Solubility, Sea Properties, Effervescence and Ballast Design

SOLUBILITY, SEA PROPERTIES, EFFERVESCENCE AND BALLAST DESIGN

FOR AN EXTRATERRESTRIAL SUBMARINE

PETER MEYERHOFER

Submitted in partial fulfillment of the requirements for the degree of

Master of Science

Mechanical Engineering

CASE WESTERN RESERVE UNIVERSITY

January 2019

CASE WESTERN RESERVE UNIVERSITY

SCHOOL OF GRADUATE STUDIES

We hereby approve the thesis of

Peter Meyerhofer

candidate for the degree of Master of Science.

Committee Chair

Prof. Yasuhiro Kamotani

Committee Member

Prof. Paul Barnhart

Committee Member

Prof. Joseph Prahl

Date of Defense

October 24th, 2017

*We also certify that written approval has been obtained for any proprietary material

contained therein.

Table of Contents 1 Introduction ...... 14 2 Literature Review ...... 18 3 Solubility Model ...... 21 3.1 Background ...... 21 3.2 Analysis and Filtering ...... 26 3.3 Analytical Model ...... 30 3.3.1 Functional Form ...... 30 3.3.2 Curve Fit to Data ...... 31 4 Titan Sea Properties ...... 37 4.1 Solubility at Titan Temperatures ...... 37 4.2 Assumptions ...... 38 4.3 Governing Equations ...... 39 4.4 Chemical Potential Model...... 44 4.4.1 Imposed Temperature ...... 44 4.4.2 Constant Enthalpy...... 46 4.4.3 Constant Entropy ...... 49 4.4.4 Constant Mole Ratio ...... 49 4.5 Imposed Solubility ...... 50 4.5.1 Surface Solubility (SS) ...... 50 4.5.2 Full Solubility (FS) ...... 50 4.6 Assessment ...... 52 4.7 Properties ...... 54 4.8 Uncertainty ...... 56 5 Effervescence ...... 60 5.1 Problem Statement ...... 60 5.2 Effervescence Model ...... 62 5.2.1 Skin Temperature ...... 62 5.2.2 Applied Solubility...... 64 5.2.3 General Bubble Nucleation ...... 65 5.2.4 Bubble Incipience and Nucleation Site Density ...... 66 5.2.5 Bubble Growth ...... 67 5.2.6 Volume and Area Coverage ...... 70 5.2.7 Pressure Drop in Propellers ...... 82 5.2.8 Solution Method ...... 85 5.3 Numerical Results ...... 86 6 Ballast Trade Study ...... 95 6.1 Submarine Literature Review ...... 95 6.2 General Considerations ...... 96 6.2.1 Tank Size and Sink Rate ...... 97 6.2.2 Uncertainty ...... 100 6.2.3 Parts ...... 103 6.3 Pump System ...... 104 6.3.1 Analysis ...... 105 6.3.2 Operations ...... 106 6.3.3 Parts ...... 108

6.4 Bladder Only ...... 109 6.4.1 Analysis ...... 109 6.4.2 Operations ...... 110 6.4.3 Parts ...... 111 6.5 Noncondensible Gas without Separator ...... 112 6.5.1 Analysis ...... 112 6.5.2 Operations ...... 116 6.5.3 Parts ...... 118 6.6 Noncondensible Gas with Separator ...... 119 6.6.1 Analysis ...... 119 6.6.1.2 CVs and Equations ...... 121 6.6.1.2.1 Ramp Stage ...... 121 6.6.1.2.2 Expansion Stage ...... 123 6.6.1.3 Solution Method...... 124 6.6.1.3.1 Ramp Stage ...... 124 6.6.1.3.2 Expansion Stage ...... 126 6.6.1.4 Numerical Results and Discussion...... 127 6.6.2 Operations ...... 133 6.6.3 Parts ...... 134 6.7 Gas Purification ...... 135 6.7.1 Refrigeration ...... 137 6.7.2 Distillation ...... 139 6.7.3 Membrane ...... 142 6.7.4 Adsorbent...... 144 6.7.5 Mass Comparison ...... 146 6.8 Nitrogen Gas with Separator ...... 148 6.8.1 Unmodified Vapor System ...... 148 6.8.2. Vapor System with Heater ...... 149 6.8.3. Liquid Storage ...... 151 6.8.4. Liquid Actuator ...... 154 6.8.5 Parts...... 155 6.9 Nitrogen Pump System ...... 157 6.9.1 Assessment ...... 158 6.10 Sea Boiling System ...... 159 6.10.1 Assessment ...... 160 6.11 Comparison ...... 160 7 Phase II Submarine...... 162 8 Conclusion ...... 163 9 Future Work ...... 164 10 Bibliography ...... 166

List of Tables Table 1-1: Submarine design requirements for Phase I and Phase II...... 17 Table 3-1: Nitrogen-methane VLE data...... 23 Table 3-2 Nitrogen-ethane VLE data...... 24 Table 3-3 Nitrogen-methane-ethane VLE data...... 24 Table 3-4: Constant values for the Antoine equation for vapor pressure, evaluated in units of K and MPa. These values are fitted by NIST for nitrogen (119), methane (113) and ethane (114)...... 32 Table 3-5: Fitted coefficients for nitrogen-methane...... 33 Table 3-6: Fitted coefficients for nitrogen-ethane...... 33 Table 3-7: Fitted coefficients for nitrogen-methane-ethane...... 34 Table 3-8: Model error by mixture and temperature...... 34 Table 3-9: Values calculated from Equation 3-6 and Equation 3-7 for validation...... 36 Table 4-1: Fitted coefficients for the normal phase of nitrogen-methane-ethane at 90-100 K...... 37 Table 4-2: Standard chemical potential and potential gradient with temperature for nitrogen, methane and ethane...... 40 Table 4-3: Temperature profiles used in chemical potential modeling...... 45 Table 4-4: Property values and uncertainties with depth for ethane-rich seas (pressure, temperature, composition, density)...... 57 Table 4-5: Property values and uncertainties with depth for methane-rich seas (pressure, temperature, composition, density)...... 59 Table 5-1: Parameters for computing skin temperature and general submarine dimensions...... 63 Table 5-2: Results of the effervescence model for several cases of interest to the Titan submarine...... 93 Table 6-1: Instrumentation list for the ballast system ...... 101 Table 6-2: Baseline part mass estimates ...... 104 Table 6-3: General assumptions in the separator model...... 120 Table 6-4: General Phase I Simulation Parameters ...... 121 Table 6-5: Simulation cases run...... 128 Table 6-6: Mass estimates for the four gas separation concepts in this paper. . 148 Table 6-7: Heater-only results table ...... 151 Table 6-8: Pros and cons of the nitrogen separator concepts, compared to using a noncondensible pressurant ...... 156 Table 6-9: Mass comparison of the separator concepts, with noncondensible and nitrogen pressurant ...... 157 Table 6-10: Mass and power estimates of the options in this trade study...... 161 Table 6-11: Overall pros and cons of the options in this trade study...... 161

List of Figures Figure 1-1: Titan atmosphere (graphic: ESA)...... 14 Figure 1-2: Phase I submarine concept with one ballast tank labeled...... 15 Figure 1-3: Phase II submarine concept...... 16 Figure 1-4: Phase I operational plan...... 16 Figure 1-5: Phase II operational plan...... 16 Figure 3-1: Schematic diagram of a forced recirculation cell...... 22 Figure 3-2: Phase coverage of VLE data for a) nitrogen-methane, b) nitrogen- ethane and c) nitrogen-methane-ethane...... 28 Figure 3-3: The typical appearance of two liquid layers, with recirculated bubbles, in a solubility test. The order of the liquid layers may vary because, above 109 K, liquid ethane is denser than liquid nitrogen (see (86))...... 29 Figure 3-4: Data points where LN2 phase forms, as reported in historical literature, for a) nitrogen-ethane and b) nitrogen-methane-ethane. The solid line is the nitrogen saturation line...... 30 Figure 3-5: Parity plots for each set of fitted coefficients: a) nitrogen-methane, b) nitrogen-ethane and c) nitrogen-methane-ethane...... 34 Figure 3-6: Model equilibrium nitrogen mole fractions from Equation 3-6 and Equation 3-7 with coefficients from Table 3-5 to Table 3-7, for a) nitrogen-methane, b) nitrogen-ethane and c) nitrogen-methane-ethane with equal parts methane and ethane. The split phase is represented in parts (b) and (c) by the red strip on the nitrogen saturation line...... 35 Figure 4-1: Titan sea properties are estimated layer by layer from the surface down...... 39 Figure 4-2: Model tree for estimating sea properties...... 43 Figure 4-3: Assumed temperature profiles used for (a) ethane-rich and (b) methane-rich surface composition...... 45 Figure 4-4: Methane mole fraction for (a) ethane-rich and (b) methane-rich surface composition in the potential-temperature model...... 45 Figure 4-5: Ethane mole fraction for (a) ethane-rich and (b) methane-rich surface composition in the potential-temperature model...... 45 Figure 4-6: Nitrogen mole fraction for (a) ethane-rich and (b) methane-rich surface composition in the potential-temperature model...... 46 Figure 4-7: Density for (a) ethane-rich and (b) methane-rich surface composition in the potential-temperature model...... 46 Figure 4-8: The calculated temperature gradients in the potential-isenthalpic model...... 47 Figure 4-9: Methane mole fraction for (a) ethane-rich and (b) methane-rich surface composition in the potential-isenthalpic model...... 48 Figure 4-10: Ethane mole fraction for (a) ethane-rich and (b) methane-rich surface composition in the potential-isenthalpic model...... 48 Figure 4-11: Nitrogen mole fraction for (a) ethane-rich and (b) methane-rich surface composition in the potential-isenthalpic model...... 48 Figure 4-12: Density for (a) ethane-rich and (b) methane-rich surface composition in the potential-isenthalpic model...... 49

Figure 4-13: Mole fraction of dissolved nitrogen in the FS and SS isenthalpic models...... 51 Figure 4-14: Estimated Titan sea density profile for a) ethane-rich and b) methane-rich seas in the FS and SS isenthalpic models...... 51 Figure 4-15: Estimated Titan sea temperature profile for a) ethane-rich and b) methane-rich seas in the FS and SS isenthalpic models...... 52 Figure 4-16: The estimated minimum freezing point of the methane-ethane mixture is about 70 K, at 65-70 mol% methane...... 54 Figure 4-17: Estimated viscosity profile in the Titan seas...... 55 Figure 4-18: Estimated speed of sound profile in the Titan seas...... 55 Figure 4-19: Estimated dielectric profile in the Titan seas...... 55 Figure 5-1: Internal temperatures and parts of the submarine, with insulation lines. Blocks are shown for subsystems (Purple = science, blue = communication and data handling, red = power)...... 61 Figure 5-2: The bottom-view dimensions of the rectangle considered in the moving case. The projected area WLsub sub (see Table 5-1 for the submarine dimensions) is used in place of skin area...... 72 Figure 5-3: The reference volume considered to be occupied by bubbles after they come off the bottom of the submarine...... 74 Figure 5-4: Relative bubble area concentration on the bottom surface as a function of distance from submarine nose for the moving case...... 75 Figure 5-5: The reference area considered to be occupied by bubbles after they come off the bottom of the submarine...... 76 Figure 5-6: For the quiescent case, a) the area coverage of bubbles over the side of the submarine is based on half the vehicle width (bubbles traveling up one side do not interact with bubbles traveling up the other side) b) the volume fraction of bubbles around the propellers is based on the projected submarine height. An example panel for calculation is shown in blue in each case...... 77 Figure 5-7: Forces on a gas bubble rising through liquid...... 79 Figure 5-8: A diagram of the model used for pressure-based effervescence, with a plot of the assumed pressure distribution...... 83 Figure 5-9: An illustration of how bubbles grow in the propeller cavity...... 85 Figure 5-10: Submarine skin temperature as a function of waste heat flux for a) ethane-rich sea and b) methane-rich sea, with sea temperature 93 K and given geometry...... 87 Figure 5-11: Equilibrium solubility of nitrogen in methane and ethane for a) ethane-rich sea and b) methane-rich sea. Color represents amount of dissolved nitrogen gas. Both figures are plotted on same color scale for comparison...... 88 Figure 5-12: Supersaturation at submarine skin for a) ethane-rich sea and b) methane-rich sea, with sea temperature 93 K...... 89 Figure 5-13: Nucleation site density as a function of skin temperature and contact angle for a) ethane-rich sea and b) methane-rich sea, with sea temperature 93 K...... 90 Figure 5-14: Bubble radius (in mm) at several skin temperatures for a) ethane- rich sea and b) methane-rich sea, with sea conditions 93 K and 0.5 MPa...... 90

Figure 5-15: Volume fraction of bubbles around the propellers (log scale, at the propeller inlet) for a) ethane-rich quiescent case, b) ethane-rich moving case, c) methane-rich quiescent case and d) methane-rich moving case. For all plots, sea 0 temperature is 93 K and   15 ...... 91 Figure 5-16: Highest computed area fraction of bubbles (log scale) on the submarine skin for a) ethane-rich quiescent case, b) ethane-rich moving case, c) methane-rich quiescent case and d) methane-rich moving case, with sea temperature 93 K and   15 degrees...... 92 Figure 5-17: Volume fraction after the propellers, moving at 0.5 m/s, for a) ethane-rich sea and b) methane-rich sea...... 94 Figure 5-18: Comparison of submarine operating point to experimental heat flux values that triggered effervescence...... 95 Figure 6-1: Required Ballast Volume across the Range of Titan Sea Mole Fractions ...... 98 Figure 6-2: Estimates of a) steady-state sinking velocity and b) steady-state sinking Reynolds number of the submarine as a function of excess volume of sea liquid, for both composition extremes ...... 99 Figure 6-3: Illustration of the use of a dielectric meter to deduce composition. The temperature is 95 K and the pressure is 0.15 MPa...... 103 Figure 6-4: Conceptual layout of a pump ballast system. There is one such tank on each side of the submarine...... 104 Figure 6-5: Conceptual layout of a bladder-only ballast system. There is a single gas bottle, and one bladder on each side of the submarine...... 109 Figure 6-6: Conceptual layout of a pressurant gas system without separators. Each side of the submarine has one gas bottle and 2 tank sections...... 112 Figure 6-7: Loss percentage of a) neon, and b) helium in both composition extremes, accumulated over many dives...... 116 Figure 6-8: The proposed Phase I ballast system for the Titan submarine. The entire tank is on the exterior of the submarine, exposed to sea temperature...... 119 Figure 6-9: Schematics for the ramp phase: (a) CV 1, (b) CV 2, (c) CV 3...... 122 Figure 6-10: Schematics for the expulsion stage: (a) CV 2, (b) CV 3...... 124 Figure 6-11: Gas bottle pressure as a function of time using helium...... 129 Figure 6-12: Gas bottle pressure as a function of time using neon. The 2 lines are on top of each other, with the case 12 line ending at 175 seconds...... 129 Figure 6-13: Gas bottle temperature for helium, ethane-rich sea cases with different mass flow rates...... 130 Figure 6-14: Gas-side pressure for helium, ethane-rich sea cases with different mass flow rates...... 131 Figure 6-15: Gas-side temperature for helium, ethane-rich sea cases with different mass flow rates...... 132 Figure 6-16: Liquid-side temperature for several cases...... 133 Figure 6-17: Liquid/vapor saturation lines of a) nitrogen and b) methane in the Titan temperature range. The triple and critical points are noted for comparison...... 137 Figure 6-18: The arrangement of a cryocooler to freeze the methane component out of the atmosphere before compressing the nitrogen into the ballast tank. The

7 cryocooler penetrates the insulation of the submarine so that one end of it remains near 300 K...... 139 Figure 6-19: The arrangement of a distillation method of removing gaseous methane...... 140 Figure 6-20: The thermodynamic process as intake gas is heated and compressed, then returned to Titan sea temperature. The saturation temperature of 95.3 K (at the final pressure) is exceeded by a safe margin...... 141 Figure 6-21: Conceptual layout of gas separation by a single membrane. The concept assumes the membrane is more permeable to nitrogen than to methane.. 142 Figure 6-22: Partial conceptual layout of an adsorbent system (adsorbent is in grey)...... 144 Figure 6-23: The dependence of gas separation time on mass of adsorbent used...... 145 Figure 6-24: A nitrogen pressurant gas system with gas separation...... 148 Figure 6-25: Updated control volumes for the separator ballast concept with heaters added...... 150 Figure 6-26: Heater-only results graph. Dotted line is the current submarine waste heat available to use as a heat source...... 151 Figure 6-27: Preliminary results for the liquid nitrogen ballast concept. The properties plotted are a) pressure and b) quality in CV 1, c) pressure and d) temperature in CV 2, and e) temperature in CV 3...... 154 Figure 6-28: Schematic diagram for the liquid nitrogen actuator model...... 155 Figure 6-29: Conceptual layout of a liquid nitrogen pump system...... 157 Figure 6-30: Conceptual layout of methane/ethane gas system ...... 159 Figure 7-1: Phase II submarine design, with the ballast tanks identified. The GHe bottles are warm internal components...... 163

Acknowledgements

This work was funded through the NASA Innovative Advanced Concepts (NIAC)

Phase 2 Titan Submarine Project.

Primarily, I would like to thank Dr. Jason Hartwig, my mentor at the NASA Glenn

Research Center. This thesis is a direct consequence of my internship at Glenn, and Dr.

Hartwig steadfastly gave me everything from technical advice to career advice along the way. Through many trials, I learned much about myself and about professional life.

Some of my other colleagues at Glenn also made important contributions to the thesis content. Steve Oleson arranged the design meetings that produced the full 3D models of the submarines that appear below, and stimulated various useful discussions. Anthony

Colozza worked to develop heat transfer models to estimate submarine skin temperature.

Greg Zimmerli provided a valuable example of fitting a curve to solubility data.

Ramaswamy Balasubramaniam performed the literature review that accompanies the effervescence model in section 5.

Outside of NASA, Dr. Hartwig and I have collaborated with Ian Richardson to study solubility and freezing of Titan sea mixtures at Washington State University. Eric

Lemmon provided a long list of solubility data sources that served as a helpful starting point for my work. Ralph Lorenz engaged in many discussions on Titan sea composition and on the interaction of submarine engineering with the intended environment.

Abbreviations CV: control volume FS: full solubility GHe: gaseous helium GNe: gaseous neon LNG: liquefied natural gas MAE: mean absolute error SS: surface solubility VLE: vapor-liquid equilibrium

Nomenclature and Units Area A : m2 Mass concentration C : dimensionless

Drag coefficient CD : dimensionless

Isobaric specific heat Cp , cp : J/K, J/kg-K

Isochoric specific heat cV : J/kg-K Depth in ballast tank 푑: meters Diffusion coefficient D : m2/s Energy 퐸: J Similarity variable 푓: dimensionless Force F : N Gravitational acceleration g : m/s2 Gibbs free energy 퐺: J Grashof number 퐺푟: dimensionless Heat transfer coefficient ℎ: W/m2-K Enthalpy ℎ0: J/kg Height H : meters Thermal conductivity 푘: W/m-K Length L : meters Mole number 푛: mol 2 Nucleation site density Nn : 1/m Mass 푚: kg Molar mass 푀: kg/mol Pressure P : MPa Excess above vapor pressure P* : MPa

Vapor pressure P0 : MPa Prandtl number 푃푟: dimensionless Heat transfer Q : W Radius 푅: meters Universal gas constant 푅푢: J/kg-K Reynolds number 푅푒: dimensionless Entropy 푠: J/kg-K Supersaturation 푆: dimensionless Time 푡: seconds

Temperature 푇: K Thickness 푤: meters Width 푊: meters Internal energy 푢: J/kg Uncertainty 푈 Volume 푉: m3 Volume flow rate 푉̇ : m3/s Velocity 푣: m/s Work done 푊: J Depth or coordinate dimension 푥, 푦, 푧: meters Mole fraction 푥(): dimensionless Compressibility factor 푍: dimensionless

Greek Bubble growth constant 훽: dimensionless Surface tension 훾: N/m Quality 휒: dimensionless Dielectric factor 휀: dimensionless Chemical potential 휙: J/mol Similarity variable 휂: dimensionless Angle θ: radians Density 휌: kg/m3 Density ratio 휌∗: dimensionless Solidity ratio  : dimensionless Thrust 휏: N. Kinematic viscosity 휉: m2/s Change between steps 훥

Subscripts 0, 1, 2, etc.: count or index 12, 13, etc.: interactions between CVs N2, CH4, etc.: chemical species 푎푚: added mass force 푏: buoyancy force. 푏푎푙: liquid ballast 푏푖: binary mixture 푏푢푏: bubble 푐: characteristic length 푐푟: maximum growth time in quiescent case 푐푟푖푡: critical bubble radius 퐶푉: control volume 푑푟: drag force 푒푚: emergency bladder 푒푞: equilibrium 푒푥푝: blowdown expansion 푓: freezing point 11

푓푟표푛푡: submarine front (projected) 푔: nitrogen vapor property 푔푏: gas bottle 푖푛: flow entering control volume 푙: pure solvent property 푙푖푛푒: line of bubbles from a single nucleation site 푛푐: induced velocity due to natural convection 푛푒푡: excess of ballast in tank, above neutral buoyancy 푛푒푢: neutral buoyancy 표푢푡: flow leaving control volume 푝: low-pressure region behind the propellers 푝푟표푝: propeller dimensions 푠: speed of sound 푠푎푡: saturation 푠푒푎: bulk sea liquid 푠푒푝: separator skin : submarine skin 푠푢푏: submarine property 푠푢푟푓: liquid-vapor interface 푤ℎ: submarine waste heat 푡: ternary mixture 푡푎푛푘: capacity of ballast tank

Superscripts Dot: time derivative ′: representative length in area/volume calculations 푗: time step count

Solubility, Sea Properties, Effervescence and Ballast Design for an Extraterrestrial

Submarine

Abstract

PETER MEYERHOFER

Saturn’s largest moon, Titan, is a remarkable environment in the solar system. Data from the Cassini flybys suggests the presence of several seas, the only known accessible surface liquid outside of Earth. These seas are composed primarily of nitrogen, methane and ethane, with applications in fields from geology to organic chemistry.

This thesis assesses several preliminary design aspects of a submarine that would explore the seas of Titan. First, it gives a comprehensive literature review of the solubility of nitrogen in methane and ethane, and develops simple analytical correlations for the mole fraction of nitrogen. Second, these correlations contribute to a method of estimating the physical properties of the Titan Seas. Third, the correlation is used to predict the formation of bubbles around the submarine due to waste heat through the skin. Fourth, a trade study is conducted on several concepts for the ballast system, based on mass, power and complexity.

1 Introduction

Saturn’s giant moon Titan is a remarkable environment, with the only known liquid surface seas outside of Earth (1) (2) (3). These seas are composed of liquid methane and liquid ethane with dissolved nitrogen gas from the atmosphere, at a temperature of 93 K and a pressure of 1.5 bar (see Figure 1-1). This environment allows the exploration of organic chemistry as well as geological cycles similar to those on Earth, but with methane instead of water as the working fluid. A submarine is currently being designed as a means to explore these liquid bodies from atmosphere to seabed to shoreline (4) (5).

Figure 1-1: Titan atmosphere (graphic: ESA).

Currently, there are two designs of a submersible body to explore the seas of

Titan – a stand-alone concept without an orbiter (Phase I) and an orbiter-supported concept (Phase II), shown in Figure 1-2 and Figure 1-3, respectively. Design requirements for these two concepts are summarized in Table 1-1. The concept of operation for both systems are shown in Figure 1-4 and Figure 1-5, respectively. As shown, the Phase I concept relies on direct to Earth communication, which requires that

14 the submarine be launched at a later date to ensure that Saturn does not block the line of sight to Earth. However, the Phase I submarine has very low data transmission rates, and therefore must spend 2/3 of each Earth day at the surface. Meanwhile, at the cost of having to launch an orbiter along with the submarine, the Phase II concept (6) is much smaller and simpler, but still maintains all relevant and important science instruments.

Further, it can be launched whenever the technology is ready, as early as the mid-2020s.

Figure 1-2: Phase I submarine concept with one ballast tank labeled.

Figure 1-3: Phase II submarine concept.

Figure 1-4: Phase I operational plan.

Figure 1-5: Phase II operational plan.

Phase I Phase II Hydrocarbon composition Any methane-ethane proportion Methane-rich (Ligeia Mare) (targeted sea) (Kraken Mare, Ligeia Mare) Mass (kg) 1386 530 Power (W) ~800 ~100 Science mass (kg) 80 80 Size 6.5 m long, 1.2 m wide, 0.8 m 2 m long, 1.4 m wide, 0.7 m high, ~2.77 m3 high, ~1 m3 Entry, descent and landing ~8 m lifting body 2.7 m aeroshell Primary mission 90 days 1 year Average Speed (m/s) 1 0.1 Range (km) >4000 >2000 Depth limit (m) 1000 200 Ballast system Reclaimable GHe or GNe Consumable warm GHe Number of dives (duration) ~365 (8 hours submerged, 16 ~25 (12 days submerged, 4 days hours surfaced) surfaced) Data relayed (schedule) 1 Gb (800 bps for 16 hours per 1 Gb (100 kbps for 30 minutes day) every 5 hours) Table 1-1: Submarine design requirements for Phase I and Phase II.

Of all the submarine systems, the most difficult design challenge at present is the ballast system, which controls the vertical ascent and descent of the submarine by changing vehicle density. The main reason is that density and other sea properties at depth are largely unknown and highly variable. A submarine traversing between freshwater and saltwater on Earth must handle a 2% density change, while the density range between liquid methane and liquid ethane is 20%. Ethane is also 5 times more viscous than methane. The seas may freeze in some places (the Titan surface temperature is barely higher than the triple points of methane and ethane), but the characteristics of such “ice” are unknown. Finally, atmospheric nitrogen is highly soluble in liquid methane

(potentially accounting for over 10% of the molecules), which raises the risk of effervescence due to waste heat from the submarine power source causing bubbles dissolved in the liquid to come out of solution. (For terrestrial submarines, the solubility of air in water is negligible (7).)

