H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 103
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11.75) also neglected ullage interaction,but assumed that the temperature profile inthe stratified region maintained a similar shapeduring the heating period. Using establishedresults for laminar natural convection boundarylayers, they predicted that the thickness ofthe stratified region would beT ~ e r e isB the coefficient of thermal expnrisio~iof theliquid phase, q the heat flux, and r , c,, v, and k thefluid properties of the liquid phase.The turbulent relation is reconi~mended forGr*Pr>lOn, which will nornlally be the casein large booster tanks.Levy e t al. (ref.
11.75) also report someexperiments, from which some idea for theassumed temperature profile can be obtained.Clark and Barakat (ref. 11.77) have examined the transient heating of a liquid partlyfilling a two-dimensional channel in an esnctnumerical solution of a model problem. Theyassumed that the liquid motion is laminar,its density is a linear function of temperature,and i t is a t rest with a uniform temperature a ttime zero. The mathematical treatment ofthis problem is as a boundary-initial valueproblem.
The non-steady-state energy, continuity, and momentum equations, written interms of the stream function, were reduced todifference equations and solved numerically.The interface was assumed to be a surface ofzero shear. Upper and lower surfaces of thetank were considered to be: (1) insulated,(2) subject to a uniform heat flux, or (3) kepta t a constant temperature. Tank side wallswere: (1) subject to constant heat flus, or (2)kept a t constant temperature. Results oftheir calculations for liquid nitrogen standingapproximately 30 centimeters deep in a 15centimeter-wide tank show that temperatureis uniform across the tanks except near the tankwall. Irregular wavelike streamline motionnear the irltersection of the free surface and theLIQUID PROPELLANT BEHAVIOR AT LOW AND ZERO Gtank wall mas also predicted, and this has beennoted in some experimental flow visualizationwork.Although in principle the method used byClark and Barakat is fundamentally the soundest for laminar flow, it cannot be used for turbulent flow, which seems to be of primary interestin large tanks.
Moreover, even the simplemodel problems require a large amount of computer time. Nevertheless, the "exact" solutions do provide a valuable basis for comparisonwith the approximate methods.I t appears that the most interesting heattransfer problems in propellant tanks may wellinvolve transient natural convection.
A recentstudy by Schwartz and Adelberg (ref. 11.78)showed that a very long time is requiredto obtain a steady-state natural convectionboundary layer in a typical large tank. TheSchwartz-Adelberg survey is recommended asa starting point for any designer faced withmaking a propellant heat transfer estimate.Boiling Under Low-g Conditions!iThere are a number of important boiling heattransfer problems encountered in low-g environments. A direct problem related to propellantsis that boiling may cause the liquid to be displaced from known or preferred locations by theattendant phase change.
Fortunately, burnoutis not a problem in propellant tanks, since theheat transfer rates encountered are very modest.A number of workers have investigated thevarious forces acting on bubbles as they formand grow in a nucleate pool boiling condition.Adelberg (ref. 11.79) has presented one of themore recent and comprehensive studies of thissort. He made an order-of-magnitude estimateof the various forces based on data of Ellion(ref. 11.80) for experiments using 8 wateraerosol solution. Not surprisingly, this indicates that inertial and surface forces controlthe early stages of bubble growth. Buoyantand drag forces are shown to be approximatelytwo orders smaller.Adelberg also estimated the forces as a function of At=t,,ll-t,,,.He found that at highAt's the bubbles are removed from nucleationsites by dynamic effects, while at lo~verat'sbuoyant forces take this role.
This suggests433that at low At's the g level mill affect the character of nucleate pool boiling, while at higher At'sit should not.Keshock and Siege1 (ref. 11.81) present asimilar order-of-magnitude analysis, based ontheir own data. Their experiments mere carried out at go and at reduced g conditions. Theliquids used on these experiments were distilledwater and a 60-percent water-sucrose solution.The authors indicate that when bubble growthrate is large, nucleate boiling is essentially independent of g level. In this case, bubble departure \\-as governed by inertial and surfacetension forces. For more slowly growing bubbles, departure rt-as governed by buoyant forcesand surface tension forces.
Viscous forkes wereindicated to be of little importance.Merte and Clark (ref. 11.52), Sherley (ref.11.83), Steinle (ref. 11.84), and Usiskin and Siegel (ref. 11.85) have carried out nucleate boilingexperimental studies under near-v-eightless conditions. Different liquids were used by theseinvestigators, and hence it is difficult to relatetheir results. 13011-ever,a t high heat transferrates, the f i s t tw-o studies indicate that heat fluxis essentially the same function of t,-t,,,,innear trveightlessness as under standard gravitational conditions, which agrees with the qualitative arguments just presented. Ho\\-ever,the quantitative effect of low g on nucleateboiling under conditions of low heat fluxes isnot established.
