H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 102
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For example,in pump-fed engines, the success of the combustion control depends in large measure onprecise flow-rate control, which in turn meansthat the liquid viscosity must be held nearsome preset value.Temperatures and temperature variationsdepend on many complicated factors, includingsystem geometry and flight program. Whenthe tanks are embedded within the vehicle,little difficulty in maintaining uniform andconstant temperatures is expected; but whenthe tanks are part of the external vehiclestructure, large temperature differences canbe expected, and propellant temperature withinany one tank can vary considerably. Theenergy input is a strong function of the flightprogram.
For example, a geocentrically oriented satellite in a twilight orbit interceptsabout three times the direct solar energy asone in a noon orbit. The seriousness of thepossible temperature nonuniformity is clearif one calculates the temperature variationaround a thin-walled cylinder exposed to solarradiation on one side. With typical emissivityvalues and wall thickness, temperature differences of the order of 200' F are found(ref.
11.70). Acting to smooth out these largetemperature variations are diffusion and convection, which we shall now consider.where B is the coefficient of thermal expansion(at constant pressure) and L is a characteristiclength in the direction of the temperaturegradient. If the fluid is heated from above,it is stable and convection is suppressed.
However, if it is heated from below, it becomesunstable a t Rayleigh numbers of the orderof 1000-2000 (ref. 11.71). Under typicalorbital conditions, the Rayleigh number forthe gas phase will be rather small, and henceonly diffusive mechanisms are expected tooperate. Within the liquid, however, largeRayleigh numbers are expected, especially inbig booster tanks, and, if the g field is properlyoriented with respect to the temperature field,an efficient natural conk-ection process willevolve. Weinbaum (ref. 11.72) has studiedthe natural convection within a horizontalcylindrical tube, completely filled with liquid,and subjected to a circumferential wall temperature distribution. His generaiized treatment is particularly useful for purposes ofestimating the smoothing effects of conductionand laminar natural convection on the temperature profle.
Convection is suppressedwhen the Rayleigh number, based on thecylinder radius and the maximum wall temperature variation, is about 1200.In the gas phase, where the diffusive mechanisms are controlling, theoretical formulation430THE D Y N ~ I I C BEHAVIOR OF LIQUIDSis not difficult, though solution is another story.The basic equation governing diffusion of twoideal-gas species may be written as (in onedimension) l 4dcdc,dtJ~,=-Dzl--J-D22--D~tdxdcdx(11.114b)Here c, is the local modal concentration ofspecies i, JM<is the local mass flux of species i,which is produced by gradients in its concentration, gradients in the concentration of otherspecies, and temperature gradients, and JE isthe energy flux.
Note that there can be agradient in the inert gas concentration of aparticular species, even though it is not flowing,produced by gradients in temperature or in theconcentration of the other species.These equations, plus the conservation equh' tions for energy and mass, provide the bnsis foranalysis of simultaneous diffusion of mass andenergy. Wang (ref. 11.73) has solved a, rathersimplified model problem based on approximations to the equations. Even with theseapproximations, his results are indicative of theorder of magnitude of the effects.
In particular,he finds that in a typical situation, whereUDMH vapor diffuses through helium along a1.5-meter path, the contribution of the first andsecond and third ternis of equation (11.114c)is significantly larger than the "heat" conduction term (-k dtldx). The relative importanceof these energy transport mechanisms isstrongly influenced by the ratio of the averagepartial pressure of the diffusing vapor to thetotal pressure. I n the case studied, the ratioof the diffusive energy transfer to the conduction energy transport is 0.233 and 0.037 forpressure ratios of 0.40 and 0.175, respectively.Thus, at high partial pressure fractions, thel4 Here D,i denotes the coefficient in the lineardiffusion equation for species i due to a gradient inconcent,ratiori of species j. This is somewhat nt oddswith older literature.FIGURE11.43.-Estimatingtemperature effects at low g.diffusive t,ransport may well be the controllingprocess.
This points up t,he importance ofconsidering coupled diffusive mass and energytransfer, and particularly the importance ofconsidering them properly.Temperature differences within a tank resultin differences in temperature of the free surfaceof the liquid. The surface tension decreaseswith increasing temperature, and the resultinggradient in surface tension acts to pull moleculesin the interface along the interface towardregions of colder temperature. Surface-tensiondriven motions are probably very important inlow-g environments; they are beautifully illustrated on a laboratory scale in reference 11.12.As yet, these effects have not been examinedin sufficient detail, but we can make a rathersimple estimate of the conditions under whichthey are important.