The outline of this thesis is as follows. Section 2 is a literature review of the study of

Titan. Section 3 gives the data sources and analysis that generate the solubility

17 correlation, with uncertainty values. Section 4 uses the solubility correlation to develop the properties of the Titan Seas, with estimates based equality of chemical potential.

Section 5 applies the solubility model to the question of effervescence due to submarine waste heat. Section 6 is the main trade study of ballast concepts, beginning with a literature review of submarine ballast design, and with section 6.7 describing a method of harvesting atmospheric nitrogen. Section 7 is a description of the simpler system chosen for the Phase II submarine. Finally, Sections 8 and 9 assess the implications of the present work. All fluid properties are calculated using the software REFPROP (8).

2 Literature Review

The surface of Titan has been studied since Voyager I passed by in the early 1980s.

Initial analyses suggested that atmospheric methane content is high humidity or near- saturated, which implied a liquid reservoir or ocean (9), though other assessments favored large dry regions (10) (11). A mostly-ethane sea was also proposed (12). Further studies continued into the 1990s focusing on liquid and gas dynamics. The topics of this work included: thermodynamic effects based on phase equilibrium data (13); the liquid and solid hydrocarbons that may form by chemical reaction in a liquid sea (14); the formation of methane clouds and rain particles (15); the variability of atmosphere components, especially argon (16); the methane concentration in the low atmosphere based on phase equilibrium (17); atmospheric saturation and condensation (18); whether methane raindrops are likely to evaporate before reaching the surface (19); the effect of local gravity and liquid properties on wave heights (20); and tidal winds caused by the eccentric orbit about Saturn (21). In this period, alternative concepts for a surface

18 hydrocarbon reservoir, such as solid storage in the pores of surface rock, also remained plausible (22).

Data from the Cassini orbiter, one of the most prolific missions ever within NASA which arrived at Titan in 2005, proved that the surface does indeed have liquid hydrocarbon lakes (23). These liquid bodies appear to have many Earth-like features along their shorelines suggesting dynamic hydrological processes (24). The topics of study based on these data have included composition uncertainties (25), atmosphere equilibrium models (26) (27), the surface cover needed to replenish atmospheric methane

(28), confirmation of the presence of liquid ethane (29), the use of the dielectric constant of liquid hydrocarbons in estimating depth (30) and a depth estimate of 160 meters for

Ligeia Mare (31). Several details of the liquid chemistry are the solution and deposition of solid acetylene (32), the solubility of benzene (33) and the interplay between nitrogen solution and methane-ethane evaporation (34). There are also a number of ways that the

Titan seas may have tides and currents similar to those on Earth. Recent studies examine a tidal range as large as several meters in Kraken Mare due to the influence of Saturn

(35), a variation in methane content similar to the difference between fresh water and salt water (36), possibly active tidal mixing dynamics within a narrow throat of Kraken Mare

(37), a numerical simulation of tides and angular momentum (38), the effects of solar insolation and methane precipitation (39) and the projected speed of currents driven by wind (40). Waves on the Titan sea surface are another interesting topic of study: the wind speed required to form waves and the conditions that hydrocarbons best suited for wave action (41), reflection estimates of the slope on the surface (42), and total wave height estimates (43). Other liquid-related studies include the possibility of transient seas (44).

Several areas of the surface have topological and reflective profiles that suggest evaporate deposits (45).

This submarine is not the first mission proposal for in-situ study of the Titan environment. An initial discussion (46) of several potential science vehicles (primarily targeting the atmosphere) was laid out very soon after Voyager I. One previous group of studies described an autonomous airship to explore the atmosphere (47) (48) (49).

Particular concerns with this vehicle were how to arrange an autonomous control scheme

(50) and how to use the wind in the Titan atmosphere for navigation (51). Another alternative mission concept is a floating sensor package for the sea surface, the Titan

Mare Explorer (TiME) (52). This vehicle would come with an orbiter to relay data to

Earth (53), and its science objectives would especially be exposed to the wind and tide in the sea (54) (55). TiME research also included the conceptual design of a depth sounder

(56). One proposal that combined an airship with a floating lander and an orbiter was the

Titan Saturn System Mission (TSSM), where the airship and lander would explore Titan while the orbiter would study both Titan and Enceladus (57). Neither an airship nor a floating probe, however, have the capability of a submarine to sample both atmosphere and seabed.

The Cassini data provides a more detailed estimate of the Titan sea composition: mostly methane and ethane, with traces of many other hydrocarbons (58). A simplified view is to focus only on the dominant methane and ethane, as well as the atmospheric nitrogen dissolved in them. From the most recent observations, the range of methane and ethane content is up to 87 mol% methane, down to almost pure ethane (59). (This work uses the bounds of 5% and 85% methane.) Even with the universal nitrogen-methane-

20 ethane solubility model developed in Section 3, composition and properties can only be estimated on the sea surface. The ballast system must tolerate a range of unexpected phenomena at depth, which requires it to be that much more robust.

3 Solubility Model

3.1 Background

Solubility and other measurements were first made in nitrogen-methane mixtures around 1920 (60), but it was in the 1950s that such studies increased in frequency, driven by the rising demand for LNG. The busiest time for these VLE studies in gas components was between 1952 and 1980. Three distinct experimental setups were used to collect the data in this analysis. The most common method is a forced recirculating cell, shown in

Figure 3-1. Predetermined quantities of gas-phase nitrogen, methane and ethane are added to a closed cell and cooled to a specified temperature. Pressure is a variable in this arrangement, determined by the extent to which the material liquefies at a controlled temperature. A recirculating pump accelerates the equilibrium process by forcing the vapor back through the liquid until both phases become constant, a process for which

Chang and Lu (61) allowed 2 hours with a cell volume of 100 mL. Finally, liquid and vapor samples are taken and mole fractions are measured from them.

Figure 3-1: Schematic diagram of a forced recirculation cell.

Two other methods are used for relatively few studies:

 An alternative to recirculating the vapor is to accelerate equilibrium by

agitating the mixture instead. With small volumes of fluid, it may also be feasible

to let them reach equilibrium without disturbance.

 A measurement technique used mainly in the 1950s is to force a fixed

mixture of gas into a closed cell until it liquefies, passing first through the dew

point and then the bubble point pressures for that composition and temperature.

The dew point and bubble point thresholds are taken to represent equilibrium

vapor and equilibrium liquid, respectively.

LNG is typically handled at temperatures of around 110 K, and relatively little of the available literature supports an analysis at Titan conditions, between the triple points of methane and ethane (90-91 K) and 95 K. Additionally, many studies report solubility for binary mixtures, instead of the various ternary mixtures that better represent Titan.

These issues prompted a recent campaign of ternary solubility experiments at Titan conditions (62).

To build a robust solubility correlation for nitrogen gas in liquid methane and ethane, and general methane-ethane mixtures, a rigorous review of the literature was conducted. A full list of the sources used for analysis is given in Table 3-1 to Table 3-3, which also describe the respective temperature and pressure ranges, and are sorted chronologically. In total, there are 40 sources, 1006 points for nitrogen-methane, 604 points for nitrogen-ethane (483 for the normal phase, 121 for the split phase discussed in

Section 3) and 746 points for nitrogen-methane-ethane solubility (561 for the main phase,

185 for the split phase). These are reported as VLE data.

Source Points 푻 Range (K) 푷 Range (MPa) Comments (63) 90 91 to 185 0.34 to 4.9 Dew and bubble points. (64) 111 91 to 190 0.14 to 5 Dew and bubble points. (65) 135 99 to 172 0.14 to 4.5 Forced recirculation cell. (66) 32 95 to 150 K 0.2 to 1.6 - (67) 9 137 to 175 3.4 Agitated to reach equilibrium. (68) 20 91.6 to 124 0.03 to 0.6 Agitated to reach equilibrium. (61) 26 122 to 171 0.35 to 5 Forced recirculation cell. (69) 19 130 to 135 0.8 to 4.1 Forced recirculation cell. (70) 8 130 0.8 to 3 Forced recirculation cell. (71) 11 112 0.18 to 1.3 Forced recirculation cell. (72) 48 95 to 120 0.2 to 2 Forced recirculation cell. (73) 104 114 to 183 0.28 to 5 Forced recirculation cell. (74) 83 130 to 180 0.2 to 4.9 Forced recirculation cell. (75) 14 111 0.24 to 1.4 Agitated to reach equilibrium. (76) 4 140 to 160 2 to 4 Forced recirculation cell. (77) 176 112 to 183 0.2 to 4.9 Forced recirculation cell. (78) 5 94 0.3 to 0.5 Forced recirculation cell. (79) 10 123 0.42 to 2.6 Forced recirculation cell. (80) 28 130 to 180 0.6 to 5.1 Forced recirculation cell. (81) 66 100 to 123 0.04 to 1.3 Forced recirculation cell. (62) 7 91 to 99 0.1 to 0.15 - Table 3-1: Nitrogen-methane VLE data.

Points Source Normal Split 푻 Range (K) 푷 Range (MPa) Comments Phase Phase (82) 97 7 100 to 300 0.7 to 10 Dew and bubble points. (83) 3 0 144 to 200 3.4 to 6.9 Forced recirculation cell.

(68) 2 0 93 0.02 to 0.05 Agitated to reach equilibrium. (61) 17 5 122 to 171 0.3 to 3.4 Forced recirculation cell. (84) 32 0 144 to 228 2 to 5 Agitated to reach equilibrium. (69) 23 8 114 to 170 0.28 to 5 Forced recirculation cell. (85) 14 14 113 to 134 1.8 to 4.1 Forced recirculation cell. (86) 2 2 109 1.3 Forced recirculation cell. (70) 19 13 94 to 134 0.4 to 4.1 Forced recirculation cell. (87) 48 0 139 to 194 0.3 to 13.5 Forced recirculation cell. (75) 6 7 111 0.2 to 1.6 Agitated to reach equilibrium. (88) 30 0 200 to 290 0.6 to 13 Forced recirculation cell. (89) 17 12 120 to 134 0.6 to 4.1 Forced recirculation cell. (90) 36 37 118 to 133 2.1 to 4.1 Two liquid phases. (91) 16 16 118 to 132 2.1 to 4.1 Agitated to reach equilibrium. (92) 15 0 240 to 260 1.8 to 7.5 Forced recirculation cell. (76) 8 0 140 to 220 2 to 12 Forced recirculation cell. (93) 30 0 220 to 270 0.8 to 12 Forced recirculation cell. (94) 11 0 210 to 270 1.1 to 10 Forced recirculation cell. (95) 25 0 120 to 138 0.8 to 3.3 Forced recirculation cell. (80) 25 0 150 to 270 0.5 to 10 Forced recirculation cell. (62) 6 0 91 to 99 0.1 to 0.15 Table 3-2 Nitrogen-ethane VLE data.

Points Source Normal Split 푻 Range (K) 푷 Range (MPa) Comments Phase Phase (83) 5 0 144 to 200 3.4 to 6.9 Forced recirculation cell. (61) 45 14 122 to 171 0.2 to 2.8 Forced recirculation cell. (69) 64 28 114 to 130 0.3 to 3.5 Forced recirculation cell. Two liquid phases. (86) 6 6 111 to 114 1.3 to 1.5 Forced recirculation cell. (85) 32 32 112 to 128 1.5 to 3.5 Forced recirculation cell. (96) 46 5 139 to 150 0.69 to 4.2 Forced recirculation cell. (70) 107 78 94 to 134 0.3 to 3.7 Forced recirculation cell. Two liquid phases. (97) 14 0 122 to 200 1.3 to 6.9 (98) 15 0 153 to 203 1 to 5 (99) 103 0 260 to 280 3.5 to 8.6 Forced recirculation cell. (100) 22 20 116 to 142 1.6 to 5 Agitated to reach equilibrium. Two liquid phases. (76) 77 2 140 to 220 2 to 12 Forced recirculation cell. (78) 13 0 95 0.08 to 0.2 Forced recirculation cell. (62) 12 0 91 to 95 0.1 to 0.15 - Table 3-3 Nitrogen-methane-ethane VLE data.

Regarding previous modeling efforts, a pioneering study of nitrogen gas in liquid methane was Hibbard and Evans (101). This report contained equations fit to data for

24 computing rate of solution. A more recent fit of binary solubility data is found in Battino et al. (102). The solubility model fitted for nitrogen-methane mixtures is:

T (Equation 3-1) lnxN2  0.34572  10.4175ln  0.28887 P  1.51847ln P 100 where temperatures are evaluated in degrees Kelvin and pressure in MPa. Battino’s model for nitrogen-ethane mixtures was corrected for a sign error (4):

391.49 lnx  5.7645   0.077743 P  0.89104ln P (Equation 3-2) N2 T

where xN2 is mole fraction solubility of nitrogen, temperatures are in K and pressures are in MPa. The nitrogen-methane model is valid for pressures from 0.02 to 5 MPa, and for temperatures from 91 to 187 K, with a standard deviation of 18% based on 7 data sets while the nitrogen-ethane model is valid for pressures from 0.35 to 13.5 MPa, and for temperatures from 122 to 301 K, with a standard deviation of 19% based on 6 data sets.

Zimmerli et al. (103) also recently fitted a solubility equation focused on GHe, the form of which is useful in this analysis:

22 xN2 P*exp aaTaT 0  1  2  bbTbTP 0  1  2  * , (Equation 3-3) PP  P*  0 1 MPa

Where P0 is the vapor pressure of the pure liquid solvent at the given temperature T ; ai

and bi are the coefficients to be fitted. The use of excess pressure above the vapor pressure, instead of absolute pressure, allows a better match to the physical limits of the system.

Some recent solubility work has focused on Titan. Cordier et al (104) describes a scenario where sinking currents bring about liquid-liquid-vapor equilibrium and release

25 bubble streams. Malaska et al (62) fits solubility data at Titan temperatures to an equation of the form

TT  0  xN 20 x Ae (Equation 3-4) for the two binary systems, while the ternary adaptation is

P x P 1  x P x  C2H6 0,C2H6 C2H6 0,CH4 (Equation 3-5) N 2 1xxC2H6 C2H6 KK12 13

Where K12 and K13 are temperature-dependent coefficients. Equations 4 and 5 are fitted in the range 85-100 K and for 0.15 MPa pressure based on a handful of data, and are intended solely for Titan surface analysis as opposed to representing the solubility systems in general. Malaska et al also describes the prospect of variable nitrogen content causing density fluctuations or overturning in the Titan lakes.

The author did not find any correlations in the literature for the second liquid phase in liquid-liquid-vapor equilibrium, dominated by nitrogen. The data for this phase are less numerous than for the main liquid phase.

3.2 Analysis and Filtering

The following data points and sets were excluded from the analysis:

 All points with temperature less than 91 K, corresponding to the triple points of

methane and ethane. There is a small phase space between solid, liquid and vapor

states that allows VLE to occur down to the triple point of nitrogen at 63 K, at

least in nitrogen-methane (105), but it was left out as an extreme condition.

 For several sets in nitrogen-methane and nitrogen-ethane, many or all of the

points are at temperatures below 91 K and were excluded entirely: (106), (107),

(108), (109) and (110).

 All points with pressure less than the vapor pressure of the solvent; there are only

a few such points, presumed to be poorly measured.

 All points where the liquid composition was not reported.

 Individual points that seem like anomalies within their respective sets; for

example if the errors between model and data are 5%, -12%, 15% and -60%, the

last point alone is excluded.

 Any sets with error values consistently greater than most other points with the

same temperature. These sets are: (60), (105) and (111) for nitrogen-methane (64

points removed); and (99) for nitrogen-ethane (59 points removed).

 An additional split liquid nitrogen phase in nitrogen-ethane and nitrogen-

methane-ethane is widely reported, but one paper (112) that mentions a third

liquid phase is excluded. It gives only one T-P point, with 3 liquid compositions,

as a passing observation.

Figure 3-2 shows the phase space occupied by these data in the T-P plane, a region bounded by the vapor pressure curves of the solvent (liquid methane or ethane) at the low pressure for given temperature, and of nitrogen at the corresponding high pressure. Since the hydrocarbons have a much higher critical temperature than nitrogen (190 K for methane, 305 K for ethane, 126 K for nitrogen), the gap between the critical points is governed by a critical pressure curve characteristic of each mixture.

The completeness of the historical data coverage is clear for the binary nitrogen- methane system (Figure 3-2a). The data for the binary nitrogen-ethane system (Figure

3-2b) are somewhat scarcer than for methane, most notably below 110 K. The ternary

27 system nitrogen-methane-ethane (Figure 3-2c) is poorly covered above 150 K, with most data near the vapor pressure of nitrogen.

6 14 a) Nitrogen/methane b) Nitrogen/ethane 12 5 Data N Vapor Pressure 2 Data CH Vapor Pressure 10 N Vapor Pressure 4 2 4 C H Vapor Pressure 2 6 8 3 6

P [MPa]

P [MPa] 2 4

1 2

0 0 80 100 120 140 160 180 200 50 100 150 200 250 300 350 T [K] T [K]

14 c) Nitrogen/methane/ethane 12 Data N Vapor Pressure 2 C H Vapor Pressure 10 2 6

P [MPa]

0 50 100 150 200 250 300 350 T [K]

Figure 3-2: Phase coverage of VLE data for a) nitrogen-methane, b) nitrogen-

ethane and c) nitrogen-methane-ethane.

One phenomenon noted in a number of experiments (any source in Table 3-2 or Table

3-3 where the “split phase” column is nonzero) is that, between nitrogen and ethane, in conditions that are simultaneously near the solubility and nitrogen vapor pressure limits, some of the nitrogen that cannot dissolve forms a separate liquid layer instead of returning to vapor. The typical appearance of such a test sample, based on a photograph and several sketches in (112), is shown in Figure 3-3. This occurs most frequently in

VLE experiments where vapor is recirculated through the bottom of the liquid. The vapor

28 is actively forced into the liquid to the greatest extent possible, under conditions where nitrogen liquefies as readily as it remains vapor. This liquefaction is less frequently observed in experiments that do not use agitation.

These two-layer configurations occur only in a small region of the temperature- pressure phase space, as shown in Figure 3-4, along the vapor pressure curve of nitrogen.

The composition of the separated liquid nitrogen is widely reported, in addition to the normal dissolved gas phase. Note that no such phase separation has been recorded at pressures lower than 0.4 MPa, and thus split-phase equilibria should not occur near the

Titan sea surface in the current epoch. An early Titan thermodynamics paper (13) noted this phenomenon, but similarly observed that it occurs at pressures too high to be relevant to Titan.

Figure 3-3: The typical appearance of two liquid layers, with recirculated bubbles, in a solubility test. The order of the liquid layers may vary because, above

109 K, liquid ethane is denser than liquid nitrogen (see (86)).

5 5 a) Nitrogen/ethane b) Nitrogen/methane/ethane

4 Data 4 Data N Saturation N Saturation 2 2

3 3

P [MPa]

P [MPa]P 2 2

1 1

0 0 90 100 110 120 130 140 90 100 110 120 130 140 T [K] T [K]

Figure 3-4: Data points where LN2 phase forms, as reported in historical

literature, for a) nitrogen-ethane and b) nitrogen-methane-ethane. The solid line

is the nitrogen saturation line.

3.3 Analytical Model

3.3.1 Functional Form

The general functional form used in this work for the binary systems nitrogen- methane and nitrogen-ethane is Equation 3-3, rewritten:

22 xN2,bi  aP 0*exp aTaT 1  2  PaaTaT * 3  4  5  (Equation 3-6)

The adaptation used for the ternary case is to multiply again by the methane fraction of the pure solvent to a coefficient power:

a6 22xCH4 xN2,t  aP 0*exp aTaT 1  2  PaaTaT * 3  4  5  (Equation 3-7) xxCH4 C2H6

where xCH4 and xC2H6 are the liquid mole fractions of methane and ethane, respectively.

The values a0 to a6 are the variable coefficients in the above two equations. Note that the ternary equation, Equation 3-7, collapses to the binary nitrogen-methane equation,

Equation 3-6, when the last term is dropped. The pressure variable P * is the excess

30 above vapor pressure, in MPa, of the pure solvent (in the ternary case, the vapor pressure used is that of the methane/ethane mixture), based on (103):

PPP* 0 (Equation 3-8)

The best measure of whether the system conforms to Henry’s Law ( x KP* for some constant K ) is whether the above solubility models are linear in pressure. An ideal case

would therefore be distinguished by a3 , a4 and a5 all being zero, while a realistic near-

ideal system would have those terms small compared to a0 , a1 and a2 .

3.3.2 Curve Fit to Data

Coefficients were fitted that minimize the MAE in multiple temperature ranges, rather than fitting a single model across the entire T-P space. The coefficient values for binary nitrogen-methane, binary nitrogen-ethane, and ternary nitrogen-methane-ethane mixtures are shown in Table 3-5 to Table 3-7, respectively. Note that these coefficients apply only to points that lie inside of the saturation lines, and do not include points that lie along the curve. The MAE values corresponding to each fit are presented in Table 3-8. Figure 3-5 shows parity plots for the fitted models. It is apparent that both binary systems have been measured with consistently greater accuracy than the ternary system. This is not surprising considering the lack of historical data across the ternary phase space (Figure

3-2c) and the greater complexity of mixing three fluids instead of two. Additionally, the values , and are of comparable magnitude to , and , which implies strongly nonlinear solubility effects. This occurs most dramatically in the coefficients of nitrogen-methane. A comparison of the ternary model to the binary methane model at

Titan conditions is limited to the temperature range 90 to 110 K. There are strong signs of

31 consistency as most of the coefficients from a1 to a5 have the same sign, and the leading constant has the same order of magnitude.

The value of vapor pressure P0 for the solvent (pure methane, pure ethane or a methane-ethane mixture) used in this work is from REFPROP (the pressure corresponding to zero quality at a given temperature). A good alternative, without special software, is to use a correlation of the Antoine equation (113) (114):

B log PA (Equation 3-9) 10 0 TC where A , B and C are constants. The fitted values of these constants are shown in

Table 3-4 with units and sources. To approximate for ternary solubility, a simple linear interpolation by methane mole fraction is used; this value deviates from the vapor pressure in REFPROP by less than either 20% or 0.03 MPa (whichever is larger), compared to pressures on the order of 1 MPa. This process only works at temperatures below the critical point of methane (190 K). The vapor pressure of nitrogen is also included as a reference, below the critical point at 126 K.

Fluid T Range (K) C Nitrogen (115) 63 to 126 2.7362 264.651 -6.788 Methane (116) 90 to 190 2.9895 443.028 -0.49 Ethane (117) 91 to 140 3.507 791.3 -6.422 Ethane (118) 140 to 200 2.9385 659.739 -16.719 Table 3-4: Constant values for the Antoine equation for vapor pressure, evaluated in units of K and MPa. These values are fitted by NIST for nitrogen (119), methane (113) and ethane (114).

The split liquid nitrogen phase only exists in a tiny phase space, and requires active agitation between vapor and liquid. Therefore, most users of this solubility model can safely ignore the “split phase” columns in Table 3-6 and Table 3-7. If one is exclusively

32 interested in the amount of dissolved nitrogen gas, then only the normal phase column should be used. The phase space where the split occurs in nitrogen-ethane, based on fitting a quadratic curve onto Figure 3-4a, is:

2 PTT0.0013(  94)  0.03(  94)  0.4 MPa, 91T 135 K (Equation 3-10)

The ternary T-P space where the split phase is measured, based on fitting a quadratic curve onto Figure 3-4b, is:

2 PTT0.0009(  94)  0.04(  94)  0.3 MPa, 91T 140 K (Equation 3-11)

T (K) 91-110 110-150 150-180 -1 22.5 8.934 2.336 a0 (MPa ) -1 -0.0112 -0.0205 -0.0181 a1 (K ) -2 -0.000233 -0.0000574 -0.00000592 a2 (K ) -1 17.08 9.49 4.644 a3 (MPa ) -1 -1 -0.243 -0.14 -0.0594 a4 (K -MPa ) -2 -1 0.000841 0.000517 0.000192 a5 (K -MPa ) Table 3-5: Fitted coefficients for nitrogen-methane.

(K) 90-140 (Normal) 90-140 (Split) 140-180 180-290 (MPa-1) 3.509 46.708 3.278 2.13 (K-1) -0.0307 -0.0313 -0.0423 -0.0363 (K-2) 0 0 0.0000919 0.0000749 (MPa-1) 0.556 -1.872 0.381 0.144 (K-1-MPa-1) -0.00369 0.012 -0.00451 -0.00138 -2 -1 0 0 0.0000141 0.00000375 a5 (K -MPa ) Table 3-6: Fitted coefficients for nitrogen-ethane.

(K) 90-140 (Normal) 90-140 (Split) 140-290 (MPa-1) 33.7 778.9 8.352 (K-1) -0.0403 -0.0657 -0.045 (K-2) -0.00001 0.0000359 0.0001 (MPa-1) 0.293 -1.49 0.31 (K-1-MPa-1) -0.000053 0.00932 -0.00296 -2 -1 -0.00001 0.0000127 0.00000749 a5 (K -MPa ) 0.298 -0.0998 0.313 a6

Table 3-7: Fitted coefficients for nitrogen-methane-ethane.

Mixture Temperature and MAE Nitrogen-methane 90-110 K: 6% 110-150 K: 7% 150-180 K: 7% Nitrogen-ethane 90-140 K: 8% Split phase: 2% 140-180 K: 8% 180-290 K: 4% Nitrogen-methane- 90-140 K: 17% Split phase: 9% 140-290 K: 13% ethane Table 3-8: Model error by mixture and temperature.

1 1 a) Nitrogen/methane b) Nitrogen/ethane +10% +10% 0.8 0.8 Normal Phase -10% Split Phase -10% 0.6 0.6

(model)

2 0.4 2 0.4

0.2 0.2

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 xN (data) xN (data) 2 2

1 c) Nitrogen/methane/ethane Normal Phase 0.8 Split Phase +20% 0.6 -20%

(model) 2 0.4

0.2

0 0 0.2 0.4 0.6 0.8 1 xN (data) 2

Figure 3-5: Parity plots for each set of fitted coefficients: a) nitrogen-methane, b)

nitrogen-ethane and c) nitrogen-methane-ethane.

