TFAWS Interdisciplinary Paper Session Two-Pendulum Model of Propellant Slosh in Europa Clipper PMD Tank Wanyi Ng & David Benson, NASA GSFC, 597, Civil Servants, Authors Presented By Wanyi Ng Thermal & Fluids Analysis Workshop TFAWS 2017 TFAWS August 21-25, 2017 NASA Marshall Space Flight Center MSFC ∙ 2017 Huntsville, AL
Outline • Objective • Background • Results and literature verification – Mass – Frequency – Damping ratio – Hinge location • Conclusions 2
Objective Model propellant slosh for Europa Clipper using two pendulums such that controls engineers can predict slosh behavior during the mission. 3
BACKGROUND 4
Motivation • Importance of predicting propellant slosh – Sloshing changes CM (center of mass) of spacecraft and exerts forces and torques on spacecraft – Avoid natural frequencies of structures – Size ACS (Attitude Control Systems) thrusters to counteract forces and torques • Can model sloshing fluid as two pendulums with specific parameters (mass, length, damping) 5
Background Notional tank • Europa Clipper tanks and PMD – Bipropellant system – Cylindrical with domed top and bottom – 8-vane PMD (propellant management device) CFD Simulation • CFD (computational fluid dynamics) data used as “real” slosh behavior – Have data for two propellants at three fill fractions each – Initial condition of 15 degree free surface offset, released and allowed to settle – CFD requires long computing time -> Need a computationally simple model 6
Background • Pendulum model – Model fluid movement as two pendulums Forces exerted by attached to central axis of the tank fluid on tank −𝒏𝑴 ሷ 𝜾 – For each CFD data set, find parameters: mass, frequency, damping ratio, 𝒏𝑴 ሶ 𝜾 𝟑 𝒏𝒃 attachment height 𝐷𝑁 𝑢 = 𝑛𝑀𝑡𝑗𝑜𝜄 𝑢 𝜊𝜕 𝜕 1 − 𝜊 2 sin 𝜕 1 − 𝜊 2 𝑢 + cos 𝜕 1 − 𝜊 2 𝑢 = 𝑛𝑀𝑡𝑗𝑜 𝜄 0 𝑓 −𝜊𝜕𝑢 7
Existing Literature • Sector tank mode Annular tank SP-106 (1966), SwRI (2000): (top view) mode (top view) Analytical equations and empirical correlations for PMD damping and frequency – Includes bare cylindrical (no PMD), Tank sector, and annular tanks Wall Cassini paper illustration of • Cassini slosh paper (1994): Two double pendulum model pendulum model – Slosh around PMD was modeled as combination of sector and annular slosh modes – Two separate pendulums to model two slosh modes – Static mass component at bottom that experiences little movement 8
METHODS OVERVIEW 9
Generate CFD Data • Propellants: NTO and MMH • Fill fractions: 25%, 50%, 85% • Data: CM, Force, Moment (all 3 axes) 10
Find Initial Guesses • Curve fitting by finding parameters in pendulum equation that most closely match CFD • Trying to resolve CFD into two pendulums • Peak-to-peak values • -> Initial guesses for damping and frequency of each pendulum • Note much higher damping before first peak 11
Find Parameters to Fit CM Data • Matlab’s fsolve(x) • -> Mass, damping, and frequency parameters to fit CMx CFD data • Refine and iterate 12
Compare Sum of Pendulums to CFD Data • Sum of two pendulums generates model for propellant slosh • Should match both CM and Force data 13
Mean Error in Force • Metric to quantify accuracy of fit: mean absolute difference between CFD force and pendulum model force 𝑜 1 𝑜 𝑏𝑐𝑡 𝐷𝐺𝐸 − 𝑞𝑓𝑜𝑒𝑣𝑚𝑣𝑛 1 • Select methods that minimize this 14
RESULTS AND LITERATURE COMPARISON 15
Basis for results • Coordinate system – origin at z top of tank • Parameters prioritized fitting y into page x x the behavior after the first peak • Two pendulum model is an approximation only – PMD does not create a perfectly sector nor annular tank and is only Approximate a fraction of tank height shape of PMD vanes – Parameters not constant over time – Model does not scale well with high fluid displacements 16
Mass Participation Fraction Mass participation fraction vs. fill fraction 0.16 0.14 NTO Mass fraction 0.12 Pendulum 1 0.1 MMH 0.08 Pendulum 1 0.06 0.04 NTO 0.02 Pendulum 2 0 MMH 0 0.25 0.5 0.75 1 Pendulum 2 Fill fraction • Pendulum mass as a fraction of total fluid mass • Monotonic trends • Mass fractions are identical between NTO and MMH • Piecewise linear fit – First two fill fractions – fluid partially submerges PMD, sloshing occurs between vanes – Last fill fraction – fluid completely submerges PMD, different slosh behavior 17