Using dimensional arguments, we find that the pertinent dimensionlessparameter relevant to the thermally induceddistortion of the meniscus of figure 11.43 isLIQUID PROPELLANT BEHAVIOR AT LOW AND ZERO GSo, if<<1Theeffectsaresmall(ll.ll5a)>>1The effect's are large(11.115b)Note that a t very low g, the effects are mostpronounced. At zero g, the interface wouldswing to the cold side of the tank.Surface tension gradients can also give riseto cellular convection patterns. Consider, forexample, the two-dimensional channel of figure11.44. The surface tension gradient would setup interface motion, which in turn woiddaugment the heat transfer within the liquid.This interesting kind of "natural" convectionrecently was reviewed a t length by Scrirenand Sternling (ref.
11.74).Heat Transfer in CrvosenicTanks. -Theof heat transfer betweencryogenic tank and its surroundings and withinthe tank itself is quite different than in thecase of thepropellant tank.themeanof the 'lXfacesurround in^thepropellant tank can bewithin the temperature range in which theellant ant must be used, tank outer su'acetemperatures are a t least 100' F above the useHotCold431temperature of even the warmest cryogen.This means that if pressure rise rates withinthe tanks are to be held to acceptable limits,the tank must be well insulated. I t also meansthat heat transfer is always to the fluid, unlikethe usual circumstance with storable propellants. Because of the insulation, the heattransfer to the liquid is quite uniform over thewetted area and relatively constant throughoutthe flight.I t is important to know the details of themechanisms of energy transport within acryogenic tank.
On the basis of a first examination, it might be considered that a certainamount of heat will flow into the liquid throughthe insulation and that this energy rate willresult in liquid boiloff, thus causing the pressurein the tank to rise at a certain rate which iseasily calculated. This is not the case, however; it is found that the pressure rise is considerably less than would be predicted by ananalysis which assumed equilibrium betweecthe liquid and vapor phases. Consequently, itis necessary to know something of the detailsof the energy transport mechanisms withinthe tank so that an accurate estimate of thetemperature distribution inside the tank canbe made.Levy et al.
(ref. 11-75) have used an integralboundary layerto study the stratification problem. Since their methodologyapplicable, it will beseem to beoutlined here. Consider liquid in the cylindricaltank of11.45. Forlet usassume that the free surface is plane andthat the only heating occurs on the cylindricalwalls. Energy entering the tank wall formsa buoyant boundary layer adjacent to the wall,which rises toward the free surface.
Weassume that all the energy entrained in theboundary iayer enters and remains in a stratified region at the top, where the liquid temperature is greater than that in the central core.The general method of analysis is to writean expression for the rate of mass transfer tothe stratified region. This will be of the formwhere H - A is the working height of the naturalF 1 ~ ~ ~ ~ 1 1 .
4 . - S ~ r f a c e . t e n ~ i 0 ~ - d ~ i convection~ ~ ~ ~ ~ ~ ~ boundary~ ~ t i ~ ~ ilayer.~ ~ ~ ~ ~This~ . expression432I-+,THE DYNAMIC BEHAVIOR OF LIQUIDSStratified liquid regionI-',IGasFree surface1- I.--C - C -(1-9i--t ---- (--corehum,of'I -Natural I Iconvectionboundarylayer/liquidI-FIGURE11.45.-Liquid(whereGr * =gPriq/(kv2)(11.117b),- 1,\.I\A corresponding analysis for turbulent convection boundary layers gavea tratification.would be determined from establishecl theoryand correlations for natural convection fromrertical surfaces with specified heat flus, andit allows one to determiue the rate of growth ofthe stratified layer.
An energy b:rlsncr, plussome assumption about the internal temperature distribution in the stratified layer, thensuffices to fix the interface temperuturc.The idealizatio113 employed in writing thee n e r a balance on the stratifled layer willdepend on the conlposition of the ullage gas.If inert gas pressurization is used, the liquidwill be strongly subcooled below the totalpressure, and the only place where evaporationwill occur is a t the upper surface.
One couldassume that the ullage gas is saturated withpropellant vapor but, in many cases, satisfactory results may be obtained by neglectingany energy or mass transfer from the interface.Bailey e t al. (ref. 11.76) did this, and furtherassumed that the temperature was uniformthroughout the stratified region. Levy e t al.(ref.
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