Figure 3-6 shows the solubility values for each of the 3 models as a function of temperature and pressure. In the nitrogen-ethane and nitrogen-methane-ethane cases, it includes only the normal phase. The output is limited to the domain between the vapor pressure of nitrogen, on the left, and the vapor pressure of the methane/ethane solvent, on

34 the right. The pressure values are limited to 3.25 MPa because above the critical pressure of nitrogen (3.4 MPa), the domain boundaries are less clear (see Figure 3-2). In general, methane allows much greater amounts of dissolved nitrogen than ethane, and ternary solubility is intermediate between the two binary extremes. Several reference points are also included in Table 3-9 as a check on the consistency of the fitted correlation.

Figure 3-6: Model equilibrium nitrogen mole fractions from Equation 3-6 and

Equation 3-7 with coefficients from Table 3-5 to Table 3-7, for a) nitrogen-methane,

b) nitrogen-ethane and c) nitrogen-methane-ethane with equal parts methane and

ethane. The split phase is represented in parts (b) and (c) by the red strip on the

nitrogen saturation line.

Solvent T (K) P (MPa) P * (MPa) xN2 Methane 95 0.3 0.28018505 0.41415524 Methane 120 1 0.80856992 0.30113739 Methane 170 3 0.67165117 0.06495948 Ethane 95 0.3 0.29999637 0.06059641 Ethane (split 95 0.5 0.49999637 0.8280048 phase) Ethane 160 1.5 1.47859478 0.06039009 Ethane 200 2.5 2.28276706 0.07126144 50% methane, 95 0.3 0.28818894 0.16613248 50% ethane 50% methane, 50% ethane (split 95 0.5 0.48818894 0.86355252 phase) 50% methane, 150 2.5 1.94826704 0.15563374 50% ethane Table 3-9: Values calculated from Equation 3-6 and Equation 3-7 for validation.

Given an estimate of temperature and pressure on the surface of Titan (or anywhere else of interest), the use of the coefficient values in the table with the equation form will provide a reasonable guide to how much nitrogen gas is dissolved in liquid methane and/or ethane. In a pure methane solvent at 93 K and 0.15 MPa, for example, subtracting the vapor pressure of methane from the absolute pressure leaves an excess pressure

P* 0.134 MPa. Such values of and , substituted into Equation 3-6 (with

coefficients from the second column of Table 3-5), yield an estimated xN2 of 0.18, with uncertainty equal to 7% of that value. For ternary calculations (Equation 3-7), the

additional factor with the a6 coefficient is the mole fraction of methane in the hydrocarbon solvent. If the liquid hydrocarbon is known to be 50% methane and 50%

a6 ethane (by mole fraction), that term is computed as 0.5 (with the appropriate a6 value

36 for the given temperature) and then multiplied by the other terms. The liquid mole

fractions of methane and ethane are then given by 0.5 1 xN2  ; the 0.5 varies to match the methane and ethane concentrations without nitrogen.

4 Titan Sea Properties

A difficulty in modeling the Titan seas is that only basic surface conditions (93 K,

0.15 MPa) are known from the orbital readings of Cassini. The solubility model in this work does not indicate how much nitrogen is dissolved in the bulk liquid of the seas, since it can only be applied at the Titan atmosphere/sea interface. Therefore, some model must be applied to obtain properties as a function of depth. The maximum depth of Ligeia

Mare is estimated at 200 meters or less (31) while the depth of Kraken Mare is not known from current measurements, but is at least 500 meters (59).

4.1 Solubility at Titan Temperatures

For calculating properties in Titan-specific conditions, the fitted solubility coefficients can be specialized to the 35 available ternary points in the temperature range

91-100 K. These coefficients are shown in Table 4-1, and have an MAE of 16%.

-1 46.5 a0 (MPa )

-1 -0.0439 a1 (K )

-2 -0.00001 a2 (K )

-1 3.07 a3 (MPa ) -1 -1 -0.000044 a4 ( K -MPa ) -2 -1 -0.00001 a5 ( K -MPa ) 0.933 a6 Table 4-1: Fitted coefficients for the normal phase of nitrogen-methane-ethane at

90-100 K.

4.2 Assumptions

In the present work, the Titan seas are modeled as a mixture of methane, ethane and nitrogen. Many other hydrocarbons are likely part of the composition, but for simplicity these are ignored. Another simplifying assumption is that the seas are static and horizontally uniform. This is the virtual equivalent of making a single, local depth sample.

The solution proceeds layer by layer as a function of depth level zi . The surface

baseline ( z0  0 ) is 93 K and 0.15 MPa, with composition determined by the solubility model. Composition and all properties are calculated for level , then used as a basis for

the algorithm computing level zi1 (see Figure 4-1). This process begins with pressure, which accumulates hydrostatically by gravitational acceleration g 1.35 m/s2. The surface methane-ethane mole ratios chosen are: 1 part methane to 19 parts ethane (to represent an ethane-rich sea), and 17 parts methane to 3 parts ethane (similar in methane- ethane ratio to Ligeia Mare (59)). These compositions are represented in the figures and tables below by the names Kraken and Ligeia.

Figure 4-1: Titan sea properties are estimated layer by layer from the surface

down.

4.3 Governing Equations

By the Gibbs’ phase rule, the inputs necessary to calculate all desired properties of

the 3-component seas are the compositions (mole fractions of methane x1,i , ethane x2,i

and nitrogen x3,i at each depth step i ) and two independent, intensive properties. The properties used in this work are temperature-pressure and pressure-enthalpy. These 5 unknowns require 5 equations to solve for them.

Pressure is the easiest to solve:

P P  g  z i1 i i (Equation 4-1)

This relation is most accurate for steps in which the density change is small, which is acceptable here since liquids are largely incompressible. The mole fractions of all components must add to one:

3  xi  x1  x 2  x 3 1 (Equation 4-2) j1

The first law of thermodynamics, written for multiple components, may be expressed under the assumption of equilibrium:

3 0 dG  V Pi1  P i  S i T i  1  T i  j , i n j ,i  1  n j ,i  j1

Where  ji, is the chemical potential of component j at depth level i . Dividing through by volume yields

3 nn jj,i 1 ,i (Equation 4-3) 0 Pi1  P i  s i i T i 1  T i  j , i   j1 VV

Several supplemental relations contribute to the use of Equation 4-3. The number of moles per unit volume of each component is based on mole fractions and density:

nxj,, i j i   (Equation 4-4) V i 3  xMk, i k k 1

The value of chemical potential  ji, must be derived from the standard potentials for nitrogen, methane and ethane, as well as potential gradients with temperature, as given in

Table 4-2. Standard conditions are T* 298 K and P* 101 kPa, where all 3 substances are gases.

 * (J/mol) (298 K, 1 bar) j  j (J/mol-K) T Nitrogen (120) 0 -191 Methane (120) -50,800 -186 Ethane (121) -32,700 -229 Table 4-2: Standard chemical potential and potential gradient with temperature

for nitrogen, methane and ethane.

The modification for a pressure different than ambient (122), as approximated by an

Pj,0 ideal gas at constant temperature, is to add the amount RT ln , where P is the u 0 P* j,0

partial pressure of component j in the atmosphere, T0  93 K and Ru is the universal gas constant. The modification for a temperature different than ambient follows by

 j adding the amount TT * . Then, at the pressure and temperature corresponding to T 0 vapor-liquid equilibrium, the potentials are equal and the surface potential of component j is

Pjj,0  * RTTT ln   * (Equation 4-5) j,0 j u 0PT*  0

The chemical potential of component j at depth level i 1 is calculated from the potential of the level above it:

V   R Tln x  P (Equation 4-6) j, i 1 j , i u j n j

The temperature term is based on (123) and the pressure term represents the effect of gravity through the hydrostatic column. Therefore the total differential of chemical potential between layers is

xj, i 1 x j , iVV n j , i 1 n j , i jiji,   , 1   jiuii , R T  1  Tln x jiui ,  R T  P i  1  P i  P i xj,,,, i n j i n j i n j i

n ji, After substituting Equation 4-5 for , this comes out in terms of mole fractions: V

3  xMk, i k xx1,ii 1 1, k 1 1,i R u T i 1  T iln x 1, i  R u T i  P i 1  P i   xx1,i i 1, i

33 xk, i M k i 1 x 1, i 1 x k , i M k kk11 Pi 1 (Equation 4-7) iix1, ix1, i M 3 x 1. i 1 M 1  M 3  x 2. i 1 M 2  M 3  

3  xMk, i k xx2,ii 1 2, k1 2,i R u T i 1  T iln x 2, i  R u T i  P i 1  P i   xx2,i i 2, i

33 xk, i M k i 1 x 2, i 1 x k , i M k kk11 Pi 1 iix2, ix2, i M 3 x 1. i 1 M 1  M 3  x 2. i 1 M 2  M 3  

3 xM 1x  x  x  k, i k  1,i 1 2, i 1 3, i k1 3,i R u T i 1  T iln x 3, i  R u T i  P i 1  P i   xx3,i i 3, i

33 xk, i M k i 11 x 1, i  1 x 2, i  1 x k , i M k kk11 Pi 1 iix3, ix3, i M 3 x 1. i 1 M 1  M 3  x 2. i 1 M 2  M 3  

The potential at each subsequent layer is based on the previous layer and current mole

fractions, as estimated by j, i 1  j , i    j , i .

The first law, Equation 4-3, can now be fully expressed in accessible variables, using

Equation 4-4:

3  xk,, i k i 3,ixx 1,1 i  1, i   3, i  2,1 i  2, i   3, i  0 P  P  s T  T    k 1 i11 i i i i i i M x M  M  x M  M 3 3 1,1ii 1 3 2,1 2 3  xM  k, i k k 1

This equation is linear in x1,i 1 , x2,i 1 , and isolating gives

BM  x B M  M     3 3,i 1,1 i  1 3  1, i 3, i  x2,i 1  (Equation 4-8) 2,ii 3, BMM 2  3  where

3  xk,, i k i PPi11 i k B si T i1  T i    3 i  xMk, i k k1

In addition to Equations 4-1, 4-2, and 4-8, two more equations (constraints) are still needed to fully specify the system. There are six conceivable ways to constrain the system. The first four presuppose no constraint on solubility at depth, and use an equi- chemical potential model to determine mole fractions at depth, along with another constraint on how properties vary layer to layer (isenthalpic, isentropic, pre-supposed temperature gradient, or pre-supposed methane to ethane mole fraction ratio). The last two ways impose either the surface solubility (SS) model where solubility does not change with depth or the full solubility (FS) model where solubility is calculated at each depth, along with the isenthalpic assumption layer to layer. The logic tree is summarized in Figure 4-2.

Figure 4-2: Model tree for estimating sea properties.

The first 4 variations impose a constant chemical potential, as an expression of equilibrium:

33 j, inn j , i j , i 1 j , i 1 jj11

Or, considering the moles per unit volume and substituting Equation 4-4

3 x   k,, i k i xx        k1 3,1i 1,1 i  1,1 i  3,1 i  2,1 i  2,1 i  3,1 i   ii3  1 (Equation 4-9) M3 x 1,1ii M 1  M 3  x 2,1 M 2  M 3   xMk, i k k1

4.4 Chemical Potential Model

4.4.1 Imposed Temperature

This variation declares a temperature profile with depth (which avoids solving for the

TTii1  in Equation 4-8). The solution procedure begins by computing pressure with

Equation 4-1, leaving only the mole fractions unknown. Specifying a value of x1,i 1

determines x2,i 1 and x3,i 1 by Equations 4-8 and 4-2 in sequence, and iterating on allows Equation 4-9 to converge.

The assumed temperature profiles are shown in Table 4-3 and Figure 4-3. The goal is to test a temperature increase, a temperature decrease, and an isothermal profile to see which is most plausible. The Ligeia simulations are run to a depth of 200 meters, and the

Kraken simulations go to 750 meters (a moderate excess over the lower bound of 500 meters).

95 95 b) 85% Methane 94.5 a) 5% Methane 94.5 Falling Falling Isothermal Isothermal 94 94 Rising Rising 93.5 93.5

93 93

T [K]

T [K] 92.5 92.5

92 92

91.5 91.5

91 91 0 100 200 300 400 500 600 700 0 50 100 150 200 z [m] z [m]

Figure 4-3: Assumed temperature profiles used for (a) ethane-rich and (b) methane-rich surface composition.

Direction Slope Falling -1 K/450 m Isothermal 0 Rising 1 K/450 m Table 4-3: Temperature profiles used in chemical potential modeling.

Figure 4-4 to Figure 4-6 show the mole fractions that result from the simulation with the assumed temperature gradients. Figure 4-7 shows the mixture density at the given composition, temperature, and accumulated pressure.

0.14 0.752 b) 85% Methane a) 5% Methane 0.12 Falling Falling 0.75 Isothermal Isothermal Rising Rising 0.1 0.748

4 0.08

xCH xCH 0.746 0.06

0.744 0.04

0.02 0.742 0 100 200 300 400 500 600 700 0 50 100 150 200 z [m] z [m]

Figure 4-4: Methane mole fraction for (a) ethane-rich and (b) methane-rich surface composition in the potential-temperature model.

1 0.135

0.95 0.13

0.9

6 0.125

xC 0.85 xC a) 5% Methane 0.12 Falling b) 85% Methane 0.8 Isothermal 0.115 Falling Rising Isothermal Rising 0.75 0.11 0 100 200 300 400 500 600 700 0 50 100 150 200 z [m] z [m]

Figure 4-5: Ethane mole fraction for (a) ethane-rich and (b) methane-rich surface composition in the potential-temperature model.

0.1 0.138 a) 5% Methane 0.08 Falling 0.136 Isothermal Rising 0.134 0.06

2 2 0.132

xN 0.04 0.13 b) 85% Methane 0.02 Falling 0.128 Isothermal Rising

0 0.126 0 100 200 300 400 500 600 700 0 50 100 150 200 z [m] z [m]

Figure 4-6: Nitrogen mole fraction for (a) ethane-rich and (b) methane-rich surface composition in the potential-temperature model.

646 523.5

645 523 644

]

3 522.5 643

[kg/m



 522 642 a) 5% Methane b) 85% Methane Falling Isothermal 521.5 Falling 641 Isothermal Rising Rising

640 521 0 100 200 300 400 500 600 700 0 50 100 150 200 z [m] z [m]

Figure 4-7: Density for (a) ethane-rich and (b) methane-rich surface composition in the potential-temperature model.

The Kraken density curves in Figure 4-7a suggest that the most plausible temperature gradient is one that falls with depth. The Ligeia density curves in Figure 4-7b are decreasing with depth, which is not intuitive in a gravitational field.

4.4.2 Constant Enthalpy

This variation is like the imposed-temperature solution, except that the difference

TTii1  is derived from setting one of the Maxwell relations (124) for enthalpy equal to zero:

V  0 Cp dT  V  T dP T P

If enthalpy is constant at all depths ( dH  0 ), then rewriting the differential as a finite difference and dividing out by mass yields the temperature increment

 PPii1  11  TTTi1  i  i   (Equation 4-10) cTpi  TPii,

The derivative in Equation 4-10 is estimated numerically with a temperature difference of

0.1 K. The quantities inside the brackets are evaluated at level i , with values assumed uniform across the distance step.

The calculated temperature gradient from this method is shown in Figure 4-8, followed by the mole fractions in Figure 4-9 to Figure 4-11 and the density in Figure

4-12.

93.1

5% Methane 85% Methane 93

92.9

T [K] 92.8

92.7

92.6 0 100 200 300 400 500 600 700 z [m]

Figure 4-8: The calculated temperature gradients in the potential-isenthalpic model.

0.78 0.08 b) 85% Methane a) 5% Methane 0.775 0.075 0.77 0.07 0.765

4 0.065 0.76

xCH xCH 0.06 0.755

0.055 0.75

0.05 0.745

0.045 0.74 0 100 200 300 400 500 600 700 0 50 100 150 200 z [m] z [m]

Figure 4-9: Methane mole fraction for (a) ethane-rich and (b) methane-rich surface composition in the potential-isenthalpic model.

0.14 0.95 b) 85% Methane a) 5% Methane 0.94 0.12

0.93 0.1

6 0.92 H

xC 0.91 0.08

0.9 0.06 0.89

0.88 0.04 0 100 200 300 400 500 600 700 0 50 100 150 200 z [m] z [m]

Figure 4-10: Ethane mole fraction for (a) ethane-rich and (b) methane-rich surface composition in the potential-isenthalpic model.

0.04 0.17 a) 5% Methane b) 85% Methane 0.035 0.16 0.03 0.15

2 0.025 2

xN 0.14 0.02

0.015 0.13

0.01 0.12 0 100 200 300 400 500 600 700 0 50 100 150 200 z [m] z [m]

Figure 4-11: Nitrogen mole fraction for (a) ethane-rich and (b) methane-rich surface composition in the potential-isenthalpic model.

647 524.5 a) 5% Methane b) 85% Methane 646.5

646 524

]

3 3 645.5

[kg/m [kg/m 645



 523.5 644.5

644

643.5 523 0 100 200 300 400 500 600 700 0 50 100 150 200 z [m] z [m]

Figure 4-12: Density for (a) ethane-rich and (b) methane-rich surface composition in the potential-isenthalpic model.

Imposing constant enthalpy at increasing pressures tends to cause a falling temperature gradient, consistent with Equation 4-10. Kraken and Ligeia show similar trends in mole fraction and density to Figure 4-7, but with regular jumps for Ligeia

(possibly imposed by the isenthalpic condition). The overall trend is toward increasing density with depth.

4.4.3 Constant Entropy

This variation takes the premise of constant entropy (adiabatic and reversible).

Because any change in the condition of the seas generates entropy, however, this is disregarded as highly unlikely.

4.4.4 Constant Mole Ratio

This variation imposes, in place of any predetermined temperature difference, a

constant ratio of methane to ethane ( x2,i to x1,i ) at all depths. This method was not considered for results because it does not allow all mole fractions to be estimated independently (as Equation 4-8 would suggest).

4.5 Imposed Solubility

4.5.1 Surface Solubility (SS)

This variation calculates the sea composition once using the solubility model, at the surface, and fixes it constant for all other levels. This essentially constrains the three mole

fractions at depth. After the pressure Pi is computed using Equation 4-1 and the surface enthalpy is imposed, all other properties are known through REFPROP. Such a description assumes that the Titan seas are well-mixed by wind and tides (38) (40). The results for this variation are presented alongside the next one below.

4.5.2 Full Solubility (FS)

This variation is like the SS model except that composition is recalculated for each depth level using the solubility model. The value of mole fractions comes from the

solubility model using and Ti , which can result in very large nitrogen fractions at depth (especially for methane-rich seas). Physically realizing this arrangement may require an active nitrogen source within the sea.

Figure 4-13 shows the estimated concentrations of nitrogen under the FS and SS assumptions. Figure 4-14 and Figure 4-15 show estimated trends for density and temperature in the Titan seas, respectively.

0.5

5% Methane, SS 5% Methane, FS 0.4 85% Methane, SS 85% Methane, FS

0.3

xN 0.2

0.1

0 0 100 200 300 400 500 600 700 z [m]

Figure 4-13: Mole fraction of dissolved nitrogen in the FS and SS isenthalpic models.

658 640

656 a) 5% Methane b) 85% Methane 620 SS SS 654 FS FS 600

] 652

]

3 650 580

kg/m

[kg/m

 648  560 646 540 644

642 520 0 100 200 300 400 500 600 700 0 50 100 150 200 z [m] z [m]

Figure 4-14: Estimated Titan sea density profile for a) ethane-rich and b) methane-rich seas in the FS and SS isenthalpic models.

94 94

93 93

92 92

91 91

T [K]

90 90 a) 5% Methane b) 85% Methane SS 89 FS 89 SS FS

88 88 0 100 200 300 400 500 600 700 0 50 100 150 200 z [m] z [m]

Figure 4-15: Estimated Titan sea temperature profile for a) ethane-rich and b) methane-rich seas in the FS and SS isenthalpic models.

In the FS model, with large changes in nitrogen fraction, the temperature and density change significantly. Note that the Kraken, FS model ends near 300 meters when

REFPROP fails to compute liquid properties (perhaps a sign of freezing).

4.6 Assessment

All the density results in Figures 4-7, 4-12 and 4-14 (excluding the FS model) suggest changes of less than 1% over the simulated depth range. Therefore, we conclude that the

Titan seas are, to first order, incompressible, which makes the constant-density assumption in Equation 4-1 valid.

One requisite for a static, physically stable fluid body in gravity is that density increase with depth; otherwise, the body tends to overturn. By this measure, Kraken Mare is expected to be stable, but the case of Ligeia Mare (based on the chemical potential- temperature model) is uncertain. The potential-isenthalpic model may not be smooth with depth, but the overall trend in density is increasing, so it meets the criterion for stability.

Therefore, the potential-isenthalpic model will be used for properties (it is preferred over the solubility-isenthalpic model because chemical potential is a more rigorous treatment).

From the density extremes in Figures 4-7, 4-12 and 4-14, the expected density range of the Titan seas is 515 to 650 kg/m3. These numbers include some margin around the highest and lowest estimates in the Figures. The higher density values correlate with higher ethane concentrations, since pure ethane (650 kg/m3) is denser than pure methane

(420 kg/m3).

The limits of this description at greater depths (primarily in Kraken Mare), or what occurs at those limits, is more speculative. One possibility is dramatic stratification, between a methane-rich surface layer (including from stream inflows) and a deep layer of ethane or heavier liquid hydrocarbons. Judging from the temperatures near the triple point suggested in (39), freezing is also a distinct possibility. This could take the form either of a clear divide (“ice”) or of gradually thickening “slush,” and it is not clear whether the solid would sink or float (125) (126). Raulin (14) proposed seabed deposits composed of acetylene or cyanide, or related polymers.

A first cut at mixture freezing temperatures can be made based on the chemical equilibrium of a solution between liquid and solid states (123). A discrete form of the equation is

R u ln x 11M solvent  (Equation 4-11) TTEff

This is a discrete, linearized form based on the point xsolvent 1 (“solvent” being “CH4”

or “C2H6”), where Tf is the freezing point, E f is the heat of fusion (with values from

(127)) and M is the molar mass (all properties of the pure solvent). For a methane- ethane mixture, Equation 4-11 can be used 2 different ways, with either methane or ethane having the role of solvent. The two lines that result can be plotted against a common horizontal axis of one of the mole fractions (see Figure 4-16), and the intersection of the lines predicts the minimum freezing point. In the case of methane and ethane, that prediction is about 70 K at 65-70 mol% methane.

T [K]

70 Methane solvent Ethane solvent

65 0 0.2 0.4 0.6 0.8 1 xCH 4

Figure 4-16: The estimated minimum freezing point of the methane-ethane mixture

is about 70 K, at 65-70 mol% methane.

All of the above models should be treated with caution because of the many simplifying assumptions in the 1D modeling processes. Earth’s oceans circulate in a variety of ways, including the thermohaline cycle in the North Atlantic and the El Nino shifts in warm Pacific waters (128). There are also intruding flows such as precipitation and river outlets, which on Titan would probably be methane-rich. More specific to Titan, the balance of heat from space and heat from the interior, such as core tidal friction, is difficult to determine.

4.7 Properties

The additional properties needed for the submarine, based on the chemical potential- isenthalpic model, are viscosity, speed of sound, and dielectric factor. These are shown, respectively, is Figures 4-17 to 4-19.

1200

1000

800

Pa-s] 600

 400

200 5% Methane 85% Methane

0 0 100 200 300 400 500 600 700 z [m]

Figure 4-17: Estimated viscosity profile in the Titan seas.

2000

1900

1800

1700

[m/s] s 1600

1500

1400 5% Methane 85% Methane 1300 0 100 200 300 400 500 600 700 z [m]

Figure 4-18: Estimated speed of sound profile in the Titan seas.

1.95

1.9

1.85

1.8

 1.75

1.7

1.65 5% Methane 85% Methane 1.6 0 100 200 300 400 500 600 700 z [m]

Figure 4-19: Estimated dielectric profile in the Titan seas.

The properties shown here vary much more over the range of composition than over the range of depth. Viscosity, in particular, is both an important parameter in the design of parts such as valves, propellers and pumps; and the variation in its value may reach a factor of 5. This makes it a key design point for the submarine.

4.8 Uncertainty

The REFPROP manual (8) gives the uncertainty of an LNG mixture density as less than 0.5%, and the uncertainty in liquid speed of sound as less than 2%. Uncertainty in viscosity and dielectric factor are assumed to be like speed of sound, but are not separately mentioned in the manual. The uncertainty in nitrogen fraction is the MAE of the coefficients, 16%, while the methane and ethane mole fractions are divided in the same proportion as on the surface. The assumed uncertainties in temperature and pressure are less than 0.05 K (129) and 0.05% of 150 psia (or 0.0005 MPa).

