H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 93
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T o determine theenergy change associated with a change ofstate in a capillary system, consider the twocones with a cylindrical mass of liquid stretchedbetween, as sholvn in figure ll.lO(a). Thedotted lines denote the boundaries of a systemof fixed mass, and we imagine that theseboundaries are free to move in a manner whichkeeps the system volume constant. The shapeof the cones is adjusted so that the liquid retainsits cylindrical shape, el-en if the contact anglechanges.When the separation between the cones isincreased by an infinitesimal amount dx, thework done on this system ( P d V work is zero forthis constant volume system) will be dW=Fdx.The instantaneous force, F, may be determinedby a force balance on a second system, definedin figure ll.lO(b).
The pressure differenceacross the interfnce is P,-Po=a/r, so the forceF is a . The surface tension force on thesystem is 2 m r , and, therefore, the force balanceindicates that the cone force, F, is 2aur-rar-rar. The work done is thereforeThe radius may be espressed t i s lt function ofthe volume using the constraint thitt the liquidvolume remain fixed. We finddV=2nrl dr-rFd1+2rr2dr -0-tan 8(11.17)The length change dl is related to dx and dr byCombinirlg equations (11.16) and (1 1.17), onefindsdJV=arr(dl-2drltan 8)(11.19)This can be put in a more convenient. form byobserving that the incremental change in theinterface area is d A = 2 ~ r d l - - 2 r l d r , and theincremental change in the wetted ttretlis dAW=4rrdr/sin 8. Combining equations(11.17) and (11.19), we can obtainThe case of constant e is of particular interest.For this case, we define the capillunj area,IDr0LIThe work is then related to the cttpillary area byn---------- ---Pi:~+vrFdW=adA,,'------------Consider'/I(b)FIGURE11.10.-Computing(11.22)pthe energy change in acapillary eyetem.now u, reversible change of statefor which the entropy change is related tothe energy transfer as heat to the capillarysystem bydS=dQ/TLIQUID PROPELLANT BEHAVIOR AT LOW AND ZERO GAn energy balance on the system givesdU=d&+dW(11.24)where U is the system internal energy.
Combining these two equations, we obtain theGibbs equation of the capillary system (ref.11.15)We next differentiate the functional relationship U(T, A,)Substituting into equation (11.25), \ve obtainFrom which it follo~vsthatXow, consider the Helmholt,~free energy,F= U - TS. Different'iating,Upon combination I\-ith equation (11.25), wefinddF=adA,-S d T(11.30)Since dF' is exact, we hare the IMax\vell relationso t,hat equation (11.28) becomesi?!,Iliquid, and is a funct,ion only of temperature( r e .
11.16). The termrepresents the energy associated with the capillary forces. Note that the capillary energy perunit of capillary area is also a function only oftemperature.Int,egrating equation (11.31), a t constant,T, we finds=(-$)A.+SO(T)(11.35)where Sorepresents the entropy of the liquidphase (ref. 11.16). -(da/dT), the capillaryentropy per unit of capillary area, is also nfunction solely of temperature. Since dald Tis negative, the entropy increases during anisothermal stretching.The Helmholt~zfree energy is thenwhere Fo=Lio-TS,,.We see that the surfacetension may be interpreted as t,he capillaryHelmholtz free energy per unit of capillaryarea; this is somet,imes used as n basicdefinition of u.Now, let us consider s capillary system whichreaches equilibrium irreversibly without a workinteraction with its environment, such as thereorientation of propellant in a system transisting from a high-g t,o a zero-g st,ate.
Thefirst law is then simplyand the second la\\- givesMeasurements indicate that the surfacetension is a function only of temperature, andhenceforth we consider this to be the case.Integrating equation (11.32) a t fixed T, wehaveu=(u- T -$)A,+u.(T)395(11.33)where Uo represents the system energy whenA,=O, that is, the internal energy of t,heThe change in ent,ropy is related to the inst,antaneous state ( U and A3 of the system throught,he Gibbs equation, equation (11.25).
