GP-B Post-Flight Analysis—Final Report - Gravity Probe B - Stanford ...

the time scales under consideration, it is not reasonable to assume that it is mechanically decoupled. As notedabove, over short time scales the evaporating liquid will adiabatically compress the vapor, and this will causeboth a density and a temperature increase throughout the bulk of the ullage vapor. In fact, an estimate (based onthermodynamic data) of the effect of heating 1.85 K liquid by 10 mK indicates that the vapor will becompressionally heated by 24 mK, or over twice as much as the liquid! It is for this reason that the vapor is mostlikely to be convectively stable, as was asserted previously. In order to test this hypothesis, the HPM data werere-analyzed a third time under the assumption of adiabatic compression of the vapor. Since this assumptiontakes the vapor off the saturated vapor pressure curve, the density and specific internal energy changes in thevapor were calculated using the adiabatic compressibility parameter derived from tabulated thermodynamicdata. [Ref.: V.D. Arp, and R.D. McCarty, NIST Technical Note 1334, 1989.] The results of the analyses of theHPM data using each of the three assumptions together with the actual depletion date are shown in Figure 12-9.It can be seen that the case where adiabatic compression has been assumed yields results that are intermediatebetween the other two. This is not surprising since adiabatic compression by a given pressure increment takesless energy than equilibrium compression (which is more nearly isothermal) thereby causing the analysis toproduce a larger liquid mass estimate. The reduction in mission duration prediction error achieved by assumingadiabatic compression rather than equilibrium compression is fairly modest, however, -3.8% instead of -5.3%. Itis not clear why the systematic error is this large (implied error of -31% for HPM #6 after correcting for the 13%scale factor error), but it is clear from the algebraic expression used to calculate the amount of liquid that theproblem is likely to involve an error in the presumed properties of the vapor, either the density or the specificinternal energy, or both.12.9 HPM Results: ConclusionsThe following appear to be reasonable conclusions based on our experience with the HPM technique:1. The calorimetric HPM technique is well (and uniquely) suited for use in measuring the quantity ofremaining superfluid liquid helium and in estimating time of depletion. Unlike an integrated flow metertechnique, the end-of-life estimate obtained from a sequence of HPM operations is insensitive to scalefactor(e.g., heater calibration) errors. If the initial quantity of superfluid is known with confidence, it canbe used to detect and correct errors of this sort.2. For 1.8 K operation, the largest source of systematic error in the EOL estimate appears to be inaccounting for the effect of the ulllage vapor. We analyzed our data using three different assumptionsregarding the post-HPM state of the vapor: a) the vapor stays in complete equilibrium with the liquid; b)the vapor is completely decoupled from the liquid and does not change state as a result of heat applied tothe liquid; and c) the vapor is adiabatically compressed by the increase in vapor pressure (i.e., the vaporis mechanically coupled to the liquid, but is thermally isolated). The first assumption is implausiblebecause the thermal equilibration time of the vapor is much longer than the measurement time. It led toa 5.3% underestimate of lifetime. The second assumption is implausible because mechanical couplingdoes cause rapid change in the state of the vapor when the liquid is heated. This assumption led to a 2.3%overestimate of lifetime. The last assumption is plausible if the ullage volume is large enough that thethermal diffusion time is long compared to the measurement time. This assumption led to a 3.8%underestimate of lifetime. It should be noted that one would logically expect that the actual result shouldlie somewhere between the predictions base on the first and third assumptions since, in reality, thesystem should be somewhere between isothermal and adiabatic. The fact of the matter, however, is thatthe actual result was between the second and third assumptions. Why this should be the case is not clear.3. Ultimately the vapor does re-equilibrate with the liquid so that, in principle, it should be possible to waitlong enough after the HPM operation for the equilibrium assumption to be true. Unfortunately this israrely a viable option. In a passive vent system (the most common configuration), the change in flow ratecaused by the HPM will cause long-term instability in the vent system (including the vapor-cooledGravity Probe B — Post Flight Analysis • Final Report March 2007 347