The propagation of uncertainty is expressed (130), with density as an example, as

2 22         UUUU2  2  2   2 (Equation 4-12)  xN 2   T   P xN 2   T    P 

Other properties are treated similarly. The first three terms arise from numerical derivatives with respect to the corresponding values, while the last term is the direct

REFPROP uncertainty. The results are shown in for Table 4-4 for ethane-rich and Table

4-5 for methane-rich seas. The total uncertainty is broadly higher in methane-rich than in ethane-rich seas, due to the higher concentration of nitrogen. The most uncertain property, especially at depth, is viscosity.

z (m) x x x T (K)  (kg/m3) U CH4 CH26 N2  (kg/m3) 0 0.0495 0.9403 0.0102 93.00 643.56 3.22 20 0.0504 0.9386 0.0110 92.99 643.66 3.23

40 0.0513 0.9369 0.0118 92.98 643.76 3.25 60 0.0521 0.9353 0.0126 92.97 643.86 3.27 80 0.0530 0.9336 0.0134 92.96 643.96 3.30 100 0.0538 0.9320 0.0142 92.95 644.05 3.33 120 0.0547 0.9303 0.0150 92.94 644.15 3.37 140 0.0555 0.9287 0.0157 92.93 644.24 3.41 160 0.0564 0.9271 0.0165 92.92 644.34 3.46 180 0.0572 0.9255 0.0173 92.91 644.43 3.52 200 0.0581 0.9239 0.0180 92.90 644.52 3.58 220 0.0589 0.9224 0.0188 92.89 644.62 3.64 240 0.0597 0.9208 0.0195 92.88 644.71 3.71 260 0.0605 0.9192 0.0203 92.88 644.80 3.78 280 0.0614 0.9177 0.0210 92.87 644.89 3.85 300 0.0622 0.9161 0.0217 92.86 644.98 3.93 320 0.0630 0.9146 0.0225 92.85 645.08 4.01 340 0.0638 0.9130 0.0232 92.84 645.17 4.10 360 0.0646 0.9115 0.0239 92.83 645.26 4.18 380 0.0654 0.9100 0.0246 92.82 645.35 4.27 400 0.0662 0.9084 0.0254 92.81 645.44 4.36 420 0.0670 0.9069 0.0261 92.80 645.52 4.46 440 0.0678 0.9054 0.0268 92.79 645.61 4.55 460 0.0686 0.9039 0.0275 92.78 645.70 4.65 480 0.0694 0.9024 0.0282 92.77 645.79 4.75 500 0.0702 0.9009 0.0289 92.76 645.88 4.86 520 0.0710 0.8994 0.0296 92.75 645.97 4.96 540 0.0718 0.8979 0.0303 92.74 646.05 5.06 560 0.0725 0.8964 0.0310 92.73 646.14 5.17 580 0.0733 0.8950 0.0317 92.72 646.23 5.28 600 0.0741 0.8935 0.0324 92.71 646.32 5.39 620 0.0749 0.8920 0.0331 92.70 646.40 5.50 640 0.0757 0.8906 0.0338 92.69 646.49 5.61 660 0.0764 0.8891 0.0345 92.68 646.58 5.72 680 0.0772 0.8876 0.0352 92.68 646.66 5.83 700 0.0780 0.8862 0.0358 92.67 646.75 5.95 720 0.0788 0.8847 0.0365 92.66 646.83 6.06 740 0.0795 0.8833 0.0372 92.65 646.92 6.18 Table 4-4: Property values and uncertainties with depth for ethane-rich seas

(pressure, temperature, composition, density).

 (μPa-s) U (μPa-s) v (m/s) U (m/s)  U z (m)  s vs  0 1014.81 36.86 1973.3 39.54 1.9274 0.0386

20 1011.64 37.75 1972.1 39.52 1.9269 0.0385 40 1008.52 38.65 1971 39.5 1.9265 0.0385 60 1005.44 39.58 1969.9 39.48 1.926 0.0385 80 1002.4 40.53 1968.8 39.47 1.9256 0.0385 100 999.41 41.49 1967.7 39.45 1.9251 0.0385 120 996.45 42.46 1966.6 39.43 1.9247 0.0385 140 993.52 43.44 1965.5 39.41 1.9243 0.0385 160 990.63 44.43 1964.5 39.39 1.9238 0.0385 180 987.77 45.43 1963.4 39.37 1.9234 0.0385 200 984.94 46.44 1962.3 39.36 1.923 0.0385 220 982.13 47.45 1961.3 39.34 1.9225 0.0385 240 979.35 48.46 1960.2 39.32 1.9221 0.0384 260 976.6 49.48 1959.1 39.3 1.9217 0.0384 280 973.88 50.5 1958.1 39.28 1.9213 0.0384 300 971.17 51.53 1957 39.26 1.9209 0.0384 320 968.5 52.55 1956 39.24 1.9204 0.0384 340 965.84 53.58 1955 39.22 1.92 0.0384 360 963.2 54.6 1953.9 39.2 1.9196 0.0384 380 960.59 55.63 1952.9 39.18 1.9192 0.0384 400 958 56.65 1951.9 39.16 1.9188 0.0384 420 955.42 57.67 1950.8 39.14 1.9184 0.0384 440 952.87 58.69 1949.8 39.12 1.918 0.0384 460 950.33 59.71 1948.8 39.1 1.9176 0.0384 480 947.82 60.73 1947.8 39.08 1.9171 0.0384 500 945.32 61.74 1946.7 39.06 1.9167 0.0384 520 942.83 62.75 1945.7 39.04 1.9163 0.0384 540 940.37 63.76 1944.7 39.02 1.9159 0.0384 560 937.92 64.77 1943.7 39 1.9155 0.0384 580 935.49 65.77 1942.7 38.97 1.9151 0.0384 600 933.07 66.77 1941.7 38.95 1.9147 0.0384 620 930.67 67.76 1940.7 38.93 1.9143 0.0384 640 928.28 68.76 1939.7 38.91 1.9139 0.0384 660 925.91 69.74 1938.6 38.89 1.9135 0.0384 680 923.56 70.73 1937.6 38.86 1.9131 0.0384 700 921.21 71.71 1936.6 38.84 1.9127 0.0384 720 918.88 72.69 1935.6 38.82 1.9123 0.0384 740 916.57 73.66 1934.6 38.8 1.9119 0.0384

Table 4-4 continued: Property values and uncertainties with depth for ethane-rich seas (viscosity, speed of sound, dielectric factor).

z (m) x x x T (K)  (kg/m3) U CH4 CH26 N2  (kg/m3) 0 0.7424 0.1310 0.1266 93.00 523.10 6.02

10 0.7488 0.1171 0.1340 93.00 523.32 5.21

20 0.7488 0.1172 0.1340 92.99 523.34 5.21

30 0.7493 0.1161 0.1346 92.99 523.36 5.20

40 0.7442 0.1269 0.1288 92.99 523.21 5.68

50 0.7453 0.1246 0.1301 92.98 523.26 5.52

60 0.7523 0.1097 0.1381 92.98 523.50 5.27

70 0.7527 0.1086 0.1386 92.98 523.52 5.30

80 0.7593 0.0946 0.1461 92.97 523.75 6.41

90 0.7550 0.1038 0.1412 92.97 523.62 5.57

100 0.7559 0.1018 0.1423 92.97 523.66 5.71

110 0.7635 0.0857 0.1508 92.96 523.91 7.54

120 0.7596 0.0940 0.1464 92.96 523.80 6.48

130 0.7675 0.0773 0.1552 92.96 524.06 8.80

140 0.7701 0.0717 0.1582 92.95 524.16 9.70

150 0.7668 0.0787 0.1545 92.95 524.07 8.58

160 0.7695 0.0730 0.1575 92.95 524.16 9.49

170 0.7722 0.0673 0.1605 92.94 524.26 10.44

180 0.7752 0.0610 0.1638 92.94 524.36 11.54

190 0.7777 0.0558 0.1665 92.94 524.45 12.49

200 0.7752 0.0610 0.1638 92.93 524.39 11.54

Table 4-5: Property values and uncertainties with depth for methane-rich seas

(pressure, temperature, composition, density).

 (μPa-s) U (μPa-s) v (m/s) U (m/s)  U (m)  s vs 

0 227.88 6.19 1458 36.63 1.6843 0.0344 10 218.22 12.02 1442.3 42.42 1.6773 0.0356 20 218.31 12 1442.4 42.41 1.6773 0.0356 30 217.59 12.59 1441.2 42.91 1.6767 0.0357 40 225.25 7.42 1453.7 38.2 1.6823 0.0346 50 223.69 8.36 1451.1 39.14 1.6812 0.0348 60 213.21 16.19 1434.1 46 1.6735 0.0365 70 212.55 16.77 1433 46.5 1.673 0.0366 80 204.48 23.22 1417.1 53.87 1.6657 0.0389 90 209.72 19.12 1427.6 48.97 1.6705 0.0373 100 208.63 20.02 1425.5 50 1.6695 0.0377 110 199.73 27.12 1407.2 58.82 1.6611 0.0407 120 204.33 23.51 1416.7 54.18 1.6654 0.039 130 195.32 30.72 1397.7 63.69 1.6567 0.0425 140 192.42 33.06 1391.4 67.01 1.6538 0.0438 150 196.13 30.16 1399.4 62.88 1.6575 0.0422 160 193.16 32.56 1393 66.25 1.6545 0.0435 170 190.27 34.91 1386.6 69.65 1.6515 0.0449 180 187.09 37.47 1379.4 73.51 1.6482 0.0465 190 184.49 39.57 1373.5 76.76 1.6454 0.0479 200 187.17 37.49 1379.6 73.5 1.6482 0.0465 Table 4-5 continued: Property values and uncertainties with depth for methane- rich seas (viscosity, speed of sound, dielectric factor).

5 Effervescence

5.1 Problem Statement

The submarine will be subject to cryogenic temperatures, therefore all external equipment needs to endure, operate, and cycle at temperatures below 96K. Meanwhile, all internal systems are maintained at 290K due to the thermal management system, which is based on an energy balance between the heat generated by the radioisotope power system, and the heat loss to the surroundings through the submarine exterior (4).

Waste heat (131) is distributed from the power system at the aft end of the sub to the

60 forward end; and along with the appropriately sized insulation thickness, the science equipment is maintained at ambient. The rest of the heat is rejected into the surrounding sea. Figure 5-1 illustrates the anticipated temperature gradients between the submarine and the Titan seas. As shown, there is a significant temperature difference between the power system and the Titan sea. This temperature difference is not likely to cause local boiling of the liquid, since the seas are so close to the freezing point, but it may be enough to cause nitrogen gas that is dissolved in the liquid to come out of solution.

Figure 5-1: Internal temperatures and parts of the submarine, with insulation lines. Blocks are shown for subsystems (Purple = science, blue = communication and data handling, red = power).

On Titan, the submarine is intended for pressures up to 1 MPa (150 psia). For terrestrial submarines, the solubility of air in water at that pressure is negligible (< 0.1% mol from (7)). On Titan, however, the atmospheric pressure is 1.5 times higher and nitrogen is highly soluble. Solubility is thus expected to be a concern for all submersible designs. Specifically as it relates to the submarine operation, bubbles that come out of solution due to excessive waste heat will cause problems in two ways:

1. In a quiescent or hovering case, bubbles that form may interfere with

instrumentation such as dielectric constant measurements, turbidity, acoustic

transmission (62), sonar (56), depth, and any visualization. This would jeopardize

critical mission objectives.

2. In a moving case, bubbles that form at the forward end and along the craft may

coalesce at the aft end and cause cavitation in the propellers, potentially hindering

control and navigation of the sub.

In order to faithfully quantify effervescence, sub-models are needed for skin temperature, solubility, bubble nucleation, bubble growth, and bubble development as a function of location with the Titan Sea.

5.2 Effervescence Model

5.2.1 Skin Temperature

The parameters used in calculating the submarine skin temperature are shown in

Table 5-1, and adopted from (5). The diffusion coefficient of nitrogen in methane is the only fluid property taken as a constant, because few studies of it exist in the literature. All other properties are computed from the software REFPROP.

Quantity Units Value m 0.117 Propeller length Lprop m 0.14 Propeller width Wprop m 0.14 Propeller height H prop m 6.54* Length Lsub m 1.23* Width Wsub Height of main body (excluding m 0.78*

communication array) H sub m2 12.5* Skin area Askin K Variable Skin temperature Tskin K 93 Sea temperature Tbulk

m/s Variable Travel speed vsub W Variable Waste heat Qwh Titan gravity g m/s2 1.35 Convection coefficient h W/m2-K Variable Diffusion coefficient of nitrogen in liquid m2/s 2*109 (132) methane D *Resized from the Phase I design to allow for operation in methane-rich seas (see

Section 6.2.1).

Table 5-1: Parameters for computing skin temperature and general submarine dimensions.

The equilibrium of the waste heat flux with convection heat flux in the Titan seas, determines the skin temperature of the vehicle. The latter heat flux is expressed as

Q conv (Equation 5-1) h Tskin T bulk  Askin for the convection coefficient ℎ. In the case where the submarine is quiescent, natural convection occurs, and the correlation used for the convection coefficient ℎ is (133):

2   hH 0.387Ra 1/6 12 sub Hsub , Ra 10 (Equation 5-2) NuH  0.6  Hsub sub 9/16 8/27 k 0.559 1  Pr

In the case where the submarine is in moving through the Titan Sea, turbulent forced convection occurs and the correlation used is (133):

hL Nu sub 0.037Re4/5 Pr 1/3 , 0.6 Pr 60 (Equation 5-3) LLsubk sub

In these correlations, the Reynolds number is based on submarine length:

vL Re  sub sub (Equation 5-4) Lsub  while the Rayleigh number is based on submarine height:

g  T T H 3 Ra  l skin bulk sub Pr (Equation 5-5) Hsub  2

The required liquid properties in these computations – thermal conductivity k , thermal

expansion coefficient l , kinematic viscosity  and Prandtl number Pr - are computed at the film temperature around the vehicle:

1 TTT  (Equation 5-6) film2 skin sea

Finally, the first-law power balance between the submarine waste heat and Titan sea convection is:

QQwh conv (Equation 5-7) where the active area of the submarine surface is the same for both heat rates, implying equal heat fluxes. To determine the resultant skin temperature, Equation 5-7 is solved using an iterative, guess-and-check method:

1. Initiate a value for Tskin , around 100 K.

2. Solve the equations for all heat transfer terms.

3. Check the energy balance (Equation 5-7) and iterate until equality is met. If

QQwh conv , then the skin temperature is too high. If QQwh conv , then the skin

temperature is too low.

5.2.2 Applied Solubility

Supersaturation is defined as the ratio of the difference in the dissolved solute concentration in the liquid and the bubble surface, to that at the bubble surface:

xx S  N2,sea N2,skin (Equation 5-8) xN2,skin

64 where x is the mole fraction of the solute (dissolved gas) in the liquid, and subscript b

denotes bulk liquid (Titan sea). xN2,sea and xN2,skin are functions of the temperature of the bulk sea and the submarine temperature, respectively, as well as the sea pressure.

5.2.3 General Bubble Nucleation

Given that solubility limits can determine how much gas is available in the liquid, conditions for when a bubble forms, or nucleates, are next required. Jones et al. (134) presented a thorough review of bubble nucleation in liquids supersaturated with a gas.

Four types of nucleation are classified:

1. Classical homogenous nucleation where bubbles are formed in the bulk liquid.

2. Classical heterogeneous nucleation where bubbles form on the solid surface in

contact with the liquid.

3. Nucleation at pre-existing gas cavities on solid surfaces with a nucleation

energy barrier.

4. Nucleation at pre-existing gas cavities on solid surfaces without an energy

barrier.

For bubbles to grow, the initial nucleus size must be larger than some critical radius.

Smaller bubbles will dissolve back into solution.

Wilt et al. (135) calculated nucleation rates for Type 1 and 2 given above for water/carbon dioxide solutions and found the nucleation rates to be quite small and unlikely to lead to experimentally observable bubbles unless a) the supersaturation ratio is very high (1100 – 1700) for homogeneous nucleation, or b) the liquid is very non- wetting (liquid/solid contact angle = 176o) for a supersaturation ratio of 5. The supersaturation ratio for nitrogen gas in the liquid methane/ethane sea is less than 1 for all

65 reasonable submarine conditions. Further, the liquid methane/ethane mixture is assumed to be quite wetting (near-zero contact angle), as is the case for all cryogenic liquids in contact with metallic surfaces (136). Under these conditions, Type 1 or Type 2 nucleation is highly unlikely to occur for the Titan submarine; the most probably type of nucleation is at pre-existing gas cavities on the surface without an energy barrier (Type 4).

The active nucleation site density on surfaces in boiling systems is well researched, see for example (137). Very few quantitative studies appear to exist for gas evolution on surfaces in contact with supersaturated solutions (see for example (138)

(139)). For analysis in the current work, the model of (137) will be employed, modified for effervescence in place of boiling.

5.2.4 Bubble Incipience and Nucleation Site Density

Hibiki and Ishii (137) developed a mechanistic model for the active nucleation site density on boiling surfaces utilizing the size and cone angle distributions of cavities on the surface. They improved upon the model of (140) by adopting physically sound approximations for cavity radius and cone angle distributions. They validated their model by comparing predictions with the boiling data from numerous investigators obtained from surfaces of different materials and experimental fluids, covering a wide range of contact angles, pressure flow conditions, and number of bubbles produced. The model equations used by (137) are:

2  N N1  exp  exp f    1 (Equation 5-9) nn2   8 Rc

23 f   0.01064  0.48246    0.22712    0.05468   

  log * 

 *  lg (Equation 5-10) g

5 2 where Rcrit is the critical bubble radius for nucleation, Nn  4.72*10 sites/m from

(137) (used as m-2 since counting sites is dimensionless),  is the liquid-solid contact

6 angle at the liquid-vapor interface (rad),   0.722 rad,   2.5*10 m, l is the

saturated liquid density of the pure solvent (methane, ethane, or their mixture) and g is the saturated vapor density of the pure solute (nitrogen). Equation 5-9 is used together with an expression for critical radius to compute nucleation site density, in m-2. Hibiki and Ishii (137) originally used this set of equations for boiling; here the same model equations are used for effervescence, but the critical radius is changed to represent effervescence of dissolved gas:

2 Rcrit  (Equation 5-11) PSsea

where  is the surface tension of the pure solvent, Psea is the ambient pressure (written in

Pa, not MPa, to resolve units to meters), and S is the supersaturation (Equation 5-8).

Note that Qi and Klausner (138) use a similar expression for critical radius to predict nucleation site density of dissolved gas. Curiously, they start with the same formulation as (140) but use different approximations for the cavity size and cone angle distributions.

The constants in these probability distributions are specific to the materials comprising the nucleation surfaces, and not universal as in the model proposed by (137).

5.2.5 Bubble Growth

Scriven (141) developed a model for the spherically symmetric growth of a bubble in a liquid of infinite extent driven by both heat and mass transfer. For heat transfer driven

67 growth, the liquid is assumed to be superheated, in which a vapor bubble in introduced whose surface temperature is the equilibrium saturation temperature at the prevailing pressure. The bubble is assumed to grow from zero size. For mass transfer driven growth, the liquid is assumed to be a two component mixture (liquid solvent and vapor solute), with the solute diffusing into the bubble and causing it to grow. In this work, Scriven’s theory is restricted to the mass transfer driven growth of a single component gas. The diffusion controlled growth of multicomponent gas bubbles was later studied by (142).

Thus, Scriven’s theory is used to assess the rate of growth of nitrogen bubbles once they are nucleated on the surface of the submarine. Two assumptions are needed:

1. The effect of the submarine surface on the bubble growth is insignificant, or the

bubble grows in an infinite liquid space.

2. The effect of neighboring bubbles on the growth rate of a bubble is insignificant.

Relaxing these two assumptions will negate the applicability of Scriven’s theory of spherically symmetric bubble growth, and the analysis is non-trivial.

The model for the growth of a bubble from zero initial size can be described as

 follows. Let C  sea be the normalized mass concentration of the dissolved gas sea surf in the liquid, where  is the liquid mixture density. Denoting the diffusivity of the solute (nitrogen gas) in the solvent (liquid methane/ethane) by D , the governing equation for solute transport, assuming spherical symmetry, is:

*2 C R dR  C D 2  C * 2 2 r (Equation 5-12) t1  r dt  r r  r  r where Rt  denotes the time-dependent radius of the bubble. Two boundary conditions and two initial conditions are required to solve Equation 5-12: 68

Rt 00

C t0, r 0

C t, r R ( t ) 1

C t,0 r   

The initial conditions state that the critical radius of the bubble is zero, and the concentration is initially zero. The boundary conditions state that the concentration at the bubble radius is 1, and that far away from the bubble surface, the concentration goes to 0.

The solution for the solute concentration field is obtained using a similarity solution

C r, t  F   , where the similarity variable is:

r   (Equation 5-13) 2 Dt

Scriven defines the growth of bubble radius as:

R t  2 Dt (Equation 5-14) where  is the bubble growth constant. Rewriting Equation 5-12 using the similarity variable, and plugging in Equation 5-14, Equation 5-12 becomes:

 123* exp 2 d  2*   1  F     123* exp 2 d  2*   1 

This relation will yield the concentration as a function of time and radial distance for a given constant . To determine the value of , a mass balance is taken at the vapor/liquid interface:

* dR C g  skin1 *    bu   skin  D 1 dt r r R t which yields the following expression for  :

1  221  sea skin 2 exp2  2x  1 dx   1     0 1 x  g skin 1  1 

Using Equations 5-8 and 5-10, the right hand side can be rewritten as

sea  skin  l * xN2,skin S 1    x N2, skin S . Therefore, the preceding expression gg becomes:

1 1   (1 )x S (1 ) x S 222 exp  2x  1 dx N2,skin  N2, skin  2   1 x 1    skin 0    11skin 1   g 1 l

(Equation 5-15)

* For given values of sea , skin , and g , or equivalently for given values of  , xN2,skin , and S , Equation 5-15 can be solved numerically for .

5.2.6 Volume and Area Coverage

Now, models are needed for how the bubbles accumulate and move along the submarine to determine both the volume gas fraction in the sea and also bubble area coverage along the submarine. General assumptions used in this section are:

 Nucleation sites are uniformly concentrated according to Equation 5-9.

 The production of bubbles at each site is a continuous stream.

 The growth of each bubble is spherically symmetric and not influenced by the

surface it nucleated from, or by other bubbles, in line with Scriven’s assumptions.

Bubble growth is therefore given by Equation 5-14 whether the bubble is moving

or stationary. This is a conservative approach and will essentially lead to

overestimating the amount of gas.

 The bubble area used in area coverage calculations is the frontal area of a (bubble)

sphere facing a flat surface (the submarine), a circle of the bubble radius. In other

words, each bubble projects an area of  R2 on the bottom of the submarine. This

is also conservative, since some bubbles will block the full projection of other

bubbles onto the sub surface.

 In the moving case, bubble buoyancy is neglected and all bubbles are swept

uniformly at the specified submarine velocity to the aft end. This is a conservative

approach and over-predicts bubble coverage because it ignores how some bubbles

will rise up the submarine side due to buoyancy.

 In the nonmoving case, buoyancy will however, carry bubbles away that form on

the top side of the sub; therefore only bubbles that form on the bottom side are

considered in calculations.

 The projected area of the bottom of the submarine is used for calculations, rather

than the actual area; this again over-predicts bubble coverage.

 The influence of the boundary layer is neglected; the flow over the skin is at

speed vsub .

5.2.6.1 Moving Case

First the moving case is addressed. At each nucleation site, new bubbles form and accumulate with bubbles that formed at sites upstream. The amount of accumulation at

71 any location is determined by the submarine velocity, as well as the nucleation and growth rates.

Each bubble has had time to grow, according to the distance from its nucleation site to the position of calculation, and the total bubble coverage at any position is the sum of the coverage of all individual bubbles. Illustrated in Figure 5-2 is the coordinate system and dimensions used for computations. Bubbles move in the positive x direction at speed

vsub .

Figure 5-2: The bottom-view dimensions of the rectangle considered in the

moving case. The projected area WLsub sub (see Table 5-1 for the submarine

dimensions) is used in place of skin area.

For bubbles that nucleate at a position 0'xxon the surface section, the growth time before reaching position 푥 is:

x' tx'  vsub

L Thus, the maximum bubble diameter is estimated by the growth time t  sub . The sum Lsub vsub of the volume of all bubbles that cross position x is determined by integrating the bubble

volume due to each width “strip” Nn W sub dx ' from the submarine nose to position x:

x 4 3 V()()' x N W R t dx bub n sub x'  3 0 where the volume of each bubble is that of a sphere with radius from Equation 5-14 at

time tx ' . Evaluating this integral yields:

3/2 3 5/2 64Nn W sub D x Vxbub () 3/2 (Equation 5-16) 15vsub

The units in this equation are m3.

In the moving case, the most important location for bubble volume accumulation is at the aft end, at the entrance region of the propellers. The volume occupied by a strip of bubbles is taken to be a rectangular box as shown in Figure 5-3. The bottom surface bubbles are assumed to occupy a cross-section equal to the height times the width of the submarine, and a length equal to the propeller length. Bubbles coming off the top surface are assumed to rise above the height of the propellers and are therefore ignored; bubbles

from the bottom surface are assumed to distribute evenly over the square WLsub sub , reflecting the varying height of surface elements on the submarine bottom, as opposed to concentrating in the propeller. The fraction of that reference volume occupied by bubbles is what affects the operation of the propellers, and is given by:

3/2 3 5/2 VLbub sub  64NDLn sub Volume fraction 3/2 (Equation 5-17) Lprop H sub W sub15 v sub H sub L prop

This is a ratio of volumes, and is therefore dimensionless. The submarine width Wsub cancels out of this ratio because the bubbles from each infinitesimally-thin strip (based on

projected area) down the length Lsub occupy, at the aft end, a thin box of cross-section

H sub by Lprop ; whether more or fewer such strips are accumulated to reach is immaterial in this model.

Figure 5-3: The reference volume considered to be occupied by bubbles after

they come off the bottom of the submarine.

The total area covered of the bubbles at position 푥 along the bottom of the submarine at any moment in time is computed similarly to the bubble volume. The main difference is that, instead of multiplying by the volume per spherical bubble, one multiplies by the cross-sectional area per bubble:

x A()()' x  N W R t2 dx bub n sub x'  0

Evaluating this integral with the radius from Equation 5-14 gives, in units of m2,

22 2Nn W sub D x Axbub () (Equation 5-18) vsub

For illustration, the functional form of area coverage across the submarine bottom surface, from forward to aft end, is that of a parabola (Figure 5-4) with the highest estimated bubble coverage at 푥 = 퐿푠푢푏. Therefore, any science instruments that are sensitive to high bubble concentrations (e.g. cameras, sonars) should be installed as far toward the nose of the submarine as possible.

0.8

)

sub

(L 0.6

bub

(x)/A 0.4

bub

A 0.2

0 0 0.2 0.4 0.6 0.8 1 x/L sub

Figure 5-4: Relative bubble area concentration on the bottom surface as a

function of distance from submarine nose for the moving case.

