to be very small cornpared to the tank and located far from anysurface. The fluid velocity a t any point isThe potential of the compressed gas bubble iswhere u,=x = - ~ wsin wt is the velocity due to+the overall tank motion, U A is the velocity due+to the bubble's expansion, and u,is the velocity+due to the bubble's vertical motion. If dV ist,he element of volume, the kinetic energy ofthe fluid is1.+psj+U A.udT'+because of the assumed symmetry.The first integral in equation (8.51) is simplyJ1x2/2, where A4 is the total mass of fluid. Thesecond integral is the srtme as the nonlinearizedform of equation (8.481.
The third term can bes11on.11t ( I e q ~ ~ ~a l~ ( -(I31)342/3,wliere i is thevertic,~ll velocity of the bubble. The fourthCollecting all the terms gives41P.E. = - ~ M X +pg(a+A)az~When equations (8.52)where L = K . E . - P . E .and (8.53) are substituted into equations (8.54),the results areandaatiritegnil B - 4 ~- [(a+ A)3z]/3,according to referel1c.e 8.33.
Hence, the total kinetic energy isThe contribl~tionto the kinetic energy due tofree s ~ ~ rc*ef t i sloshing has been neglected ine q ~ ~ n t i o(8.52).r~'rhe poter~titilenergy ( ~ o ~ ~ s ofi s tthreesptlrts.'l'l~e ~>oteritit~lof the prtlrity field is'I'lle pote~iti~tlof the tillage gas above the surface1s.
. - .Eqi~otions (8.55) nnd (8.56) may now besolved siiri11lt2ir1eouslyto yield the necessaryconditio~is for sinking bubbles. However,Bleicl~(ref. 8.33) d ~ o w e dthat this was equivalent to solving n much simpler problem: findthe conditions for which a bubble will undergon smltll periodic vertical motion rtbo11t somelevel, I. The level, I, can be shown to separatethe regions for which bubbles sink and theregions for \i.hich they rise.I t is col~verlient to replace the coordinatez by X=z-1.Tlleti, because Ih/ll<<l (sincethe bubble is assumed to vibrate near the level I )and IA/aI < < I , equations (8.55) and (8.56)can be ~ a r t i a l l ylinearized to yieldL.r-,*i;. ,..... -..<-,:F-- i..:,-i.-' -..,, ' .,.-....,,,=I,-.The two pertinent forms of Lagrange's equations arewhere W A and w,are the vertical compoilents ofu a and u,.
There are ~ i ocontributions such as1:7..,,....-.-;.,A+.st:. ..'-. . - '.* ..,.I.,-,,295VERTICAL EXCITATION OF PROPELLANT TANKSin which use has been made of the relationP~=PO+P~~QI.Instead of investigating the solution ofequations (8.57), the same problem for a nonrigid tank is examined, since the rigid tanksolution is merely a special case of this. Thenecessary modifications can be performed readilyif it is noted that in eqliations (8.57) the dynamicpressureis -pix and the pressrlre gradient is.~ = - P Z .Hence, by analogy, the equationsbxfor a nonrigid tank can be written downimmediatelyThe dynamic pressure j7 must be calculatedbefore the location of stationary bubbles can befound from equations (8.58).
Since equations(8.58) are themselves somewhat approximate,a really complete pressure analysis is notwarranted. 9 relatively simple approach, verysimilar to that used in an ordinary wt~terhammer pressure analysis, is found to besatisfactory. That is, the pressure is assumedto be uniform across any horizontal sectionthrough the tank, the deflection of the tank wallis assumed to be the same as its static deflectioncaused by the same instantaneous pressure, andlongitudinal and bending deflections areneglected.First, the water-hammer wave velocity mustbe calculated; to do this the density of the,,+I.^,.--.^----:L.l..*-L V L I ~ ~ ~ ~ ~ S I L J U I~iI I Y the systemmust be determined.
The equivalent densityof the liquid-gas mixture is simplyllt.luU b t u-2MOE&1673%b3133L.gE2loo5S67Volume air I total volumeFIGURE8.24.-Water-hammer wave velocity in a liquid-airmixture.tank system is the sum of the three individualcompressibilitiesK is the reciprocal of the liquid's bulk modulus ;Kg= (rp)-' is the compressibility of an idealgas; K,=d/bE is the compressibility of thetank, where E is the modulus of elasticity ofthe tank material. By definition, the wavevelocity is c= ( p K ) - u 2 so that-112c={[(l--s)p,+sp,l[(l-s)~,+~+~])rpo bE(8.59)PZ (1 -s)p[+spgThis equation is shown graphically in figure8.24 for various values of po and a typicalvalue of d/b=39 for a small plastic test tank."where s is the gas to total volume ratio, is theliqrlid density, and P, the gas density.