Combining with the two equations above, me findwhich tells us that, during the irreversibleapproach to equilibrium,396THE DYNAMIC BEHAVIOR OF LIQUIDSSince the capillary area continually decreasesduring the process, the final equilibrium stateis a configuration of m i n i m u m capillary area.A similar result is obtained for an isothermalcapillary system when the gravitational forceis not zero. The work done on a capillarysystem b y external force when its center of massis lifted an amount dh iswork interaction with its environment, suchus the propellant reorientation from one gloading to another.
An energy balance for anelemental step in bhis process then giveswhere 111 is the liquid muss, nnd !I is the localgravitational acceleration.The total Work done On capillary system isthenclh.d w =U ~ A(11'42)Applying the first la\\- of thermodynamics~ h ChaIlge,in entropy is related to the changein state (U, A,, and h) through the Gibbseqllption,(1 1.45). Combining lViththe two equations above, we learn thnt thechanges in state which occur during- the irreversible process rnust be such thatThe energy transfer as heat and the entropychange must be such that the second law ofthermodynamics forreversible process issatisfied; that is,T dS=d(Jre,.(11.14)The tern1 A[qh is the graz.itationa1 potentialenergy of the system; sometimes the term aA,is called the capillalrj potential energy. Thestable equilibrium state is then the state ofminim i r m total potential energy, whereFor this system the Gibbs equation, describingchanges of state, is obtained by combinationof the first and second lams, and isP E = u A c + Mgh,+dS=-1Tu'MsdU-- dAc-'dhTTUpon differentiating the functional relationshipU ( T ,A,, h), and following the steps leading toequation (11.28), one finds that equation(11.28) holds for this system also, providedthat the derivatives are taken a t constantT and h.
Similarly, the Maxwell relations,equations (11.31) and (11.32), both hold if his held constant in forming the partial derivatives. Then, integrating equat,ion (11.32), onehasThe function j (h) is determined by consideringa reversible isothermal elevation a t constantA,, and is simply f(h)=Mgh. The entropy isalso given by equation (1 1.35).Now, consider a capillary syst#em which isallowed to approach equilibrium without anyand the seco~ltflun- for the irreversible processisTdS-dQ>O(11.48)(11.50)This is an extremely important result, for itprovides the basis for the determination ofstable equilibrium configurations in capillarysystems.
We shall use it frequently in thefollowing section.11.3CAPILLARY HYDROSTATICSPreferred Configurations in Zero gThe thermodynamic analysis in section 11.2indicated that the stable equilibrium state of acapillary system in an isothermal zero-g environment is the state where the potentialenergy, PE=oAC, has the least value. Theremay be several states for which small perturbations in configurat,ion result in an increase inPE; all of these states, except for the one oflowest PE, are metastable, for with a sufficientlylarge disturbance the system will pass to thestable state. This behavior is indicated infigure 11.11. We can use this idea to determinewhere liquid is most likely to be found in acont,ainer.
Since the surface tension dependsLIQUID PROPELLANT BEHAVIOR AT LOW AND ZERO GpoA1Metastable statesPEThe most stableConfigurationFIGURE11.11.-StableWall boundDropstates.only on temperature, it is constant in anisothermal system, and we need only to investigate the capillary area -4,.
Thus the calculation is entirely geometrical.I t is important to realize that orie cannot besure from-these considerations that the sys temwill indeed be in this stable state; it may be('trapped" inametustablestate. Theanalysismerely indicates what is niost likely, whichoften is too weak it prediction for purposes ofconservn tive design.Consider first a collection of liquid dropsaway from interactiori with walls. Suppose thedrops are floatinr aimlessly itbout, and vie seekto learn if there will i ) e :my tendericy for then1to merge up011 contact with one another..Let us take the simple case where all N dropshave the same diameter, D. If V representsthe total liquid volume, thenFree(a- -- -Wall boundBubbleFreeLb)FIGURE11.12.-Capillaryareas for wall-bound and freedrops and bubbles.forming orie large bubble.
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