The reference area to which the bubble area is compared is shown in Figure 5-5, the strip with width equal to the submarine and length equal to the propeller length. This length is chosen to be consistent with how volume fraction is estimated. Therefore, the fraction of bubble area to reference area is: 75

22 ALbub sub  2NDLn sub Area fraction  (Equation 5-19) Wsub L prop v sub L prop

This is an area ratio and is therefore dimensionless. The submarine width Wsub cancels out for the same reason it cancels in the volume computation. Any area can be inserted into the numerator in Equation 5-19 to determine the amount of bubble coverage at any particular location.

Figure 5-5: The reference area considered to be occupied by bubbles after they come off the bottom of the submarine.

5.2.6.2 Quiescent Case

The bubbles generated when the submarine is stationary in the Titan seas are driven upward both by buoyancy relative to the liquid around them and by the bulk flow of that liquid due to natural convection. Such convection is induced by the submarine waste heat. Adding the bulk velocity of the liquid due to natural convection to the relative velocity of the bubble in the liquid gives the bubble velocity relative to the submarine.

For the quiescent case, two cases are examined: the distribution of bubble area coverage up either side of the submarine over the instrument section, and the volume fraction of bubbles that accumulate around the top propellers before they turn on. Figure

5-6 shows the 1D bases for area and volume estimates.

Figure 5-6: For the quiescent case, a) the area coverage of bubbles over the side of the submarine is based on half the vehicle width (bubbles traveling up one side do

not interact with bubbles traveling up the other side) b) the volume fraction of

bubbles around the propellers is based on the projected submarine height. An

example panel for calculation is shown in blue in each case.

The speed of the flow up the submarine side due to natural convection is based on a laminar similarity solution for a vertical flat plate (133):

2 f Gr 1/2 Hsub vnc  (Equation 5-20) Hsub where

g  T T H 3 Gr  l skin sea sub Hsub  2

By numerical approximation, the peak values of f (based on a function arising in the solution of the differential equation) depend on Prandtl number. ffmax ' 0.2119Pr0.302 ( Pr 0.5 )

This speed estimate will be considered constant in both space and time, neglecting boundary layer growth.

Buoyancy is not neglected in this case, it is added at the end after assuming vnc  0 for the solution process. Figure 5-7 depicts the forces on a bubble in a free stream at any time t with the reference frame chosen so the sea far from the submarine is at rest. Let y point vertically up with y  0 the point where the bubble nucleates; y  0 is not fixed relative to the submarine, because different bubbles nucleate at different points on the skin and thus grow to different sizes before leaving the submarine side. The bubble buoyancy is driven by the density difference between gas and liquid over the assumed spherical volume:

4 3 Fb g l g  R  t (Equation 5-21) 3

where l is the mixture liquid density and g is the density of the saturated nitrogen vapor in the bubble. The resistance is assumed to come from drag due to flow around the bubble, and the associated area is a plane circle with the radius of the bubble:

2 1 dy 2 Fdr l v nc C D R  t 2 dt

2 where CD is the drag coefficient of a sphere based on the projected frontal area  R t .

The drag coefficient used is that for shear-free flow (of the liquid over the bubble),

CDR 48 / Re2 (143), where the Reynolds number of the individual bubble is based on diameter:

dy  vnc 2 R t dt Re  2R  and  is the kinematic viscosity of the sea liquid. Therefore the drag force on the bubble is

dy Fdr12 R  t v nc (Equation 5-22) dt

Figure 5-7: Forces on a gas bubble rising through liquid.

The acceleration of the bubble mass (based on gas density) through the liquid is expressed by Newton’s second law:

2 4 3 dy F F  F  R  t  (Equation 5-23) b dr am3 g dt 2

The “added mass” force (144) as the bubble grows and pushes liquid out of the way is:

2 1 43 d y 3  4  3 dvnc 1  dy  d  4  3  Fam  lRRR t 2   l t   l v nc   t 2 3 dt 2  3  dt 2  dt  dt  3 

2 3 2 3 d y R t  dy  lR tv 2  l nc  3 dt t dt

(Equation 5-24)

The resulting equation of motion is:

2 22 d y99   D1 dy 2lg       D v l g  l nc (Equation 5-25) 2 22 dt2l 4  g DD t dtlg 2  2  l 4  g   t

dy with initial conditions y 00  and 0 finite. The analytical solution that fits these dt criteria, by the integrating factor method, is:

2 2lg   D y t gt2 v t   2 nc (Equation 5-26) 9 5 lg 4   D

The time tcr for the solution to reach a height yH (taking the positive root of the quadratic equation) is

22 9 5 l  4  g DD 8  l   g   t  v  v2  gH (Equation 5-27) cr4g  22 D nc nc 9   5   4   D  l g  l g 

dy Note that 0  v as the bubble is swept up in the flow. dt nc

The method of turning the time from Equation 5-27 into bubble area or volume estimates is similar to that used in the moving case. The submarine sides are discretized

1 W into n panels with representative length L ' (which cancels out later) and area L' sub n 2

1 for area coverage and LH' 1.3  for volume coverage calculations, and the bubble n sub

80 rise time is calculated from the center of each panel. The number of bubbles from each

panel is based on the site density given by Equation 5-9; using tcr and the radius Rt cr  from Equation 5-14 and, the areas and volumes of bubbles from each section are added to estimate the total coverage at any point. The area fraction of bubbles is taken relative to the width of the side scan sonar (1/8 of the submarine width); the volume fraction of bubbles is taken relative to a rectangular prism based on the propeller size. Therefore the area and volume fractions are:

n W ' 2 Nn L'*  R t cr  n  2 in1 n 8' NW Area fraction Rt cr  (Equation 5-28) W nW  L' sub sub i1 8

n H '4 3 Nn L'* R t cr  n  3 in1 n 3 4' NH Volume fraction Rt cr  (Equation 5-29) L'3 Hprop W prop nH prop W prop i1

In this computation the heights used are WW' 0.5 sub (area coverage) and HH' 1.3 sub

(volume fraction) according to Figue 5-6.

For comparison, the detailed solution is checked against a quasi-steady solution

obtained by assuming no acceleration and no bubble growth, FFb dr . When this expression is expanded and integrated from y 00  , the result is:

2    2D  lg 2 y t  gt v t (Equation 5-30) 9 nc

The time to reach yH (taking the positive root of the quadratic equation) is

2 9 8lg   D t  v  v2  gH (Equation 5-31) cr4gD   2  nc nc 9  lg 

The deviation in area and volume fraction between Equations 5-27 and 5-31 is less than

3% within the range of submarine operation. The detailed solution, Equation 5-27, is used for the figures below.

5.2.7 Pressure Drop in Propellers

The last major component of effervescence on the submarine is that due to the pressure drop through the propellers. Boiling-based cavitation is not considered because the Titan seas are near freezing. Such a pressure drop reduces nitrogen solubility further than the thermally driven case alone, and as a result allows more bubbles to come out of solution at a nucleation site. This nucleation occurs on the aft surface of the propeller blade, and any bubbles so formed travel at the forward speed of the submarine (Figure 5-

8). The bubbles travel at the speed vsub through half the propeller length, while not interacting with any walls around the propeller (so bubbles come from the propellers

exclusively). The pressure is assumed to decrease by an amount PP01 between the freestream and the propeller disk. The surface area for nucleation used for this estimate is

 some fraction  of the total disk area H 2 , which is determined by the propeller 4 prop blade size. The solidity ratio of the propeller , , accounts for the fact that bubbles will only nucleate on the blade and not in the open area portion of the propeller cavity; at

fixed H prop , is larger for more/wider blades (i.e. looks more like a solid disk). It is also assumed that bubbles do not nucleate on the side walls of the cavity, behind the blade, because the pressure drop is concentrated around the hydrodynamic surface of the propeller itself.

Figure 5-8: A diagram of the model used for pressure-based effervescence, with

a plot of the assumed pressure distribution.

The pressure P1 is estimated according to the required thrust of the submarine using the Bernoulli method upstream of the propeller. The thrust is assumed to divide equally among the 4 propellers, while the sum of all propellers equals the vehicle drag. It is also related, by impulse-momentum, to the total velocity change v through the propeller:

112   v   lv sub A front* C D    l A p  v sub    v (Equation 5-32) 2 4   2 

 where AH 2 is the area of the propeller disk. Since C  0.4 and A  0.5 prop4 prop D front m2 for the submarine (5), one can solve for the required velocity change:

A v  v 11   C front (Equation 5-33) sub D 4A prop

Finally, the pressure drop corresponding to half of such velocity change (half of the total cavity length) is approximated by the Bernoulli relation along a streamline leading to the propeller:

2 1 v 2 P01 P l v sub   v sub 22 (Equation 5-34) 2 11AA     v2  1  1  Cfront     1  1  C front  2l sub D 4AA  4  D 4  prop   prop 

The first step in the estimate of additional volume is to obtain the nucleation site density of new bubbles on the low-pressure side of the propeller. The equilibrium

nitrogen solubility xN2,skin is updated for the reduced pressure, at the skin temperature,

then used to get supersaturation by Equation 5-8. Substituting the pressure P1 into

Equation 5-11, with the new supersaturation, gives the new critical radius. Finally, the new critical radius goes into Equation 5-9 to estimate the total nucleation site density at

the reduced pressure, Nn,p . This includes, however, the nucleation sites that would already be active at the sea pressure, so the count of new bubbles behind the propeller is based on the difference between low- and high-pressure concentrations:

 2 #bubbles  HNNprop n,p n  (Equation 5-35) 4

All the new bubbles, which nucleate only on the propeller blade, have a growth time equivalent to the time taken to travel half the length of the housing. In this time, the bubbles grow to a size determined by Equation 5-14, and integrating such growth along the streamline from propeller surface to open sea (see Figure 5-9) gives the volume of this line of bubbles in m3:

2 0.5LLprop 0.5 prop 22 2 x  DL V R() t dx  R dx  prop (Equation 5-36) line x   00vvsub2 sub

Finally, the volume to which the bubble stream is compared, is the disk filling the aft

 half of the propeller casing: 0.5LH2 . Adding the total volume fraction of the prop4 prop bubbles behind the propeller to the volume fraction of bubbles at the propeller blade, the result in the moving case is

#bubbles*V Volume fraction Volume fraction line p atblade  0.5LH2 prop4 prop

3/2 3 5/2 NNDL  2 64NDLn sub  n, p n prop 3/2 (Equation 5-37) 15vsub H sub L prop v sub

In the quiescent case, the only flow through the propellers is entrained flow, which is more specific than this model covers; this work does not give volume fraction behind the propeller in such cases. If the entrained flow speed is known, this speed can be used in

place of vsub with the volume fraction from Section 2.5.2.

Figure 5-9: An illustration of how bubbles grow in the propeller cavity.

5.2.8 Solution Method

The method of solution is as follows:

Thermally driven effervescence

1. Choose whether the submarine is quiescent or moving. This specifies the Nusselt

number correlation to use for skin temperature. For the moving case, also specify

the submarine velocity.

2. Choose a sea (e.g. Ligeia Mare, Kraken Mare) and a location within the seas. This

determines P , Tbulk , xCH4 , and xC2H6 ; these four variables uniquely specify  *

and D .

3. Use section 5.2.1 to compute the submarine skin temperature as a function of

waste heat flux.

4. Determine the critical radius from Equation 5-11, then use Equation 5-9 to

determine the number of nucleation sites per square meter.

5. Use Equation 3-7, with the coefficients in Table 4-1, to determine xN2,skin and

xN2,sea , and thus S from Equation 5-8.

6. Solve Equation 5-15 for the bubble growth constant  .

7. Finally estimate bubble area and volume fractions due to thermal effects. In the

moving case, these are given by Equations 5-19 and 5-17, respectively. In the

quiescent case, follow the numerical solution of Section 2.5.2, leading to Equation

5-28 for area and Equation 5-29 for volume.

Pressure driven effervescence

In the moving case, include the volume fraction effect of the pressure drop through the propellers according to Equation 5-37.

5.3 Numerical Results

Simulations were run for two different sea concentrations representative of Ligeia and Kraken Mare. The diffusion coefficient value 퐷 used in the calculations is quoted in

Table 5-1. 100 panels are used in the quiescent case, which was shown to achieve a grid and time step independent solution. The present design of the submarine has waste heat

2 2 of 3800 W and skin area Askin 10.34m , for an average heat flux of 370 W/m . At this value of heat flux, the skin temperature is expected to be less than 96 K (see Figure 5-10).

Note, however, that this is an average value – typically on real hardware there are sites

(e.g. instrument penetrations through insulation) where local heat fluxes may be significantly higher than this.

First, Figure 5-10 presents submarine-specific values of skin temperature, as a function of waste heat flux. Figure 5-10 shows that, in comparing moving to quiescent cases, forced convection clearly cools the submarine skin more effectively than natural convection. The skin is also noticeably warmer in ethane, because in the lower viscosity of a methane sea, Reynolds and Rayleigh numbers are substantially higher, which leads to higher heat transfer coefficients.

104 104 a) 5% Methane b) 85% Methane 102 102 Quiescent Quiescent Moving, 0.5 m/s Moving, 0.5 m/s 100 100

[K]

skin 98 skin 98

96 96

94 94

0 200 400 600 800 1000 1200 1400 1600 0 200 400 600 800 1000 1200 1400 1600 Q /A [W/m2] Q /A [W/m2] wh skin wh skin

Figure 5-10: Submarine skin temperature as a function of waste heat flux for a)

ethane-rich sea and b) methane-rich sea, with sea temperature 93 K and given

geometry.

The solubility limit of section 5.2.2 is shown graphically in Figure 5-11 for two bounds on the methane to ethane mole ratio: 5 to 95 and 85 to 15 (the latter is based on

(59) to represent Ligeia Mare, the former is a generic low bound to represent Kraken

Mare). The trends are that solubility rises with increasing pressure, decreasing temperature, and increasing methane mole fraction. If the bulk liquid is already saturated, heating the submarine surface will reduce solubility and cause supersaturation.

Figure 5-11: Equilibrium solubility of nitrogen in methane and ethane for a)

ethane-rich sea and b) methane-rich sea. Color represents amount of dissolved

nitrogen gas. Both figures are plotted on same color scale for comparison.

Figure 5-12 plots supersaturation as a function of submarine skin temperature and liquid pressure for two different sea concentrations. Clearly, a hotter surface generates a greater excess of gas over the solubility limit. The reason for the substantially greater supersaturation at the bottom of Figure 5-12b is that, at lower pressures, the solubility values are lower. Therefore, the difference in solubility is greater relative to the value at the skin temperature, which raises the fractional supersaturation. The supersaturation of the liquid mixtures is relatively small, but the surface could generate a high concentration of bubbles anyway depending on the contact angle between the liquid and the submarine surface (Figure 5-13).

Figure 5-12: Supersaturation at submarine skin for a) ethane-rich sea and b)

methane-rich sea, with sea temperature 93 K.

Next, nucleation site density as a function of skin temperature and contact angle is plotted in Figure 5-13. Contact angle is highly dependent on surface properties of the submarine and results show that the number of nucleation sites is highly sensitive to contact angle. While the contact angle between cryogenic liquids and metallic surfaces is expected to be small (136), it is still necessary to quantify the effect of non-zero contact angle. Results show that the number of sites changes by several orders of magnitude between contact angles of 1o and 15o; Figure 5-15 to Figure 5-17 use an angle of 15 o, to show the worst case, while Table 5-2 shows the difference at Titan conditions between 1o and 15o. Comparing plots, the number of sites is higher in the methane-rich sea over the ethane-rich case.

Bubble growth rate is plotted in Figure 5-14 (changed from meters to millimeters to display fewer zeros). Bubbles grow more quickly at warmer skin temperatures and higher methane concentrations because both conditions raise the supersaturation of sea liquid against the submarine skin. This in turn raises the value of the growth constant  .

5 5 10 10 a) 5% Methane 4 4 =1 deg 10 10 =15 deg

]

] 2 1000

2 1000

[1/m

N 100

N 100 b) 85% Methane

10 10 =1 deg =15 deg

1 1 93 94 95 96 97 98 99 100 93 94 95 96 97 98 99 100 T [K] T [K] skin skin

Figure 5-13: Nucleation site density as a function of skin temperature and contact angle for a) ethane-rich sea and b) methane-rich sea, with sea temperature

93 K.

1.2 1.2 a) 5% Methane b) 85% Methane 1 T = 94 K skin 1 T = 94 K T = 96 K skin skin 96 K T = 98 K 0.8 skin 0.8 98 K

0.6 0.6

R(t) [mm]

R(t) [mm] 0.4 0.4

0.2 0.2

0 0 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Time [s] Time [s]

Figure 5-14: Bubble radius (in mm) at several skin temperatures for a) ethane-

rich sea and b) methane-rich sea, with sea conditions 93 K and 0.5 MPa.

Figures 5-15 and 5-16 plot the volume fraction (at the propeller inlet) and area coverage, respectively. Table 5-2 presents results of simulations at various vehicle velocities, contact angles, and sea temperature and pressure (which simulates the effect of depth). For both the moving and non-moving case, the volume fraction is computed for a

rectangular prism of liquid the size of a propeller LHWprop prop prop . The area fraction is taken at the point where gas coverage is maximum. For the moving case, this occurs at

the aft end of the submarine using an area equal to the area of the propeller LWsub prop , and

90 for the quiescent case, an area from the bottom centerline of the submarine to the end of

W the ballast tank, using any representative length, L' (from the side scan sonar). 8

Figure 5-15: Volume fraction of bubbles around the propellers (log scale, at the

propeller inlet) for a) ethane-rich quiescent case, b) ethane-rich moving case, c)

methane-rich quiescent case and d) methane-rich moving case. For all plots, sea

temperature is 93 K and   150 .

Figure 5-16: Highest computed area fraction of bubbles (log scale) on the

submarine skin for a) ethane-rich quiescent case, b) ethane-rich moving case, c) methane-rich quiescent case and d) methane-rich moving case, with sea temperature

93 K and   15 degrees.

Results in Table 5-2 show that effervescence has a much greater propensity in methane-rich seas than it is in ethane-rich seas (skin temperature being equal). This is attributed to the fact that methane rich seas have a higher supersaturation than ethane rich seas, which amplifies both the number of nucleation sites and the bubble growth rate.

Comparing quiescent to moving cases, the highest accumulations are in the quiescent cases. The highest area coverage for the cases in Table 5-2 is 5.5%, and the highest volume fraction is 0.0758%. These extremes occur in the quiescent, methane-rich sea,

92 high pressure,    degree case; other cases (especially where   degree) are orders of magnitude smaller.

CH4 P  Area Volume Volume vsub Qwh Tskin Nn fraction (MPa (deg fraction fraction fraction (m/s (K) (1/m2 A ) ) (before (after prop) ) skin ) (W/m2) prop) 0 0.85 370 0.15 1 95.5 34.2 6.00E-05 6.74E-07 NA 0 0.85 370 0.3 1 95.5 84.2 0.000247 3.40E-06 NA 0 0.05 370 0.15 1 96.3 21.3 1.15E-06 3.06E-09 NA 0 0.05 370 0.3 1 96.2 43.6 8.16E-06 3.88E-08 NA 0.5 0.85 370 0.15 1 93.7 6.22 1.78E-05 7.86E-09 7.89E-09 1 0.85 370 0.15 1 93.4 3.39 1.64E-06 5.14E-10 5.33E-10 0.5 0.85 370 0.3 1 93.6 11.3 7.65E-05 3.38E-08 3.40E-08 1 0.85 370 0.3 1 93.4 6.00 8.11E-06 2.54E-09 2.62E-09 0.5 0.05 370 0.15 1 94.1 5.48 5.03E-09 2.23E-12 2.24E-12 1 0.05 370 0.15 1 93.6 3.02 1.39E-09 4.33E-13 4.50E-13 0.5 0.05 370 0.3 1 94.0 9.73 8.94E-09 3.95E-12 3.96E-12 1 0.05 370 0.3 1 93.6 5.23 2.4E-09 7.51E-13 7.71E-13 0 0.85 370 0.15 15 95.5 7628 0.0134 0.000150 NA 0 0.85 370 0.3 15 95.5 18790 0.0551 0.000758 NA 0 0.05 370 0.15 15 96.3 4754 0.000257 6.83E-07 NA 0 0.05 370 0.3 15 96.2 9736 0.00182 8.66E-06 NA 0.5 0.85 370 0.15 15 93.7 1389 0.003964 1.75E-06 1.76E-06 1 0.85 370 0.15 15 93.4 756 0.000367 1.15E-07 1.19E-07 0.5 0.85 370 0.3 15 93.6 2523 0.017074 7.55E-06 7.58E-06 1 0.85 370 0.3 15 93.4 1340 0.001811 5.66E-07 5.85E-07 0.5 0.05 370 0.15 15 94.1 1223 1.12E-06 4.97E-10 4.99E-10 1 0.05 370 0.15 15 93.6 673 3.09E-07 9.67E-11 1.00E-10 0.5 0.05 370 0.3 15 94.0 2170 1.99E-06 8.82E-10 8.85E-10 1 0.05 370 0.3 15 93.6 1167 5.36E-07 1.68E-10 1.72E-10 Table 5-2: Results of the effervescence model for several cases of interest to the

Titan submarine.

The gas volume fraction after the propellers is determined using baseline propeller parameters. Based on pictures of the submarine (5), the solidity of the propellers is about

0.5. The results of this assumption are shown in Figure 5-17. Compared to Figure 5-15(b) and (d), the difference is small (on the order of 1% of the value before the propellers).

The conclusion that methane produces more bubbles than ethane still holds.

Figure 5-17: Volume fraction after the propellers, moving at 0.5 m/s, for a)

ethane-rich sea and b) methane-rich sea.

Finally, this model can be compared to recent data on the heat flux values that trigger effervescence in similar mixtures (Figure 5-18) (145). The submarine operating point, about 370 W/m2, is an order of magnitude less than the data, suggesting a safety factor of at least 5 for the present design. As noted in (131), fluxes of the order of 90 W/m2, as encountered with the Huygens probe, are enough to cause detectable environmental perturbations on Titan, but considerably larger fluxes (~50,000 W/m2) are needed to cause actual boiling of liquid methane.

105

]

104

Data, Richardson et. al. 1000 Submarine Operating Point

Waste Heat Flux [W/m

100 0 0.05 0.1 0.15 0.2 0.25 0.3 xN 2

Figure 5-18: Comparison of submarine operating point to experimental heat flux values that triggered effervescence.

6 Ballast Trade Study

The following section begins with a submarine literature review, then describes each of the 7 ballast concept for the extraterrestrial submarine. For those concepts that are fully developed, the parts in the schematics are labeled with a letter and a number; the letters used are P for pump, V for valve and C for compressor, and the numbers are 1, 2, etc.

6.1 Submarine Literature Review

The purpose of the submarine ballast system is to control the weight, and thus the density, of the vehicle, so that it can rise and sink at will. This involves a balance of the vehicle mass and the mass of the displaced fluid (both subject to the same gravity):

seaVm sub sub (Equation 6-1)

In general, a design can work along two different lines: change the mass msub of the

vehicle for a fixed volume Vsub , or fix within a flexible hull that increases or

decreases Vsub as necessary. Terrestrial submarines change the left side of Equation 2-1, among other methods, by using diesel fuel to force denser water in or out of the bottom of the tank (146); several control logic schemes for such designs have been developed (147)

(148) (149) (150). Another design method based on liquid ballast, which controls weight distribution but not total weight, involves shifting fluid between trim tanks at the forward and aft ends of the submarine (151). One means of changing the right side of Equation 2-

1 is by shifting a fluid between an internal rigid tank and an external bladder that expands

95 when filled. This is the method used by many autonomous submarines on Earth (152). A scheme that controls trim and ballast simultaneously is to have a ballast tank on each end of the vehicle, so that filling one more than the other changes the mass balance (153).

Many of the ballast designs that function well on Earth cannot be applied to the Titan submarine. Naval submarines (154) commonly flood their main ballast tanks at the surface to achieve neutral buoyancy, then make small modifications at depth using “hard” tanks that operate at water pressure; only in emergencies do they blow out the main tanks using compressed air. Such vehicles benefit from relatively small density changes with salinity and temperature, but on Titan, the possibility of very large changes (between methane-rich and ethane-rich seas) would require bigger hard tanks and offer reason for a different concept entirely. The other benefit submarines enjoy on Earth is that compressed air can easily push water from the ballast tanks directly; the Titan atmosphere, however, is both near condensation by itself and is likely to dissolve in the sea in large quantities, which demands modifications.

Another Earth example (155) is the method of carrying large weights that may be dropped to quickly rise. This is a common tactic in historic deep submersibles (156), but it requires a support ship at the surface to replace the shot for each dive. This is no difficulty on Earth research expeditions, but the Titan submarine must be completely self- contained; any weight it drops can be used only once as an emergency. One ballast tank on each end has also been used for simultaneous trim control (153).

6.2 General Considerations

Several aspects of the ballast design are common to all the trade study options considered here. Based on upper and lower density bounds, the ballast tank can be sized

96 to allow full operation and the amount of control resolution determines how quickly the vehicle will rise or sink (based on Reynolds number, and thereby viscosity). Most of the required parts, and the various uncertainties involved in the design, are also shared between the concepts.

6.2.1 Tank Size and Sink Rate

After sea properties, the next step toward designing the ballast system is to decide the requisite tank volume, as well as the resolution with which small amounts of liquid can be added or removed. The total weight of the Phase I submarine was estimated at 1386 kg, and its total volume was roughly 2 m3. For safety, the vehicle has to float in the lightest sea, nitrogen-methane (~500 kg/m3), which places the required volume at

3 3 Vsub  2.77 m . The volume threshold for sinking in a nitrogen-ethane sea, ~650 kg/m , is

3 Vsub  2.12 m or less. The volume that the tank system must take in to sink is that which covers the surplus of displaced mass, or the difference between displaced and actual mass divided by the density as shown in Figure 6-1. As a representation of conditions slightly below the sea surface, the pressure (for density calculations) is taken to be 0.2 MPa, with the dissolved nitrogen concentration assumed to be the same as at the surface (pressure of

0.15 MPa). The ballast volume required is greatest for the densest possible composition, pure liquid ethane with dissolved nitrogen, at about 0.65 m3. For Ligeia Mare, with 85% methane, the required ballast volume is about 0.12 m3. These numbers are the total ballast volume between all tanks. As a margin of error, and also to compensate for any future increases in submarine weight or displacement, the ballast tanks should be baselined at 1 m3 capacity.