Also,the effective compressibility of the gas-liquid-l1 The compressibility of the gas, K,, varies throughout the liquid, but it is sufficient here to evaluate itcaused byfor the ullage gas and neglect anydepth of submersion of the gas bubbles.296THE DYNAMIC BEHAVIOR OF LIQUIDSEquation (8.59) gives only a correct order ofmagnitude value of the wave velocity, chieflybecause the compressibility K t is overestimated.Furthermore, the exact quantity of gas isdifficult to measure, and it is not homogeneouslydistributed throughout the liquid as assumedimplicitly in equation (8.59) ; hence, an experimentally determined value of c should beused in numerical calculations.
But, equation(8.59) does give a correct qualitative picture,and rather small values of the speed of sound,c, should be expected.The fluid pressure is now determined bytreating the pressure wave as a one-dimensionalacoustic wave. Thus, if f(z, t) are the fluidparticle displacements measured from theirequilibrium positions, thenThe critical value of I, that is, the depth forwhich the bubble will remain stationary, canbe found from t,he first of equations (8.58) byrequiring that the average buoyancy over onecycle of motion be zero. That is(2 )Average of (a4-3A) - -+gwhich reduces toag+ Average of;']=0-.-where A, is the value of A corresponding to thecritical value of z=l.
Consequently, the critical bubble depth I can be calculated from112[+ e+clrJw2xo7The boundary conditions require thatr(h, t) =xo cos wt and bt/bz(O, t) =O; the bottomof the tank is z=h, the free surface is z=0.An appropriate form for t(z, t) is=O201sin -=[%(I+")]'"pglThis equation can be further simplified by usingthe nondimensional variables $=2wh/c andar=l/h. The final result isWcos - Zf (2, t ) = sC-COS wtcos hCSince the pressure is j5=pc2bf/bz, it can beseen thatwsin - zC~ = - p ~~ o cos ot(8.60)cos hCThe second of equations (8.58) can now besolved for A by using equation (8.60), with theresult that-Wsin - z-*COSwtA = Z [ + J [cos~ - ~-C ~h, ~ ~ Iwhere Q2=3rp1/a2pis the square of the naturalfrequency of bubble pulsations. Since Q isgenerally very large, the ratio w 2 p 2 can beneglected in comparison to unity.(Eq.
(61) can be specialized to the case of urigid tank and incompressible liquid-air mixtureby setting the bracketed term on the left-handside equal to one; that is, by letting +cor$+O.)Actual test results (ref. 8.35) are comparedwith equation (8.61) in figure 8.25. As can beseen, the correlation is very good, although anexperimental value for c was employed.
Itmay be noted that $ = T corresponds to a firstmode resonance. For values of $ greater oreven slightly less than T, bubbles do not sinkcompletely to the bottom no matter how largethe input vibration is, but, instead, they collecta t some lower depth. Bubbles inserted belowthis depth will rise to this level, as is indicatedin the figure by the fact that for some values ofI) there exist two critical values of 1 for a givenw'x,,/g. As a specific example, consider the$=4.0 curve with 6 g's acceleration. Bubblesabove a depth ratio a=0.16 return to the freeVERTICAL EXCITATION OF PROPELLANT TAM(SI1.0FIGURE8.26.-Bubble pattern at higher frequency.o00.20.40.80.6Bubble ueprh ratioQ=1.01-hFIGURE8.25.pVariation of required input acceleration withdepth ratio (ref. 8.35).iI3.-1.0 .6 .4.-1,,. .
. ."- -- j1..+.i-i- .+--.-.1...I-%*..<--:,4surface, while bubbles below this depth collectat a depth a=0.64.For values of + less than about T ,the bubblesalways sink completely to the tank bottom ifd q / g is large enough. For $)T, the bubblebehavior becomes increasingly complex since,for these values of 9, equation (8.61) has increasingly more values of 1 as roots. This isillustrated in figure 8.26 for a relatively largevalue of 4. Several bubble equilibrium planes(critical values of 1) are shown, with the upperone being unstable (bubbles move away from itwhc:: they are =l;crh+ly dis~!~cprl),the secondY--3--level stable, the third unstable, and so on.Bubble migration directions are indicated bythe arrows.A slightly different theory for sinking bubbleswas proposed by Buchanan et al.
(ref. 8.31).The essential result is thatwhere C is an experimentally determined constant. This equation is compared to testresults in figure 8.27, with correlation beingfairly good for C = l . However, equation(8.62) is not as general as the preceding equation (8.61).Explanation of Overall Bubble BehaviorWith the aid of the preceding analysis, it ispossible to explain completely the bubble behavior described in figures 8.19 through 8.22.At the start of the process, air bubbles arethrown into the body of the liquid by the freesurface sloshing motions.
The bubbles thatsink a t least to a dept.h 1 (the root of eq. (8.61)for the given input vibration conditions) willcontinue to sink to the tank bottom. As moreand more air bubbles are entrained and thensink, the wave velocity decreases (see fig. 8.24),so that $, which was initially considerablylessthan r , increases continuaiiy. bventuaiiy, avalue of $ equal tb T is obt,ained, a resonanceoccurs, and the cluster motion is very violent.As a result of the large quantity of air that isentrained during resonance, a value of $>rultimately results.
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