Frequency Frequencies vs. Fill Fraction 0.120 NTO Sector 0.100 Frequency (Hz) 0.080 MMH Sector 0.060 NTO Annular tank 0.040 0.020 MMH Annular 0.000 Tank 0 0.2 0.4 0.6 0.8 1 Fill fraction • Function of pendulum’s length and acceleration • Monotonic trends • Frequencies are identical between NTO and MMH • Frequencies for the two pendulums converge as fill fraction increases – Sector and annular slosh modes become less distinct as PMD becomes fully submerged 18
Frequency - Literature Comparison 1 Frequencies vs. Fill Fraction Cassini Paper Frequencies vs. Fill Fraction 0.120 NTO Sector 0.100 Frequency (Hz) MMH Sector 0.080 0.060 SP-106 (Bare Tank) Analytical 0.040 Bare Tank NTO Annular (Annular Tank) 0.020 tank 0.000 MMH Annular 0 0.5 1 Tank Fill fraction • Left: Cassini paper referenced SP-106 for an analytical equation for slosh frequency in a bare tank (cylindrical tank with no PMD) and compared it to the frequencies of their two pendulums • Right: Similar trends to Cassini found in Europa pendulum model frequencies • Sector and annular slosh modes converge towards bare tank frequency as PMD becomes more submerged (fully submerged at 85% fill fraction for Europa tank) 19
Frequency – Literature Comparison 2 Frequencies vs. Fill Fraction Frequencies vs. Fill Fraction 0.14 0.14 NTO Sector 0.12 0.12 Frequency (Hz) Frequency (Hz) MMH Sector SP-106 Analytical 0.10 0.10 Sector Tank 0.08 0.08 NTO Annular tank 0.06 0.06 0.04 0.04 MMH Annular Tank 0.02 0.02 SP-106 Analytical SP-106 Analytical 0.00 0.00 Sector Tank Annular Tank 0 0 0.5 0.5 1 1 SP-106 Analytical Fill fraction Fill fraction Annular Tank • SP-106 references tables (Bauer, 1963) for an analytical equations for sector and annular slosh frequency • Function of acceleration, geometry, and fluid height • Pendulum frequencies are close to analytical equation frequencies • Differences between analytical and pendulum fits due to: – PMD is not exactly a sector/annular tank – Half-dome bottom approximated as flat bottom – at 25% fill fraction, sloshing fluid is almost entirely in the dome – PMD doesn’t include entire height of tank – at 85% fill fraction, PMD is completely submerged 20
Damping Ratio Damping Ratio vs. Fill Fraction 0.4 NTO 0.35 Damping Ratio Pendulum 1 0.3 0.25 MMH 0.2 Pendulum 1 0.15 NTO 0.1 Pendulum 2 0.05 0 MMH 0 0.25 0.5 0.75 1 Pendulum 2 Fill fraction • Monotonic trends • Slightly higher damping ratio for higher dynamic viscosity (MMH) 21
Damping Ratio – Comparison 1 Damping Ratio vs. Fill Fraction Damping Ratio vs. Fill Fraction 0.40 0.40 NTO Pendulum 1 0.35 0.35 Frequency (rad/s) Frequency (rad/s) MMH Pendulum 1 NTO SwRI Theoretical 0.30 0.30 Bare Tank 0.25 0.25 NTO Pendulum 2 0.20 0.20 0.15 0.15 MMH Pendulum 2 0.10 0.10 MMH SwRI Theoretical NTO SwRI Theoretical 0.05 0.05 Bare Tank Bare Tank 0.00 0.00 MMH SwRI Theoretical 0 0 0.25 0.25 0.5 0.5 0.75 0.75 1 1 Bare Tank Fill fraction Fill fraction • Mikishev and Dorozhkin found correlation for damping in a bare tank • Function of geometry, acceleration, viscosity, and fluid height • Scales by correction coefficient for domed bottom • Pendulum damping within order of magnitude of analytical prediction • Pendulum damping less sensitive to viscosity than analytical prediction – viscous vs. drag forces 22
Length and Hinge Location Hinge height vs. fill fraction 1.5 1 NTO Pendulum 1 Hinge Height (m) 0.5 MMH Pendulum 1 0 NTO Pendulum 2 0 0.25 0.5 0.75 1 -0.5 MMH Pendulum 2 -1 NTO Static Mass -1.5 MMH Static Mass Fill fraction • Origin is top of tank • Pendulum bobs stay within fluid • Monotonic values for pendulum heights • NTO and MMH heights are close but not identical 23
Length and Hinge Location NTO 25% fill NTO 85% fill NTO 50% fill Approximate tank wall Pendulum at 15 degree offset 24
PLOTS COMPARING PENDULUM MODELS AND CFD DATA 25
NTO 25% Fill Fraction 26
NTO 25% Fill Fraction 27
NTO 25% Fill Fraction 28
NTO 25% Fill Fraction 29
NTO 50% Fill Fraction 30
NTO 50% Fill Fraction 31
NTO 50% Fill Fraction 32
NTO 50% Fill Fraction 33
NTO 85% Fill Fraction 34
NTO 85% Fill Fraction 35
NTO 85% Fill Fraction 36
NTO 85% Fill Fraction 37
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