0.7

0.6

0.5

] 3 0.4

neu 0.3

0.2

0.1

0 0 0.2 0.4 0.6 0.8 LCH Fraction 4

Figure 6-1: Required Ballast Volume across the Range of Titan Sea Mole

Fractions

Another concern for operation is the submarine sink rate. This determines how tightly the flow through the valves must be controlled. The rate of sinking was calculated from a force balance of gravity and drag:

1 V g v2 C A , VVV (Equation 6-2) sea net2 sea D sub net bal neu

2 where g 1.35 m/s is the gravity on Titan, the density sea is for the liquid mixture and

v is the sinking velocity. The value of the drag coefficient is that for a cylinder: CD 1.2

 vL for Reynolds numbers Re  sea c (where  is the viscosity and L is the  c characteristic length) below about 500,000 (157). The bottom-facing area of the

2 2/3 submarine is about 6.5 m in Phase I, but here is scaled up by the factor V to Asub  8.1 m2. This scaling accounts for the increased volume required to float in Ligeia Mare. Also,

1/3 vehicle length is scaled up by the volume factor V to Lc 1.23 meters.

Figure 6-2 shows the results obtained from setting Vnet as a parameter and solving for velocity 푣 in each case. Results are for ternary seas with 1 mole methane to 19 moles ethane and with 17 moles methane to 3 moles ethane, the Titan composition extremes representing Kraken Mare and Ligeia Mare, respectively. The velocities so obtained for rising or sinking are virtually identical for both compositional extremes, but the Reynolds numbers are much higher for the 85% methane case. Due to the risk of tripping to turbulence near the critical Reynolds number, which would cause the submarine to rise or

3 sink significantly faster, the desired excess ballast while diving is 0.01Vnet 0.05 m , corresponding to about 0.1 m/s of depth change. At that rate, the vehicle would rise or sink 360 meters in an hour. Therefore, the control resolution of the all concepts should be

0.01 m3 of liquid.

0.16 4 b) Reynolds number 0.14 3.5 85% Methane 5% Methane 0.12 3

0.1 2.5

)

5 0.08 2

v [m/s]

0.06 (x10 Re 1.5

0.04 a) Equilibrium rising/sinking velocity 1 85% Methane 0.02 5% Methane 0.5

0 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 V [m3] V [m3] net net

Figure 6-2: Estimates of a) steady-state sinking velocity and b) steady-state

sinking Reynolds number of the submarine as a function of excess volume of sea

liquid, for both composition extremes

It should be noted that sinking quickly in a methane sea carries a potential risk of the flow around the submarine tripping to turbulence, causing a rapid acceleration and making control more difficult. This is because the turbulence transition could occur at a

99 smaller Reynolds number than 500,000. If the vehicle sinks rapidly, it may hit the bottom with significant force. This can be distinguished by the rate of sea pressure change, which at a rising/sinking speed of 0.14 m/s varies between 95 and 120 Pa/s. A speed of 0.1 m/s, for comparison, creates a pressure change of 68-86 Pa/s. Therefore, the control system should operate gradually enough that it does not allow more than 90 Pa/s of pressure change.

The arrangement of ballast tank volume relative to the submarine body is also important. Because weight distribution influences vehicle pitch and roll, the same liquid ballast can be used to cause or mitigate any such variation that is desired. One alternative is to have separate internal, or trim tanks, that shuttle a liquid between several positions inside the submarine, but that would be an additional system. A simple implementation of this method is to have 4 discrete tanks, sharing some of their hardware, at each of the forward and aft, port, and starboard corners. Selectively filling one tank first causes that area of the vehicle to angle down, at which point filling the other tanks would stabilize the attitude.

6.2.2 Uncertainty

One important aspect of the operating scheme is the uncertainty within which its parameters can be measured. Current commercial flow meters for LNG (158) (159) use ultrasonic measurement and generally have an uncertainty less than 0.5% of the reading; they could likely be miniaturized for the submarine. Cryogenic pressure transducers (160) can have uncertainties well within 1% of full scale. A densimeter is not necessary: before the dive, the chemistry analysis package onboard the submarine can report the composition with negligible error, and the processor can use REFPROP (8) (along with

100 temperature and pressure) to compute mixture density. This software has an internal density uncertainty of 0.5% for LNG mixtures. On the next dive, the control system then matches the vehicle density to this estimate of sea density, with physical checks such as liquid level sensing. Including a measured temperature uncertainty of 0.022 K for a silicon diode near 77 K (129) does not change this procedure significantly.

The instrumentation needed to operate the ballast system in general is shown in

Table 6-1. Inclinometers (161) can detect whether the submarine is pitching or rolling.

The sea liquid level against the submarine side is detected on the surface by diode rakes forward and aft, which both span the vehicle height from top to bottom. While the vehicle is fully submerged, a sonar beacon on the top surface pings the sea surface to give a proxy for depth based on a method used in tanks (124); if this proxy suddenly or rapidly increases, the submarine may be sinking. Commercial towable sonar units for Earth applications have mass about 20 kg (162), though the unit designed for Titan will likely be significantly lighter (due to structural integration). Depth sounding accuracy of 1% of reading is a plausible design goal (56).

Measurement Location Function Range Uncertainty Comments Tilt 1 (161) Inside Detect pitch 60 ° 0.05 ° Adapted for insulation Titan gravity Tilt 2 (161) Inside Detect roll 60 ° 0.05 ° Adapted for insulation Titan gravity Top sonar Top surface Ping the surface 10 to 1000 1% of Plausible design to test m reading goals (56); rising/sinking Earth unit is ~20 kg (162) Diode rake Spanning top Detect liquid 0 to 100% 0.5 cm Includes silicon to bottom level submerged diodes surfaces, forward and aft Silicon diode On diode Sea temperature; 1.4 to 500 K 0.022 K (at 77 --- (129) rake Detect liquid K) level Table 6-1: Instrumentation list for the ballast system

101

One alternative to the sonar- and pressure-based control scheme described in the concept sections below is the use of temperature and dielectric constant to infer sea density, and thereby estimate the degree of control action required. A common uncertainty in dielectric factor, for a strictly room-temperature apparatus, is 2% of the reading (163). Doubling this relative uncertainty to 4% for the cryogenic conditions on

Titan with several probable dissolved components, and using a notional value of 2 (see

Figure 4-19), the absolute dielectric uncertainty is ~0.08. At the measured temperature and pressure, the absolute dielectric factor corresponds to the nitrogen-methane-ethane composition as shown in Figure 6-3, based on (164) using the solubility model to fix the nitrogen content at each methane/ethane ratio. The approximate mole fraction uncertainty

(of methane or ethane) is then ~0.25, which via REFPROP implies a density uncertainty of 10%. Therefore, the projected uncertainty in neutral ballast volume is 35% for ethane seas, rising to 100% (i.e. making the calculation meaningless) above 60 mol% methane.

Unless a superior dielectric meter is available, with an uncertainty of perhaps 0.1% at

Titan conditions, this control method is not worth pursuing.

1.95 95 K, 0.15 MPa

1.9 Nitrogen Methane Ethane 1.85

1.8

 1.75

1.7

1.65

1.6 0 0.2 0.4 0.6 0.8 1 x

102

Figure 6-3: Illustration of the use of a dielectric meter to deduce composition.

The temperature is 95 K and the pressure is 0.15 MPa.

6.2.3 Parts

Compared to most LNG applications on Earth, the Titan seas are expected to hold many other hydrocarbons such as benzene as well as solid particulates. Any system that takes sea liquid into the vehicle and puts it through pumps or valves must filter out solid particulates, which would otherwise clog the hardware and possibly cause rapid failure.

Such a filter could then be flushed out, for example, with a shot of pressurant gas.

One potential complication with gas storage is leakage through the bottle

(especially of helium) over the 8-year duration of the mission, consisting of 7 years in transit from Earth and 1 year operating on Titan. Pressure vessels can generally be built and inspected to a helium loss rate of 106 scc of helium per second (165). Over 8 years, that adds up to 250 scc (0.05 grams) of helium. For neon gas the equivalent mass is 0.25 grams, which is an upper bound since neon should diffuse through the bottle wall less effectively than helium. In summary, the gas bottle should not be compromised by diffusion loss during transit.

The component parts of the concepts below are presented, along with the baseline mass estimates, in Table 6-2. The parts to be used in the final submarine will be custom- built, but a first-order estimate of the masses can be made by comparison to existing commercial products that reach similar specifications. The gas bottles are sized to meet 2 conditions: (1) maximum pressure at 300 K on Earth is 30 MPa; and (2) after filling the required volume for the density at 93 K and 1 MPa, the bottle pressure at 93 K remains

103

1.1 MPa. If the gas is helium, the fully charged pressure at 93 K is 9.4 MPa; for neon, it is 7.83 MPa. The isentropic efficiency of hardware such as pumps is assumed to be 80%.

Part Unit Mass Comments Ballast tank 26 kg 2 tanks, made of titanium (166) with safety factor of 2, cylinder with volume 0.5 m3 and length 5 meters, thickness 1 mm (for 1 MPa external pressure) Valve 5 kg Based on commercially available cryogenic valves Pump 9 kg 1 MPa baseline (167), scaled in proportion to pressure, 80% efficiency Compressor 9 kg Similar to pumps, but baselined at 0.85 MPa (168) (169), 80% efficiency Gas 30 kg See section 6.7. Includes compressor, heat exchanger and 3 Purification additional valves Gas bottle Var Spherical geometry, thickness based on ASME standard (170) with 30 MPa internal pressure in a cylindrical shell, uses yield strength from (124) with safety factor of 2 --- Liquid Cv Standard method for values (171), 50 kPa pressure loss --- Gas Cv Standard method for values (171), 7% pressure loss Bladder Var Spherical, 1 mm stainless steel (~8000 kg/m3) Separator Var Planar, 1 mm stainless steel (~8000 kg/m3). Table 6-2: Baseline part mass estimates

6.3 Pump System

This is the simplest ballast concept; it uses only valves and a pump as shown in

Figure 6-4. However, it is limited in its operations compared to other concepts. Its main feature is that the tank sets neutral buoyancy at the surface, then establishes small positive buoyancy so that the vehicle floats if power fails.

Figure 6-4: Conceptual layout of a pump ballast system. There is one such tank on each side of the submarine.

104

6.3.1 Analysis

In the following, the change in gas pressure with ullage volume in the tank is treated as ideal and isothermal (so PV is constant) and the total tank volume is assumed to be 1 m3. For nitrogen at Titan conditions, the compressibility factor is Z  0.96 , decreasing to Z  0.88 near the saturation pressure (0.462 MPa at 93 K). The atmosphere is assumed to be pure nitrogen because the possible condensation of methane above Titan surface pressure removes only a minor gas component.

The greatest operational risk in the pump concept is that the sea density might decrease while the submarine travels submerged. Such an event would result from an increase in methane concentration, and the resulting lack of buoyancy may compromise the mission. This is a concern mainly for the pump concept because without access to the atmosphere, removing liquid ballast draws a partial vacuum in the tank. The opposite change, the sea becoming denser, is safe because the submarine will simply float. The worst case is a transition from the high end of the density range, 650 kg/m3, to the low end, 500 kg/m3.

To sink at the high density value, the submarine needs to take on 0.65 m3 of sea ballast. If the atmosphere is vented while taking on liquid, any liquid removal while submerged will risk structural failure as the tank collapses on itself. Therefore, the vent valve should remain closed until the tank pressure reaches 0.45 MPa, just below the saturation pressure. Because the total tank volume is 1 m3, the pumping of 0.65 m3 of liquid ballast is enough to approach this threshold.

The removal of all of this ballast in an emergency should, in principle, restore the tank to 0.15 MPa (without replenishing ullage from the atmosphere). However, the 5%

105 methane content of the atmosphere might have been lost to condensation at the elevated pressure, which would put the empty tank below the surface pressure. It is therefore advisable to cover such uncertainties by building in a source of emergency ballast that allows the submarine to function as a boat even if the pumps fail.

The purpose of emergency ballast is to inflate the submarine volume, to make up for the loss in sea density. Because the submarine displaced volume has to remain the same, the balance is

12VVVsub sub em  , therefore

d 12 VVem sub , d  (Equation 6-3) 1 d 1

where 1 and 2 are the initial/higher and final/lower densities, respectively, and Vem is

3 the added amount of emergency submarine volume required. Using 1  650 kg/m as

3 the top of the density range and 2  500 kg/m as the bottom of the range, the value of

3 d is at most 0.23 and Vem is at most 0.83 m . This is the volume that the single emergency gas and bladder should be able to fill, to cover all plausible requirements.

6.3.2 Operations

The operational procedure to dive is:

 Measure the sea composition and use section 6.1.1 to estimate how much liquid

ballast is required for neutral buoyancy. The derived uncertainty of this estimate

is less than 1.5% in high-ethane seas, rising to 7.5% in high-methane seas.

 Activate P1 to start flooding the tank. A flow meter will track the total liquid

volume into the tank.

106

 At 80% of the estimated required ballast, partly close V1 to slow down the tank

flooding; this is well short of the uncertainty limit. Turn off P1 to cease tank

flooding when the diode rakes sensors record full submergence.

 Turn on P1 for 2 minutes (section 4.1.3) to remove enough liquid to establish

slight positive buoyancy.

 Only open V2 if the tank pressure reaches 0.45 MPa during the previous 2 steps.

This threshold is 3.6% less than the nitrogen vapor pressure at 93 K, against 1%

measurement uncertainty.

 With positive buoyancy established, the propulsion system may be used to drive

the vehicle down as needed. The submarine will rise by itself if propulsion fails or

is turned off.

 Turn off the propulsion and use the top sonar every minute to verify that the

submarine is not sinking.

 When the submarine has resurfaced, open V2 and pump the liquid out. The

atmosphere fills the tank volume as liquid is removed.

The other instructions needed are those to recognize and respond to an emergency loss of buoyancy. The emergency system of choice for the submarine is a cryogenic bladder inflated with a bottle of neon.

 Sinking is recognized if the external liquid pressure is rising faster than 40 Pa/s

and the top sonar indicates a rising distance from the surface. The latter should be

corrected for the submarine not being level, to send the sonar beam perpendicular

to the sea surface.

107

 If sinking is recognized after propulsion has been turned off, remove ballast to

0.15 MPa tank pressure or to empty the tank, and abort the dive.

If the tank pressure has fallen to 0.15 MPa and the top sonar indicates the submarine is still sinking, inflate the emergency bladder.

6.3.3 Parts

The largest flow rate that would need to be actively pumped out occurs at depth, to cover possible emergencies. The pump may have to remove 0.1 m3 (neutral 0.05 m3) in

4 minutes for a total volume flow rate V  0.000833 m3/s. At this rate, it would take 28 minutes to empty 0.72 m3 (ethane-rich sea). The total power requirement is 160 W and

139 W per pump for 5% methane or 85% methane sea, respectively.

For the liquid valve, an acceptable pressure drop is 50 kPa, which implies a Cv

for the liquid valve in each tank of 1.8 in an 85% methane sea or a Cv of 2 in a 5% methane sea.

The gas mass let out through the top valve in each tank, based on a volume of 0.5 m3 and nitrogen density, is 2.8 kg. If the whole gas mass is let out in 10 minutes, then the

mass flow rate is 16.8 kg/hour and the required valve size is Cv  0.9 .

The emergency bladder would need to have a volume capacity of 0.83 m3 with a minimum bladder and bottle pressure of 1.1 MPa, which requires 4.6 kg of helium. The corresponding bottle volume is 0.126 m3, holding 0.7 kg of residual gas. Such a bladder will only be inflated once. If the emergency bladder fills in 1 minute, then the flow rate

through the valve is 276 kg/hour and the required Cv value is 5.2. The approximate bottle weight assuming titanium is 75 kg. The bladder mass is about 9.8 kg. Each of the two

108 ballast tanks has 1 pump and 2 valves, while there is one emergency system on the submarine, for a total mass of 180 kg.

6.4 Bladder Only

The emergency concept for the pump system, a gas-inflated cryogenic bellows to

increase Vsub , can also be used directly to make the vehicle rise or sink. This ballast concept is illustrated in Figure 6-5. The gas used is a non-condensable such as helium or neon.

Figure 6-5: Conceptual layout of a bladder-only ballast system. There is a single gas bottle, and one bladder on each side of the submarine.

6.4.1 Analysis

This analysis assumes an ideal, non-condensible gas such as neon or helium as the working fluid in the tank and bladder. The transfer of this gas between the two volumes is considered to be isothermal, because the effect of adiabatic heating is to exaggerate any changes in pressure. The isothermal pressure rise as gas is compressed into the bottle may

be estimated from the sea/bladder pressure Psea and the change in bladder volume VV12

. The change is gas mass inside the bottle is then:

109

VVP  m 12sea (Equation 6-4) gb R u T M where M is the molar mass for the working fluid used and T is the sea temperature.

Adding this mass difference to the mass already in the gas bottle, which produces

pressure P1 according to the ideal gas law, the new isothermal pressure is:

RT VV12 P2 mgb ,1   m gb  P 1  P sea (Equation 6-5) VVgb gb

where Vgb is the volume of the gas bottle.

3 The volume required for this bladder comes from Equation 6-3 with 1  650 kg/m ,

3 3 2  500 kg/m , and VVsub em 2.77 m . That means the smallest submarine volume, with the bladders fully contracted, is 2.12 m3 with the total volume of the 2 bladders being about 0.65 m3; all gas is stored in a single bottle. Since the helium density corresponding to 93 K and 1 MPa is 5.1 kg/m3, the amount of gas in the bladder is 3.3 kg.

6.4.2 Operations

The submarine should launch and travel to Titan with the bladder fully expanded, so the submarine can float on arrival. The operating procedure for a dive is:

 Before diving, compute a sea density estimate sea from composition and

temperature measurements (within 0.5%), the submarine mass msub and the

maximum submarine volume Vsub from system data. The amount of volume

decrease required to achieve neutral buoyancy follows as:

msub VVV1 2  sub  (Equation 6-6) sea

110

 Turn on C1 to retract the bladder boundaries and reduce vehicle volume for

slightly positive buoyancy according to Equation 6-6; from that state, drive down

with propulsion. Turn C1 off when the bottle pressure has increased from its

initial value P1 to a final value P2 estimated from Equation 6-5, or when all but

the top surface of the submarine is submerged, whichever happens sooner. During

the transfer of gas from bladder to bottle, the pressure may spike to a higher value

than this, but with effective heat transfer that spike will fade quickly to the

isothermal pressure.

 Open and close V1 as necessary while diving to match the gas pressure and sea

pressure to maintain bladder volume. In particular, each dive must operate from

lowest depth to highest, sinking and rising monotonically.

 To return to the surface, open V1; with the sea pressure already matched, the

added gas expands the bladder volume. Close V1 when the bladder reaches full

volume at 1 MPa gas pressure, and hold that status for the remainder of surface

time.

To initiate any subsequent dive, turn on C1 to reduce gas pressure in the bladder, then to retract bladder volume. The stopping criterion is the same as on the first dive.

6.4.3 Parts

The gas bottle is required to hold 3.3 kg of helium, plus 0.5 kg of residual. The minimum volume required is 0.09 m3, and the bottle mass is about 53.2 kg. With a similar arrangement of 2 gas valves, the mass flow rate per valve is 50 kg/hour, and the

required Cv is 0.94. When the vehicle starts a dive, the 2 compressors may take 65 minutes to take in 3.3 kg of helium, for a flow rate of 0.00042 kg/s. With a final pressure

111 of 9.4 MPa, the required power to each compressor is about 385 W. The mass of each bladder is ~20 kg. The gas bottle is shared between both sides of the submarine; each side individually has one bladder, one valve and one compressor. The mass of the whole ballast system (both sides and the gas bottle between them) is ~210 kg.

6.5 Noncondensible Gas without Separator

The principal characteristic of this ballast concept, shown in Figure 6-6, is that the submarine carries a bottle of neon or helium with it, and blows down the bottle directly against the liquid surface every time the submarine returns to the surface. Due to the solubility of the pressurant gas in the liquid, a small amount of gas will be dissolved and thus lost during each dive. Therefore, the pressurant gas bottle must be sized to expel liquid at pressure, but also have enough mass to overcome the residual loss over the mission.

Figure 6-6: Conceptual layout of a pressurant gas system without separators.

Each side of the submarine has one gas bottle and 2 tank sections.

6.5.1 Analysis

The estimate for amount of gas leaked is based on the small amount of gas that diffuses into the ballast liquid on each dive due to solubility. The loss accumulates and is tracked over the total number of dives. Since the first dive is operated differently and sets

112 up every dive to follow, the losses at that stage are ignored. The assumptions in this model are:

 The ballast tank is described as a cylinder, 6 meters long, with 1 m3 of volume,

and a diameter of 0.46 meters. Based on length L  6 meters and radius R  0.23

meters, the liquid volume based on the smaller central angle θ, which is oriented

upward if the tank is more than half full, is

1 1 VRL2 ( sin ) (V  0.5 m3) or VRL1 2 (  sin ) (V  0.5 m3) bal 2 bal bal 2 bal

(Equation 6-7)

With the central angle calculated numerically from ballast volume Vbal , the surface area of the liquid body and maximum depth immediately follow:

 Asurf  2 RL sin (Equation 6-8) 2

 3  3 dRbal 1 cos (Vbal  0.5 m ) or dRbal 1 cos (Vbal  0.5 m ) 2 2

(Equation 6-9)

 The volume of ballast liquid taken into all 4 tanks combined ranges from 0.13 m3

(85% methane) to 0.65 m3 (5% methane), assumed to vary linearly with

composition. At the surface, the assumed liquid volume is 0.005 m3 in a recess at

the bottom of the tank, with 0.1 m2 surface area and 0.05 m depth.

 Pure diffusion of the gas occurs through the liquid surface through the dive time

of 8 hours. The amount of gas lost is the full soluble amount, if 8 hours exceeds

the time for dissolved gas to reach equilibrium solubility, or an equivalent fraction

of the full soluble amount to 8 hours as a fraction of that time. A similar

113

calculation holds over 16 hours for the small liquid volume retained at the surface.

The approximate time required for equilibrium is based on Fick’s first law of

diffusion:

n C JDeq (Equation 6-10) Asurf t eq  z where D  2*109 m2/s is the approximate diffusion coefficient of helium in methane at

Titan conditions extrapolated from (172), Asurf is the contact area between liquid and gas,

and neq is the number of moles of gas dissolved at equilibrium. If the concentration

xnNe bal gradient is Cdeq/ bal , where Ceq  , xNe is the dissolved mole fraction of gas (neon Vbal

or helium), Vbal is the amount of liquid in the tank and dbal is the depth of the liquid, then

solving for teq gives:

3ndeq bal teq  (Equation 6-11) DAsurf C eq

The proportionality constant 3 accounts for the spread of dissolved gas in 3 dimensions, which takes longer than in 1 dimension due to lateral effects. Assumptions needed are:

 The additional solubility loss that occurs during ascent or descent is neglected.

 The effect of any change in bottle pressure conditions as gas is let out, due mostly

to expansion cooling, is negligible, because that operation is only done

occasionally.

114

 Gas pressure is treated as ideal and isothermal; this is justified because the

compressibility factors are, at 92 K (temperature assumed constant) and 0.15

MPa, Z  0.9986 for neon and Z 1.002 for helium.

 The solubility of gaseous helium in liquid methane (103) is:

2 xHe  P*exp(  13.76  0.0748 T  0.00137 T  0.015 P *) (Equation 6-12) where the temperature T is in Kelvin and P * is the pressure minus the vapor pressure of the solvent (in MPa). The data for helium in ethane below 160 K (173) (84) and neon in methane (174) are fitted in this work, and no data for neon in ethane was found. A correlation for mole fraction of helium in liquid ethane below 160 K is:

7 xHe (8.11*10 ) P *exp(0.0419 T  0.00686 P *  0.000126 TP *) (Equation 6-13)

A correlation for mole fraction of neon in liquid methane is:

4 xNe (3.99*10 ) P *exp(0.0194 T  0.0395 P *  0.000239 TP *) (Equation 6-14)

For the case of helium in methane-ethane mixtures, the solubility was interpolated linearly between the two binary cases. For neon, the solubility in pure methane was taken as constant for all compositions due to lack of data.

The key to guaranteeing that the submarine can surface is that the minimum gas pressure at full blowdown is 1 MPa, sufficiently high for any design point in the sea.

Given this requirement, Figure 6-7 shows the percentage of gas lost for neon and helium in both composition extremes, accumulating over 30 dives. The losses are much greater for neon than for helium, due to higher solubility. Solubility is also the cause of the reversal in loss trend with composition; according to the relations described above, neon is more soluble in methane while helium is more soluble in ethane. Such a difference may

115 result, in part, from the limited solubility data; nonetheless, all estimates are presented here as an order-of-magnitude guide for designing the remainder of the ballast system.

20 0.0008 a) Neon b) Helium 0.0007 5% LCH4 5% LCH4 85% LCH4 15 0.0006 85% LCH4

0.0005

10 0.0004

0.0003

Precent of Lost of Gas Precent Precent of Lost of Gas Precent 5 0.0002

0.0001

0 0 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Number of Dives Number of Dives

Figure 6-7: Loss percentage of a) neon, and b) helium in both composition

extremes, accumulated over many dives.

Maintaining 1 MPa of neon gas pressure after 30 days of solubility losses requires a mass of neon in the ballast tank of 26.7 kg. Since 5.3 kg of gas have to be replenished every 30 dives (therefore this replenishment mass must be provided 12 times), the total requirement for the whole 365-day mission is to have 90 kg of neon. For helium, a starting gas mass of 5.4 kg is adequate for the entire mission, because the losses are inconsequential. A loss of 0.0007% of the gas to dissolution, repeated 12 times, implies that the ballast tanks lose only half a gram during the entire year of operation.

6.5.2 Operations

Upon arrival to Titan, the ballast tank contains 0.1 MPa of gas and the gas bottle is at 9.4 MPa (for helium) or 7.8 MPa (for neon).