Since this is the most important region for boiling in propellant tanks,experiments in this range are badly needed.Burnout heat transfer rates are indicated bymost investigators to be significantl? reducedby reduction in acceleration level. Merte andClark (ref. 11.82) tentatively give the zero-gto 1-g burnout heat fluxratio as 0.41. I t shouldZe zotec! thnt t E s mas for 9. spherical heateron which a stable film can be sustained underzero g.
However, on cylinders the film tendsto be unstable, even a t zero g, due to the surfacetension effects and, hence, less drastic reductionsin burnout heat flux are expected.Rex and Knight (ref. 11.86) report an experimental study to propane in a spherical tankunder nearly zero g. The nucleate boiling heatt,ransfer rate for fixed At was approximately30 percent of the value at 1 go.434THE DYNAMIC BEHAVIOR OF LIQUIDSThere is a rather large body of literaturedealing with condensing in zero g. However,as this work is aimed primarily a t tube condensers, and is therefore not particularly relatedto propellant problems, we will omit its discussion here.11.7EXPERIMENTAL SIMULATION OF LOW-g ENVIRONMENTSSimulation RequirementsBefore we investigate the various meanswhich one can use to simulate low gravityenvironments, let us examine the conditionswhich one would like to simulate.We first consider hydrostatic configurations.U ,t,he pertinentThe Bond number, B O = ~ ~ L ' / isdimensionless parameter characterizing hydrostatics.
In addittion, the contact angle mustbe maintained for proper modeling. Thesimplest means for maintaining contact angleis to use the same fluids and surfaces in thetest which would normally be employed in thespace system. Then, in order to model environmental condit'ions of a prototype with a scalemodel, the g imposed on the model would haveto be selected to maintain the proper value ofthe Bond number.
If the same fluid is used forboth model and prototype, a KO-scale modelgoof a system designed to operate inshould therefore be tested in ago environment. Modeling of the Bond number is not.so important if its value is significantly lessthan unity (i.e., in the range of "zero-g hydrostatics"). For example, if the prototype willoperate with the Bond number equal to 0.001and a t lo-' go, testing the KO-scale model a tlo-' go(Bo = 0.01) would likely provide adequatemodeling.
However, if the Bond number ofthe prototype will be of order unity, modelingof the Bond number is essential.The relative time response for the model willbe determined by the length and gravity scaling.In the gravity dominated regime (Bo> l ) , thetime constant varies as(see eq. (11.lla)).If the same fluid is employed, and if g will haveto appropriately be increased to maintain themodel Bo, the response times of the prototypeund model will be related bymg>A :io-scale model would therefore have a timeconstant of the order of KO of the prototype's.I n the capillary-dominated regime (Bo< 1),the time constant varies as L3l2 for a givenfluid.
Therefore, in this regime the model andprototype time constants would also be relatedby equation (11.119). These arguments suggest that, in' the region Bo .= 1, where bothcapillary and gravity forces are important, thetime constant should also scale like L3/' for ngiven fluid. This provides a strong motivationfor using the design fluid in any model tests.The desirability of visual observntion in themodel tests may well exclude the use of solidsurfaces.
I n the event that transparent containers are used in the tests, the fluid should beselected to provide the proper contact angle.The appropriate dimensionless parameterswould then have to be employed to determinethe proper scaling.I n hydrodynamic situnt ions, the Webernumber, Reynolds number, and Bond numbershould be maintained to provide accuratescaling. While scaling of the Bond numbercan be accomplished by proper selection of gfor the model test, the Reynolds and Webernumbers cannot both be maintained if thedesign fluid is employed, since velocity andcharacteristic length appear in different maysin the dimensionless group. I n situationswhere interface stability is of prime concern,it would appear best to forego maintenance ofthe Reynolds number and keep the properWeber number.
This should be reasonable aslong as the resulting Reynolds number is suchthat the model and prototype operate in thesame flow regime (i.e., laminar or turbulent).We shall now discuss some experimentaltechniques currently used to investigate hydrostatics and hydrodynamics a t low g.<Bench TestingI t may seem strange to suggest testingspace system on a bench at 1 go. However, inlarge vehicles subjected to very small loadings,the Bond number may be sufficiently high thatLIQUID PROPELLANT BEHAVIOR AT LOW AND ZERO Ga bench test on a small-scale model will give arealistic appraisal of the prototype performance.For example, suppose we are interested inknowing the sloshing frequencies of the propellant in a large tank of peculiar configurationa t loe4 go. Neglecting viscous effects, thedimensionless frequency, 02=pL3wZ/a,is a function only of the Bond number.
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