 Follow the sinking instructions for the pump concept.

 Drive the submarine down using propulsion from near-neutral buoyancy.

Recalculate the sea density every 10 minutes.

116

 If it is desired to adapt to a higher sea density, then pump in the excess volume

above and beyond that loaded at the surface. This amount should be undershot by

0.5% to account for flow rate uncertainties.

 If the vehicle orientation is anything other than level and upright when surfacing,

the ballast liquid may be covering the gas bottle release valve. It is necessary,

then, to blow down one tank of the submarine before other tanks, to make it level

and upright.

 To surface from the first dive, open V2 to pressurize the gas in the ballast tank.

When the gas pressure exceeds the outside sea pressure, open V1 to expel liquid

from the tank. To prevent the pressurant gas from being expelled alongside the

liquid, V1 should be set in a recess on the bottom of the tank; when a liquid level

sensor observes when that small volume has begun to empty, close V1 and V2.

 To surface from subsequent dives, the initial action of the pump would compress

the gas in the tank to above sea pressure, so just open V2. The stopping criterion

is the same as on the first dive.

 If, at any point while expelling liquid on subsequent dives, the gas pressure in the

tank falls below 1 MPa, open V2 and close it when that pressure reaches 1.05

MPa. Such a threshold may be reached in an isentropic spike, which settles below

1 MPa after equilibrium; in that case, repeat the procedure.

The relief valve V3 should be set to open at 4 MPa, the same limit as the ballast tank is designed to withstand.

117

6.5.3 Parts

In this concept, the two ballast tanks hold 4 MPa, with a resulting mass of 56 kg between them. The reason for this high pressure is that the ullage is compressed by pumping in liquid to sink; in the most extreme case, a pure ethane sea, the 0.72 m3 of liquid reduces the ullage from the whole tank volume (1 m3) to ¼ of that volume. That leads the pressure to rise by a factor of about 4 from the baseline 1 MPa, which is required to expel the ballast liquid.

If neon is the pressurant gas, then each bottle has to release 45 kg (half of the total requirement) while remaining at 1.1 MPa. The minimum volume to meet this requirement is 0.247 m3, with 7.1 kg of residual gas, and the bottle mass is 146 kg. The gas valves for a neon system split are considered to split the gas 2 ways from each bottle. If the full load of 90 kg is assumed to pass through the valve in 5 minutes, the flow rate is then 270

kg/hour per valve and the Cv is 2.2.

If helium is the pressurant gas, then each bottle has to release 2.7 kg (half the total requirement) while remaining at 1.1 MPa. The minimum volume to meet this requirement is 0.074 m3, with 0.4 kg of residual gas, and the bottle mass is 44 kg. Splitting this amount of helium 4 ways, and assuming 5 minutes to pass the full load of 5.4 kg, yields a

mass flow rate of 16.2 kg/hour per valve. The required Cv is 0.3.

The liquid valves also split the sea volume into the tanks 4 ways. Taking in 0.72

3 3 m of liquid in 5 minutes results in a flow rate of 2.16 m /hour per valve, and a Cv of

2.9. The 4 pumps must raise liquid pressure to 4 MPa for this concept instead of 1 MPa.

Moving a liquid volume of 0.18 m3 across each pump in a time of 1 hour (V  0.18 m3/hr

118 per pump requires an input power of 185 W per pump. (In fact, the pumping can proceed faster because the peak power only occurs at the end of the process).

Each of the two ballast tanks has 1 gas bottle, 4 valves (2 liquid, 2 gas) and 2 pumps.

The total mass of both tanks is 664 kg for neon or 362 kg for helium.

6.6 Noncondensible Gas with Separator

This concept, shown in Figure 6-8, is a compressed neon gas system but with a physical separator between pressurant and liquid tanks. This concept was baselined for the Phase I design in 2014 (5). The advantage of including a mechanical separator between a pressurant gas such as neon and the liquid ballast is that the gas can be recovered into the storage bottle when high pressure is not needed. Therefore, the ballast tank need not store the maximum amount of gas permanently, and may be built to much lower pressure specifications. The trade-off is the need to include a compressor to recover the gas, which adds weight.

Figure 6-8: The proposed Phase I ballast system for the Titan submarine. The

entire tank is on the exterior of the submarine, exposed to sea temperature.

6.6.1 Analysis

6.6.1.1 Model Assumptions

The recompression operation can take place over 12-16 hours after each dive, so it is assumed to be isothermal, because any temperature difference would easily be

119 dissipated by conduction on that time scale. Blowdown, however, is much faster, so the temperature change matters. The model presented here is based on the thermodynamic evolution of the 4 fluid CVs in Figure 6-8. Other general assumptions are listed in Table

6-3. The following analysis is based on the volume-integrated forms of the mass and energy conservation equations: dm CV mm dt in out

dECV 1122    QW   mhin in  vgz in  in   mh out  out  v out  gz out  (Equation 6-15) dt 22   

Assumption Implication The fluid properties of CVs are uniform and fluid temperatures are equal to the respective surface temperatures. Potential and kinetic energy are negligible. 1 h v2  gz  h (mass transfer) 002

ECV  mu (mass in CV) Only the liquid mass in CV 3 is considered, and assumed to be fully in contact with the separator. Gas heat storage/transfer is negligible. Ballast liquid is incompressible. CV 4 is a temperature reservoir. T4  93 K All fluid CVs begin the ramp stage at 93 K. Fixed mass flow rate from gas bottle. dm dm 21   m dt dt 1 All volumes are constant during the ramp stage. V constant CV 1 is fixed-volume in general. 1 V2 , V3 constant (ramp)

Sink rate – the liquid expelled is V dt . 1 2  liq g V dt v C A sea liq2 sea D sub

ALWsub sub sub , CD  1 Table 6-3: General assumptions in the separator model.

Two stages of blowdown are considered in this analysis. The first is the ramp stage, when the gas let out of the storage bottle raises the gas-side pressure up to the maximum local sea pressure (1 MPa). The second is the expansion stage, where additional stored gas is used to move the separator, thereby expelling ballast liquid. Only

120 one half of one ballast tank is modeled, due to symmetry; the right hand side of the tank shown in Figure 6-8 is considered. Common parameters for both stages are listed in

Table 6-4. The area A24 is given by the lateral surface area of a cylinder with the ballast

tank radius and volume equal to CV 2. The area A34 is given by the lateral surface area of a cylinder with the ballast tank radius and volume equal to CV 3, plus an end piece with area equal to the separator. The volume of the gas bottle is derived from the requirement that, when the full ballast volume is filled with gas to a pressure of 1 MPa, the bottle has a residual pressure of 1.1 MPa. See section 6.5.3 for the mass estimates of parts and the total system.

Quantity Symbol Value Gas bottle volume (1/2 ballast tank) 0.0365 m3 Gas bottle surface area (sphere) 0.53 m2 A12 Gas bottle thickness (for 30 MPa) 1.2 cm tgb Initial CV 1 pressure (reduced from 30 MPa at 300 K, on 9400 kPa (helium) P1 0 Earth, to 93 K) 7830 kPa (neon) Ballast tank radius (5-m long cylinder, 0.5 m3 volume) 0.18 m Separator area (circle, ballast tank radius) 0.1 m2 A32 CV 2 initial volume 0.05 m3 CV 3 initial volume 0.25 m3 Ballast tank thickness (1 MPa external pressure) 1 mm ttank Separator thickness 1 cm tsep Titanium thermal conductivity (gas bottle, tank, k 6.7 W/m-K separator) (166) Table 6-4: General Phase I Simulation Parameters

6.6.1.2 CVs and Equations

6.6.1.2.1 Ramp Stage

The gas bottle, CV 1, is assumed to be fully contained in CV 2 with no outside

contact as shown in Figure 6-9a. There is the mass flow rate m1 out from the bottle to the

gas side, and the heat transfer Q21 through the bottle wall. This heat transfer is modeled as pure conduction across the bottle thickness, with no other thermal resistance:

121

TT21 Q21 kA 12 (Equation 6-16) wgb

The simplified mass and energy equations become: dm 11 dt V1

du11  m 1 P 1 Q21 (Equation 6-17) dt1 V 1 1

Figure 6-9: Schematics for the ramp phase: (a) CV 1, (b) CV 2, (c) CV 3.

The gas side of the ballast tank, during the ramp stage, is shown in Figure 6-9b.

While the pressure in this CV rises, the volume is considered constant. Therefore, no work is done until the expansion stage. The values of the heat transfer into CV 2 (again, pure conduction) are

TT32 Q32 kA 32 (Equation 6-18) wsep

122

TT42 Q42 kA 42 (Equation 6-19) wtank

The simplified mass and energy equations for CV 2 become: dm 21 dt V2

du2 1 Q42  Q 32  Q 21  m 1 h 0,1  u 2  (Equation 6-20) dt22 V

The liquid volume in CV 3 is shown in Figure 6-9c. The heat transfer term from the sea (at 93 K) to the liquid ballast is assumed to be conduction only, through the ballast tank:

TT34 Q34 kA 34 (Equation 6-21) wtank

The resulting energy equation (as there is no mass transfer during the ramp stage) reduces by the incompressibility of the liquid: dT Q Q 3 34 32 (Equation 6-22) dtliq V3 c V

where V3 is the volume of ballast liquid.

6.6.1.2.2 Expansion Stage

After CV 2 reaches 1 MPa, it expands to expel ballast liquid. Its volume is no longer constant, and now it does work against the separator to cause a volume change V as shown in Figure 6-10a.

123

Figure 6-10: Schematics for the expulsion stage: (a) CV 2, (b) CV 3.

The work done is pressure-volume work so that WPV 2 liq . The new mass and energy equations become:

V dm21 liq 2 dt V22 V

du2 1 Q42  Q 32  Q 21  PV 2liq  m 1 h 0,1  u 2  (Equation 6-23) dt22 V

The processes in effect in CV 3 during the expansion step are shown in Figure

6-10b. Note that the work done by the gas side is done to the liquid side. The mass balance of the liquid-side changes based on liquid density:

mVliq  liq liq (Equation 6-24) and the resulting energy equation is

dT3 Vliq PV2 liq liq V s h 0,3  Q 34  Q 32 T3 (Equation 6-25) dt V33liq V c V

6.6.1.3 Solution Method

6.6.1.3.1 Ramp Stage

The known quantities before blowdown begins are T1 0 , P1 0 , T2 0 , P2 0 ,

T3 0 , which specify the initial states in all CVs. The geometry (volumes and surface

124 areas), time step size t , blowdown mass flow rate m1 and sea composition (hence the

ballast liquid volume V3 ) are also specified. There are 5 quantities that need to be solved

for at each time step: T1 , P1 , T2 , P2 ,T3 . The 5 equations provided for solution are the mass equations for CVs 1, 2 and the energy equations for CVs 1, 2, 3. First, the mass equations can be discretized to get the new densities at step j :

jj1 mt1 11 (Equation 6-26) V1

jj1 mt1 22 (Equation 6-27) V2

With both densities determined, the next step is to iterate the pressures P1 , P2 by the bisection method. All other properties are determined as necessary using REFPROP, with

pressure and density serving as inputs. The method is to iterate P1 as a loop within P2 ;

for each single iteration of P2 , full convergence is obtained for P1 . Other properties follow from density and pressure, and the equations are: uujj 1 1 11 jj (Equation 6-28) j Q21  m 1 u 1  h 0,1  tV11 uujj 1 1 22 jj (Equation 6-29) j Q42  Q 32  Q 21  m 1 h 0,1  u 2  tV22

Finally, the ballast liquid temperature can be estimated using the present value of T2 .

After expanding the heat transfer terms in Equation 6-23 and rearranging to isolate the

present value of T3 , the result is:

jj1 j wwtanksep 3 VCT 3 V 3 kwATtkw sep 34 4   tank AT 32 2  t T3  (Equation 6-30) wwtanksep 3 VC 3 V kwA sep 34  tkwA  tank 32  t

125

6.6.1.3.2 Expansion Stage

The known quantities after the pressure in CV 2 reaches 1 MPa, before expansion

begins, are the fluid states of all 3 CVs (T1 , P1 , T2 , P2 ,T3 ) and the same 5 equations from

the ramp stage are still available (modified in CVs 2 and 3). The mass flow rate m1 will be the same user-prescribed constant as before. Now, however, 2 more unknowns are

introduced: the rate at which the separator expels liquid, Vliq , and the mass flow of liquid

out of the ballast tank, mliq . The rise rate v , and total expansion time texp are user- prescribed. and are related by assuming the submarine begins blowdown in a neutrally buoyant state:

1 gV t v2 C A (Equation 6-31) sea liqexp 2 sea D sub

Thus, the total expansion amount determines the velocity at which the submarine

rises. In Equation 6-31 the increase in V2 , Vtliq exp , is assumed to be linear in time, which

makes Vliq a derived constant. The liquid flow rate mliq then follows from Equation 6-24 so the system of equations is closed.

The constant value of m1 assumes use of a “bang-bang” control valve that is either fully open or fully closed. The change in density in the bottle in CV 1 is the same as Equation 6-26, at time step j .The change in density in CV 2 follows immediately from the discretized and rearranged form of Equation 6-23:

jj11 j 2V 2 m 1 t 2  j1 (Equation 6-32) V2  Vliq t

126

The pressures in CVs 1, 2 are found by iterating the discretized forms of Equations 6-20 and 6-23: uujj 1 1 11 jj (Equation 6-33) j Q21  m 1 u 1  h 0,1  tV11 uujj 1 1 22 j j j (Equation 6-34) jjQ42  Q 32  Q 21  P 2 Vliq  m 1 h 0,1  u 2  tV22

The same kind of iterated bisection method is used as in the ramp stage solution, based on choosing entropy values for both CVs (property values are computed in

REFPROP from density and entropy). Finally, the new liquid ballast temperature is determined by a discretized version of Equation 6-25 where liquid density and specific heat are assumed constant:

TTjj 1 1 Pjj V V h  Q  Q 33TVj 2liq 3 liq 0,3 34 32 (Equation 6-35) j 3 liq t V33 cV

j j This relation is solved by bisection on T3 at constant density, with properties h0,3 ,

3 and cV computed in the saturated liquid state at that temperature.

6.6.1.4 Numerical Results and Discussion

The 12 cases run in this simulation are listed in Table 6-5, for two different pressurant gases (GHe and GNe), 2 different seas (5% and 85% CH4), and several

different values for m1 , desired v , and texp . The volume expansion rate column Vliq is computed by Equation 6-31. In the case of a methane-rich sea, the ballast tank never takes in very much liquid anyway, so it is emptied before the submarine reaches any significant thresholds in desired v . The “mass out” column gives the total amount of pressurant mass that leaves the bottle during both the ramp stage and the expansion stage.

127

After trial and error, the time step at which results become time step-independent is 0.05 seconds or less, so the step used for all simulations is t 0.05 seconds.

Case Gas Sea Mass out (kg) v (m/s) texp (s) m1 (kg/hr) 3 Vliq (m /s)

1 GHe 5% CH4 0.1 90 2.5 0.279 6.67e-5 2 GHe 5% CH4 0.1 90 3.75 0.31 6.67e-5 3 GHe 5% CH4 0.1 90 7.5 0.404 6.67e-5 4 GHe 5% CH4 0.25 150 5 0.425 2.49e-4 5 GHe 5% CH4 0.25 150 7.5 0.529 2.49e-4 6 GHe 5% CH4 0.25 150 10 0.841 2.49e-4 7 GHe 5% CH4 0.35 200 5 0.494 3.66e-4 8 GHe 5% CH4 0.35 200 7.5 0.633 3.66e-4 9 GHe 5% CH4 0.35 200 10 0.772 3.66e-4 10 GHe 85% CH4 Empty 100 10 0.49 7.2e-4 11 GNe 5% CH4 0.35 200 53.3 4.08 3.66e-4 12 GNe 85% CH4 Empty 100 53.3 2.6 7.2e-4 Table 6-5: Simulation cases run.

To examine the effect of pressurant gas mass flow rate on P1 , Figure 6-11 plots gas bottle pressure as a function of time for cases 4 through 6 using GHe in an ethane- rich sea, for a v  0.25 m/s in 150 s. As shown, the faster the pressurant is released from the bottle, the faster the pressure of the remaining fluid in the bottle drops, with no discernible slope change between ramp and expansion stages. Releasing more mass causes the ramp stage to be completed more quickly, but the total pressure drop is higher for the same v requirement. Still, the lowest predicted bottle pressure is well above the design threshold of 1.1 MPa. Also, the “mass out” values are at most half of the gas in the bottle (1.5 kg of helium or 7.5 kg of neon per half of the ballast tank – see section 6.6.3).

To examine the effect of varying the pressurant gas type and flow rate on bottle pressure, Figure 6-12 plots the GNe cases 11 and 12. As shown, the bottle pressure is little affected by sea composition and expansion rate.

128

1x 104 Bang-bang GHe, 5% LCH Sea 4 9000 v =0.25 m/s, 150 s rise m =5 kg/hr (case 4) 1 8000 7.5 kg/hr (case 5) 10 kg/hr (case 6)

[kPa]

P 7000

6000

5000 0 50 100 150 200 250 300 350 Time [s]

Figure 6-11: Gas bottle pressure as a function of time using helium.

8000

7000

6000

5000

4000

[kPa]

1 P 3000 Bang-bang GNe, m =53 kg/s 2000 1 Case 11 1000 Case 12

0 0 50 100 150 200 250 300 Time [s]

Figure 6-12: Gas bottle pressure as a function of time using neon. The 2 lines are

on top of each other, with the case 12 line ending at 175 seconds.

Figure 6-13 plots the gas temperature inside the bottle as a function of time for cases 4 through 6. As gas leaves the bottle, the temperature of the gas tends to decrease along with the pressure; this decrease is greater the faster the gas is released. Past a certain decrease, however, thermal contact with the gas-side prevents any further

129 temperature change. The “knee” in the curves represents the effect of the decrease in T2 as the separator starts expanding (see Figure 6-15), which reduces the heat conduction

back to the bottle that had been preventing T1 from decreasing further.

93.5

Bang-bang GHe, 5% LCH Sea m =5 kg/hr (case 4) 93 4 1 7.5 kg/hr (case 5) v =0.25 m/s, 150 s 10 kg/hr (case 6) rise 92.5

[K]

1 T 92

91.5

91 0 50 100 150 200 250 300 350 Time [s]

Figure 6-13: Gas bottle temperature for helium, ethane-rich sea cases with

different mass flow rates.

To examine the thermodynamic state of the gas in the gas-side of the ballast tank,

Figure 6-14 and Figure 6-15 plot the pressure and temperature as a function of time for 3 different mass gas flow rates, for cases 4 through 6, respectively. As gas enters the ballast tank under its own pressure, it raises the pressure of the gas side smoothly, and more rapidly at a higher rate of mass transfer. At the stated threshold of 1 MPa, the mass transfer rate interacts with the volume expansion rate to determine what the pressure will

be thereafter. Below a certain value of m1 , the expansion causes density to fall, reducing pressure. Near this value, the gas-side pressure stays nearly constant at 1 MPa, as long as the expansion rate is constant (the 5 kg/hour line). Above this value, the pressure

130 continues to rise as the increase in mass exceeds the increase in volume (the 7.5 kg/hour and 10 kg/hour lines).

1600

1400

1200

1000 Bang-bang 800 GHe, LC H Sea

[kPa]

2 2 6 P v =0.25 m/s, 150 s 600 rise

m =5 kg/hr (case 4) 400 1 7.5 kg/hr (case 5) 200 10 kg/hr (case 6)

0 0 50 100 150 200 250 300 350 Time [s]

Figure 6-14: Gas-side pressure for helium, ethane-rich sea cases with different

mass flow rates.

The temperature of the gas-side is largely constant but shows 2 distinct sudden changes in Figure 6-15. The first occurs initially, when gas from the bottle first raises CV

2 density. The second occurs after the expansion stage begins and the gas is cooled by doing work against the separator. Both of these spikes fade relatively quickly because the model assumes solid thermal contact between all CVs, so that such temperature anomalies are quickly distributed through the system. The cold Titan seas provide an effective thermal bath at 93 K.

131

93.15 Bang-bang GHe, 5% LCH Sea 4 93.1 v =0.25 m/s, 150 s rise

m =5 kg/hr (case 4) 93.05 1 7.5 kg/hr (case 5) 10 kg/hr (case 6)

[K]

2 T 93

92.95

92.9 0 50 100 150 200 250 300 350 Time [s]

Figure 6-15: Gas-side temperature for helium, ethane-rich sea cases with

different mass flow rates.

Figure 6-16 plots the temperature of the liquid-side of the ballast tank for different displaced volumes and expansion time. As shown, there is little temperature change in the liquid ballast during the ramp stage. However, as it absorbs the work done by the pressurant gas, its temperature rises substantially (Figure 6-16). This temperature rise is fastest for case 9 where the largest volume of liquid needs to be expelled.

[K]

T 95 Bang-bang, GHe m =10 kg/hr 94 1 5% LCH sea, case 6 4 5% LCH sea, case 9 93 4 85% LCH sea, case 10 4 92 0 50 100 150 200 250 300 Time [s]

132

Figure 6-16: Liquid-side temperature for several cases.

In summary, the separator concept should function well in the Titan seas. The bottle cools down by less than 2 K from a starting point of 93 K, which does not cause issues with helium or neon. Because the bottle is wholly contained in the gas side, it is little affected by changes in CV 3. The ballast liquid may warm significantly when the separator does work on it, but that change, too, is limited to a few Kelvin, and the rise would have to reach 110 K to risk boiling the liquid. Dissolved nitrogen may come out as bubbles due to reduced solubility at the higher temperatures, but these bubbles would not prevent the liquid from being expelled out the bottom valve in Figure 6-8. The model describes solid thermal contact between all regions of the ballast system. As such, temperature changes are small because any imbalance is quickly distributed to another

CV or to the Titan Sea. Therefore, the concept is feasible and meets all requirements, assuming the hardware can cycle at cryogenic temperatures.

6.6.2 Operations

The submarine will arrive on Titan with the gas bottle full, the ballast volume filled with nitrogen, and the separator fully retracted toward the compressor.

 Follow the sinking instructions for the pump concept (section 6.2.2).

 Drive down using propulsion.

 To adjust for a lower density, open V1 and V3 (moving the separator) until the

flowmeter indicates that the desired liquid volume change has been achieved.

 To surface, fully open V1 and V3 (moving the separator) to blow down the

compressed gas and remove ballast liquid, until the separator is as far as possible

133

from the compressor. Record the liquid volume remaining in the onboard

processor.

To recompress the pressurant gas and retract the separator to the compressor, open V2 and turn on C1 until the volume of CV 2 is at its smallest and matches the surface pressure of 0.15 MPa.

6.6.3 Parts

For the gas valve that vents to the atmosphere, an operation time of 5 minutes to fill the entire 0.25 m3 sea volume (per ballast tank) will be adequate. The corresponding vent

flow rate is 17 kg/hour, and the required Cv is 0.9. The liquid valves also split the sea volume into the tanks 4 ways. Taking in 0.72 m3 of liquid in 5 minutes results in a flow

3 rate of 2.16 m /hour per valve, and the required Cv is 2.9. The required amount of neon gas in each ballast tank is enough to fill 0.5 m3, or 13.2 kg. The minimum volume required for this requirement is 0.0724 m3, with 2.1 kg of residual. The bottle mass is about 42.8 kg. For the gas valve that empties the pressurant tank of neon, an operation time of 5 minutes is adequate. The flow out of the pressurant bottle is split two ways, for

a flow rate per valve of 80 kg/hour, and the required Cv is 0.7. The 4 compressors may take up to 12 hours (5) to process 26.3 kg of neon, for a flow rate of 0.00015 kg/s to each compressor. With the associate temperature change neglected, the required power to each compressor is about 70 W. If helium is the pressurant chosen, then the required amount of helium gas in each tank is enough to fill 0.5 m3, or 2.55 kg. The minimum volume required for this requirement is 0.0695 m3, with a residual of 0.4 kg. Then the bottle mass is about 41.1 kg. For the gas valve that empties the pressurant tank of helium, the flow out of the pressurant bottle (split two ways, 5 minute operating time) is 15 kg/hour per

134 valve. The required Cv is 0.3. In the helium version of the concept, the 4 compressors have the same 12 hours to process 5.1 kg, for a flow rate of 30 mg/s per compressor. If the associated temperature change is assumed to be negligible, the required power to each compressor is about 75 W. The mass of each separator is about 1 kg. The whole ballast system has 2 tanks, 2 loaded gas bottles, 12 valves, 4 separators and 4 compressors; the total mass is 560 kg for neon or 600 kg for helium. There is no shared hardware between the tanks on different sides of the submarine.

6.7 Gas Purification

Before developing the concept where nitrogen is the pressurant, the process of acquiring that nitrogen from the Titan atmosphere is described. Phase I of the Titan submarine project (5) selected the ballast design of section 6.6. The principal drawback is leakage, because the reliability of cryogenic seals for a piston is unknown and because polymer bladders have only been tested to dozens of cryogenic cycles (175), significantly less than what is needed for the Titan mission. Any GHe or GNe that leaks is lost and cannot be replaced, a loss that would accumulate over the mission.

One alternative to using a noncondensible pressurant would be harvesting gas from the thick Titan atmosphere. In-space resource utilization (ISRU) concepts such as this may lead to a reduction in system mass and complexity. A scheme which could harness that supply of nitrogen as the pressurant would eliminate the risk pf accumulated leakage and simplify the ballast system. The initial pressurant could be harvested in-situ and then replenished in-situ as necessary. However, the methane component should first be removed by a purification unit, as shown in Figure 6-24. This is because (1) as the gas temperature in the bottle falls during blowdown, the methane may freeze and (2) the

135 mixture saturated vapor pressure at 93 K is 0.25 MPa, compared to 0.46 MPa for pure nitrogen, which magnifies operational uncertainties.

The amount of nitrogen that the separation system would need to collect depends on the operating conditions. The depth effects in the Titan seas are unknown, so the surface temperature (93 K) is a good baseline. The pressure, and thereby depth, is limited by the saturation pressure of nitrogen at 93 K, 0.462 MPa, above which the pressurant will condense instead of expelling liquid. The density of nitrogen in that state is 19.1 kg/m3, and the design liquid ballast volume in 1 m3, for a mass of 19.1 kg.

There is also a residual nitrogen mass in the bottle to account for, when the ballast tank is fully expelled. As a first estimate, the bottle will store its full contents as a saturated liquid at 93 K (saturation density 730 kg/m3, with at least 20% ullage) while the residual will be saturated vapor at 93 K. Then a residual mass of 0.9 kg and a bottle volume of 0.047 m3 are adequate. Applying a safety factor of 1.25 to the 20kg total, the required nitrogen mass is 25 kg. This is the initial run to fill the pressurant bottle before the submarine begins its first dive. If any of the nitrogen then leaks during the mission, it can be replenished as needed between dives. Because such secondary operations will require much less flow through the purification hardware, all estimates are done for the initial fill.

The components and operational methods for four kinds of nitrogen-methane separation systems are described in this section: refrigeration, distillation, membrane, and adsorption concepts. Then estimates of the mass and power of each system is given.

Finally, the most feasible gas separation concept is identified.

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6.7.1 Refrigeration

The separation of nitrogen and methane in the gas phase is a well-known problem in the natural gas processing industry, and several systems to accomplish this have been devised. One method is to exploit the tendency for methane to condense before nitrogen as the mixture cools. The goal is to select a compression ratio such that, with partial pressure of each gas so multiplied, the methane boiling temperature is higher than the nitrogen boiling temperature. The pressure and temperature ranges corresponding to vapor-liquid saturation are shown in Figure 6-17 for nitrogen and methane. Below the triple point pressure, methane is predicted to freeze rather than condense.

1.2 0.06

a) Nitrogen b) Methane 1 0.05 Triple point: 63 K, 0.012 MPa Triple point: 90.7 K, 0.012 MPa Critical point: 126 K, 3.4 MPa Critical point: 190 K, 4.6 MPa 0.8 0.04

0.6

[MPa]

sat

sat 0.03

P 0.4

0.02 0.2

0 0.01 80 85 90 95 100 105 92 94 96 98 100 102 104 T [K] T [K] Figure 6-17: Liquid/vapor saturation lines of a) nitrogen and b) methane in the

Titan temperature range. The triple and critical points are noted for comparison.

In the atmosphere near the Titan seas, the partial pressure of nitrogen is 0.1425 MPa, with a saturation temperature of 80.4 K; the partial pressure of methane is 0.0075 MPa, which is likely to freeze out below the triple point. Therefore, if the atmosphere were cooled at constant pressure, reducing the gas temperature to perhaps 85 K will be adequate. This requires the installation of a cooling system.

The power needed to chill the atmosphere down to 85 K depends on the flow rate.

The difference in combined enthalpy for the Titan atmosphere composition (at 0.15 MPa)

137 between 93 K and 85 K is 24 kJ per kg of total atmosphere. A nitrogen mass of 25 kg implies 0.89 kmol, so the quantity of methane (1/19th of that) is 0.047 kmol, for a total gas mass of 25.8 kg. Therefore, the total energy that must be extracted to cool the requisite amount of atmosphere is 619 kJ. Based on constraints from the submarine’s heating system, the power to the cryocooler is limited to 200 W in stationary surface mode (5), and a small commercial cryocooler (176) will accept 180 W of electrical power. This cooler has a specific power of 18:1 around 85 K, so it will lift 10 W of heat from the gas. Therefore the required freezing time is 17.2 hours.

Since the operation of cryocoolers depends on rejecting heat near 300 K, one end needs to be exposed to the internal submarine temperature while the other end is in contact with the atmosphere. One way to do this is to have the room-temperature end of the cooler inside the insulation while the cold end is exposed to Titan temperatures. To commence harvesting, the intake valve is opened, while turning on the cooler, and blower. As atmosphere flows past the cold finger, the methane freezes onto the finger

(see Figure 6-18). The intake is closed and the gas in the cavity is circulated by the blower until the amount of frozen methane is unchanging, at which time the valve to the ballast tank is opened and the nitrogen is sent to the bottle. Then the tank valve is closed, the vent valve is opened and the frozen methane is vaporized, either by a resistance heater

(as shown) or by heat transfer through the inactive cryocooler. This arrangement repeats as many times as necessary.

138

Figure 6-18: The arrangement of a cryocooler to freeze the methane component

out of the atmosphere before compressing the nitrogen into the ballast tank. The cryocooler penetrates the insulation of the submarine so that one end of it remains

near 300 K.

The amount of heat it takes to vaporize the methane comes from the heat of sublimation, 575 kJ/kg (177). Applied to 0.8 kg of methane, the energy requirement is

460 kJ and the venting time is about 40 minutes at 200 W. Therefore, the total operating time is less than one Earth day. The biggest challenge in this design is to shape the cavity and place the blower so that the whole intake mass is circulated past the cold finger.

6.7.2 Distillation

An alternative option that does not require refrigeration is to put the atmosphere through a liquefaction compressor that raises the pressure so that one gas component condenses out at 93 K, while the other remains a gas because it condenses below 93 K.

The compression to achieve this pressure is expected to raise the temperature of the gas, but that can be quickly undone by bringing it back in contact with the atmosphere or sea through a heat exchanger. It is therefore relatively simple to achieve the higher pressure,

93 K state for the process to work.

139

The most important question in designing this system is what pressure to use in the compression process. Below a pressure of 0.312 MPa, both gas components (5% methane and 95% nitrogen) have saturation temperatures below 93 K, and so both remain gas.

Similarly, above a pressure of 0.486 MPa, both components have saturation temperatures above 93 K, and so both would liquefy. Therefore, an optimal point for distillation by gas compression is a pressure between 0.312 MPa and 0.486 MPa. For this work, a pressure of 0.413 MPa is assumed (where the methane component condenses at 95.3 K and the nitrogen component condenses at 91 K); this corresponds to a compression ratio of 2.75.

The equipment to accomplish this distillation is a portion of a cycle for gas liquefaction (124), shown in Figure 6-19. The compressor raises the pressure and thereby the temperature of the atmosphere before the flow is immediately exposed to sea liquid through a heat exchanger, which brings the temperature down and condenses the methane component. That methane component is then removed by a pump, to keep the now- purified nitrogen isolated, before the nitrogen is sent though the compressor in the main ballast tank.

Figure 6-19: The arrangement of a distillation method of removing gaseous

methane.

140

The excess power requirement of this system is either that of the pump, or of the distillation compressor. The power limit is 200W, as for the refrigeration concept. If the compressor has an isentropic efficiency of 80%, and raises the atmosphere from an initial state of 93 K and 0.15 MPa to 0.413 MPa, then the corresponding mass flow to this power limit is 0.005 kg/s, and the gas is heated to 130 K (see Figure 6-20 for the thermodynamic process used). This flow rate implies that the initial charge time for the pressurant tank is about 1.5 hours. The pressurized liquid methane can then be expelled through a valve to the Titan sea; since the density of this (assumed saturated) liquid is

390 kg/m3, the 0.8 kg of methane to be removed occupies about 0.002 m3 in total.

0.45 Ballast Tank 0.4 Heat Exchanger

0.35

0.3

0.25

P [MPa]

0.2

0.15 Intake 0.1 90 100 110 120 130 140 T [K] Figure 6-20: The thermodynamic process as intake gas is heated and compressed, then returned to Titan sea temperature. The saturation temperature of

95.3 K (at the final pressure) is exceeded by a safe margin.

In total, distillation is an attractive separation method that would take a couple hours complete, and has moderate power requirements. The principal difficulties are (1) designing the heat exchanger that reduces the gas temperature by more than 35 K as well as separating the liquid, (2) maintaining the space between the compressors near 0.413

MPa and (3) monitoring the discharge of the liquid methane to minimize leakage.

141

6.7.3 Membrane

A membrane system works with a large pressure difference forcing the mixture from one side to the other. If the membrane is more permeable to nitrogen than to methane, it will raise the concentration of nitrogen in the “permeate” gas (which penetrates the membrane, then flows onward) while raising the concentration of methane in the

“retentate” gas (the gas that flows out the membrane housing without passing through the membrane). The passage of permeate through the membrane is enabled by a large pressure gradient, which requires a compressor to pressurize the gas intake. This process is illustrated schematically in Figure 6-21.

Figure 6-21: Conceptual layout of gas separation by a single membrane. The

concept assumes the membrane is more permeable to nitrogen than to methane.

Membranes are most common in near-room-temperature industrial settings to separate small amounts of nitrogen from methane, before the latter enters natural-gas pipelines. One typical plant arrangement (178) involves an interconnected network of two membranes supplied by a pressure difference around 3 MPa (0.5 to 3.5 MPa), which achieves a reduction of nitrogen content from 10% to 4%. Such a design is unsuitable for the submarine for several reasons:

142

 As shown in Figure 6-17, the vapor pressure of both nitrogen and methane at 93 K

is far less than 1 MPa. The entire Titan atmosphere would condense long before it

comes close to industrial operating conditions.

 To the author’s knowledge, no membrane materials have been tested at Titan

temperatures. The best performance available is deca-dodecasil 3R (DDR)

zeolites (179), which are 40 times more permeable to nitrogen than to methane

but have been tested only down to 230 K.

 Working within a pressure difference such as 0.5 MPa would make the

performance worse, all else being equal. The alternative is to build a larger

surface or another membrane unit, which would greatly increase the mass.

 Even at 3 MPa, the whole system only removes 60% of the nitrogen in the input

natural gas. This outcome is not acceptable for the submarine, which needs to

remove the methane down to 0.24% to reduce vapor pressure uncertainty to 0.012

MPa. This is equivalent to removing 95% of the methane concentration.

In total, the membrane method of gas separation is not nearly effective enough to justify in the context of the submarine. The dead weight of 3-MPa processing equipment makes the concept even less viable; the typical industrial system has a diffuser, a compressor and the piping around the membrane chambers (doubling the pressure doubles the required thickness of every wall). Every kilogram spent on an attachment to the ballast system is a kilogram taken away from propulsion, communication and science instruments.

143

6.7.4 Adsorbent

An adsorbent system is based on the equilibrium amount of gas sticking to the surface of a chosen material at a given temperature and pressure. Because this amount changes with both temperature and pressure, and because nitrogen and methane adsorb differently, a procedure that puts several gas flows through the adsorbent while cycling through high and low pressure values can produce an outflow of gas that contains less methane than the inflow. The effects of a molecular sieve to pass smaller molecules preferentially are also related to this concept. The schematic in Figure 6-22 is based on a single-bed, counter-flow process illustrated in (180).

Figure 6-22: Partial conceptual layout of an adsorbent system (adsorbent is in

grey).

A general version of the operating procedure for such a system starts with compressing intake gas through the adsorbent bed and sending it to the intermediate tank.

Then the bed is vented down to atmospheric pressure, before the vent is closed and the intermediate gas is drawn through the bed again to the compressor in the ballast tank. The bed can, perhaps, be vented again before the next round of intake gas is harnessed. The adsorbent, in that scenario, is required to adsorb methane in much larger amounts than nitrogen, so the former can be stripped out and vented.

144

To estimate the weight of adsorbent material required in this arrangement, it is necessary to know the typical difference between equilibrium adsorption rates for a given material. This difference does not normally exceed 1 mole of gas per kilogram of solid at modest temperature-pressure differences (181) (182) (183). With respect to temperature coverage, one adsorbent has been tested down to 148 K in nitrogen and methane (182).

The pressure range in which the 148 K data were taken (again, for nitrogen and methane) is 0.025 to 3 MPa, which includes a good range around the Titan atmosphere.

During filling, the 0.8 kg of methane are retained while passing the nitrogen, or 12.8 kmol, so that the total equivalent adsorbent mass is 12,800 kg. Since the time the adsorbent needs to reach equilibrium is as little as 30 seconds (183), a reasonable low- end cycle time is 2 minutes. The relation between amount of adsorbent mass and operating time to fill the bottle, if the full mass and adsorption swing are exploited, is shown in Figure 6-23.

104

1000

100

Operating minutestime,

10 1 10 100 1000 Adsorbent mass, kg

Figure 6-23: The dependence of gas separation time on mass of adsorbent used.

The total system mass is significantly larger than the adsorbent mass, because it needs separate thermal management to cycle between temperature and/or pressure values

145 that make a useful difference in equilibrium adsorption. Temperature is a special concern, because the heat flow required might easily exceed the total submarine waste heat; and achieving 148 K consistently in a 93 K environment is likely to be very difficult. The intermediate tank must also be accommodated, next to a relatively large ballast tank.

A separate concern is the effectiveness of this method to clean the atmosphere of methane. The purity standards are just as high as those in the membrane concept, and the residual gas left on the adsorbent bed after venting is expected to contribute significantly to the composition sent into the ballast tank. Indeed, one installation (180) typically reduces 15% nitrogen content to 3% or 4%, which is inadequate for the submarine. In summary, this purity issue and the lack of data near Titan conditions make the concept somewhat implausible.

6.7.5 Mass Comparison

For comparisons between the masses of the four concepts, parts that they have in common are disregarded. These parts are the intake valve, the valve into the ballast tank, the ballast tank, the compressor inside the ballast tank, the pressurant bottle inside the ballast tank and any connectors inside the ballast tank. General channels (excluding the heat exchanger in the distillation method) are also considered roughly the same between all concepts.

For the refrigeration concept, the additional important masses are the cryocooler (the same as the one cited in section 6.7.1), a blower and a vent valve (the wire used for resistance heating is assumed to be negligible). The mass of the cryocooler is 4.1 kg

(184). The blower is simply a fan, and is taken to be ~1 kg. A typical cryogenic actuated

146 valve has a mass of 5 kg, most of which represents the actuator (the valve itself can be quite lightweight). Therefore the additional mass is ~10 kg.

For the distillation concept, the additional components are a compressor and the additional length of passage to exchange heat with the Titan sea. The compressor is considered to be about 10 kg, which is within the range of commercial oil-free air compressors (169) (168) that reach 0.6 MPa output pressure. The added pipe mass in the heat exchanger is also considered to be ~5 kg. Therefore, the total distinct mass is ~15 kg.

For the membrane concept, a starting point for the mass estimate is the mass of a 3

MPa compressor. Since compressor mass is roughly proportional to exit pressure, and the compressor in the distillation concept may be based on 0.6 MPa, applying a factor of 5 gives ~50 kg. It is likely that achieving high performance out of the system would require a two-membrane system with another diffuser or compressor (178), giving ~100 kg before the weight of the membrane itself and the piping.

For the adsorbent concept, assuming an initial fill period of 1000 minutes (17 hours, to match the refrigeration concept) requires 30 kg of adsorbent and three additional valves. The subtotal is then ~45 kg, but this does not represent the weight of the temperature/pressure management system. The total could easily be twice as much, ~90 kg.

The respective mass estimates of each concept are given in Table 6-6. There is no time estimate for membrane separation because it is impractical to operate at all. The maximum power consumed by each could be arranged to be 200 W, as part of the surface stationary operating mode.

Concept Additional Parts Mass (kg) Initial Fill Time (hr)

147

Refrigeration Cryocooler, blower, vent ~15 18 valve Distillation Compressor, heat exchanger ~15 1.5 Membrane Membrane, compressor >100 N/A Separation Adsorbtion Adsorbent, compressor, ~90 17 Separation intermediate tank Table 6-6: Mass estimates for the four gas separation concepts in this paper.

6.8 Nitrogen Gas with Separator

This concept, shown in Figure 6-24, replaces the noncondensible neon or helium with nitrogen from the atmosphere. It comes with the gas separation unit selected in section

6.7.

Figure 6-24: A nitrogen pressurant gas system with gas separation.

6.8.1 Unmodified Vapor System

One version of using nitrogen is to change nothing else about how the ballast system is arranged. Such a design has three constraints. First, the pressure at 93 K when all gas has been reclaimed must not exceed 0.46 MPa. Second, the both CV 1 and CV 2 must remain vapor; this is tested using one of the cases from the noncondensable separator analysis. Third, the gas bottle pressure must be greater than the gas-side pressure at all time steps during the solution. The last two constraints are facilitated by a modification of

148 the separator simulation, where the gas bottle valve is closed whenever the gas-side pressure is more than 10 kPa above the target value.

Using the bottle volume from the neon/helium simulations of 0.0365 m3, the unmodified nitrogen system works up to 0.25 MPa. This corresponds to 135 meters depth in Ligeia or 115 meters in Kraken.

6.8.2. Vapor System with Heater

Heating the nitrogen vapor to a temperature above Titan ambient would allow it to remain vapor at higher pressure. However, the waste heat of the submarine is limited to

3800 W. The simulations that are performed to this end are conducted using the separator simulation from the GHe/GNe system, with a few differences. Gas is let out of the storage bottle, CV 1, into CV 2 at constant volume (the ramp stage); after CV 2 reaches the pressure of the surrounding sea, it pushes the liquid in CV 3 out of the sea valve by applying force across the separator (the expansion stage). Figure 6-25 shows the updated

CVs with mass and energy exchange when a heater is included around the gas-side of the separator and gas bottle. With the rate of heat transfer Q to the bottle, the numerical solution now solves the discretized energy equations: uujj 1 1 11 jj(CV 1) (Equation 6-36) j 0.5Q  Q21  m 1 u 1  h 0,1  tV11 uujj 1 1 22 jj (CV 2, ramp stage) j 0.5Q  Q42  Q 32  Q 21  m 1 h 0,1  u 2  tV22

(Equation 6-37) uujj 1 1 22 j j j (CV 2, expansion jj0.5Q  Q42  Q 32  Q 21  P 2 Vs  m 1 h 0,1  u 2  tV22 stage) (Equation 6-38)

149

Figure 6-25: Updated control volumes for the separator ballast concept with

heaters added.

The test case run to test functionality is a mass flow rate out of the bottle of 50 kg/hour and a time step of 0.05 sec. The 5 unknowns at each time step are pressure and temperature in CVs 1 and 2, and the liquid temperature in CV 3. The 5 equations used to solve the unknowns are the mass and energy equations in CVs 1 and 2, and the energy equation in CV 3.

In addition to the variables used in the simulation, there are 3 adjustable parameters: the starting temperature of the gas bottle, the volume of the bottle, and the heater power. The rule for determining starting temperature is that the saturation pressure is set to 0.2 MPa above the desired gas-side pressure. The bottle volume is increased in proportion to pressure based on the previous row in Table 6-7, then adjusted if necessary.

The goal is to determine the minimum required heater power by trial and error, as judged

150 by reaching the desired expansion pressure with near-zero superheating left over. Table

6-7 shows the results for required heater power as a function of depth in the seas. Figure

6-26 compares these power values to the submarine. A GN2 based system can be used to achieve any desired depth in either Titan Sea, at the cost of increasing heater power demand.

Pressure Ligeia depth Ethane-rich Bottle V Starting T1 Required Q (MPa) (m) depth (m) (m3) (K) (kW) 0.25 135 115 0.0365 93 0 0.3 200 170 0.0438 94.6 2 0.4 - 285 0.06 96.9 8 0.5 - 400 0.073 99.0 12 0.6 - 515 0.09 101.0 16 0.7 - 625 0.105 102.7 22 0.8 - 740 0.12 104.3 30 0.9 - 855 0.135 105.8 38 1 - 970 0.15 107.2 45 Table 6-7: Heater-only results table

Ligeia Kraken 40 Submarine

Required Heater Power [kW] Power Heater Required

0 0 200 400 600 800 1000 Depth [m]

Figure 6-26: Heater-only results graph. Dotted line is the current submarine waste heat available to use as a heat source.

6.8.3. Liquid Storage

What happens when the bottle is allowed to be a two-phase storage medium with both liquid and vapor? This version of the system might not require heating. Because pressure

151 and temperature are dependent properties at saturation, the thermodynamic state of the

bottle is given using pressure and quality instead. The mass flow rate m1 is estimated

according to the size of the gas valve Cv and the pressure difference between CVs 1 and

The appropriate energy and mass equations for the bottle are: d m1lg m 1   m 1 (Equation 6-39) dt d m1l u 1 l m 1 g u 1 g   m 1 h 0, g  Q 21 (Equation 6-40) dt

After expanding the derivatives and rewriting in terms of the quality of the fluid in the

bottle 1 , the equations are:

d dP m 1 1   1 (Equation 6-41)  dt P dt V1

Q m h  uud11   dP 21 1 0,g uu    (Equation 6-42)  dt  P  P dt V1

The numerical solution procedure is based on discrete time steps j , and is done by

j bisection on the bottle pressure P1 . First, the quality is estimated from the discretized mass equation:

j1 mj P j P j1 1  1 1 jj1 V1  P dt 11dt j1 (Equation 6-43)   

j Then the values of 1 , are checked against the discretized energy equation:

j j j11j j j Qj m j h j  uu1 1   PP 1 1 21 1 0,g uu    (Equation 6-44)  dt  P  P dt V1

152

j The other nested bisection in the whole system solution is for the gas-side pressure P2 .

Some preliminary results for the ramp stage are displayed in Figure 6-27: Figure

6-27a and Figure 6-27b show the pressure and quality in CV 1, Figure 6-27c and Figure

6-27d show the pressure and temperature in CV 2, and Figure 6-27e shows the temperature in CV 3. Bottle quality rises over time; this makes sense during discharge.

After sufficiently long operation the bottle is wholly vapor, and continuing the simulation would require a switch to the single-phase model. A larger bottle or a bottle with a lower initial quality would continue to have quality less than 1 until there is no pressure gradient between CVs 1 and 2. Then the bottle quality is near constant. Saturated nitrogen at pressure on the order of 1 MPa is significantly warmer than the Titan seas. This heat quickly transfers to CV 2, and less quickly to the liquid side, CV 3. After the initial rise, the temperature decays in the same manner as the pressure, though CV 2 remains vapor for the whole run. The model predicts no lasting pressure accumulation on the gas-side, which would prevent any effective blowdown.

1 2500 b) CV 1 Quality a) CV 1 Pressure 0.8 2000

1500 0.6



[kPa]

P 1000 0.4

500 0.2

0 0 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 Time [s] Time [s]

153

164 101 c) CV 2 Pressure d) CV 2 Temperature 162 100 99 160 98 158

[K] 97

[kPa]

2 156 T P 96 154 95

152 94

150 93 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 Time [s] Time [s]

93.02 e) CV 3 Temperature

93.02

[K] 93.01

93.01

93 0 1 2 3 4 5 6 7 8 Time [s]

Figure 6-27: Preliminary results for the liquid nitrogen ballast concept. The

properties plotted are a) pressure and b) quality in CV 1, c) pressure and d)

temperature in CV 2, and e) temperature in CV 3.

6.8.4. Liquid Actuator

One alternative is to apply force directly with the liquid nitrogen, as opposed to the nitrogen gas, as shown in Figure 6-28. This would essentially be a hydraulic actuation system, where liquid is pumped out of the bottom of the tank to a piston-cylinder that pushes liquid out of the ballast tank.

154

Figure 6-28: Schematic diagram for the liquid nitrogen actuator model.

The largest liquid ballast volume that the system should be able to expel is 0.65 m3 in an ethane-rich sea. If that takes the bottle from 10% liquid volume to 90% liquid volume, then the bottle volume needs to be 0.81 m3 (equivalent to a diameter of 106 cm).

The work done to move the liquid is WPV* for highest pressure P  1 MPa. If the efficiency of the pump is 80%, the volume to be expelled is 0.1 m3 and the power available is 500 W, then the required expulsion time is 4 to 5 minutes. This is a reasonable operating time to end a dive, though perhaps insufficient for an emergency.

6.8.5 Parts

The pros and cons of the concepts described in this work are listed in Table 6-8. The nitrogen vapor concept could function by the same principles as the Phase I GHe/GNe concept, but the risk of gas condensation severely limits the operating range of the vehicle. Also, the vapor cannot be highly compressed, which requires a very large bottle.

The concept where nitrogen is stored in two-phase is only plausible if the liquid is drawn to push against the piston (only that case is shown in the table); if the vapor is drawn, the system may be incapable of sustaining high-pressure expulsion.

Concept Pros Cons

155

GNe/GHe No fundamental operating limits. Zero leak tolerance; gas must be brought No condensation (supercritical). from Earth.

GN2 (no heater) Can accommodate leakage. Very large bottle required. Operating depth < 120 meters.

GN2 (with Can accommodate leakage. Required heating far exceeds submarine heater) capacity. Requires larger bottle.

LN2 (actuator) Can accommodate leakage. Need additional hardware to control phase change. Operating depth < 120 meters.

Table 6-8: Pros and cons of the nitrogen separator concepts, compared to using a noncondensible pressurant

The part lists and mass estimates of each concept are given in Table 6-9. The

GHe/GNe concept is by far the heaviest because it functions at high pressure; this affects both the bottle and the compressors. The other nitrogen concepts are the same order of magnitude at about 100 kg. These totals do not include the two shared ballast tanks, each with a mass of 26 kg, or the nitrogen harvesting equipment at 30 kg.

Concept Parts Mass (kg) GNe/GHe 2 gas bottles: 42 kg each. 560 (neon) Gas mass: 3 kg (helium), 15 kg (neon). 600 (helium) 4 compressors (169) (168): 9 kg each at 0.85 MPa baseline, raised in proportion to pressure. 4 separators: 1 kg each. 12 valves: 5 kg each. GN2 (no heater) 2 gas bottles: 6.3 kg each. ~100 Gas mass: ~1 kg. 4 compressors (169) (168): 9 kg each. 4 separators: 1 kg each. 12 valves: 5 kg each. GN2 (with 2 gas bottles: 18 kg each. ~135 heater) Gas mass: ~10 kg. 4 compressors (169) (168): 9 kg each. 4 separators: 1 kg each. 12 valves: 5 kg each. Heaters: 5 kg. LN2 (actuator) 2 bottles: 10 kg each. ~100 4 pumps (167): 9 kg each. 8 valves: 5 kg each.

156

Table 6-9: Mass comparison of the separator concepts, with noncondensible and nitrogen pressurant

The liquid nitrogen actuator concept may be the most reliable because hydraulic systems are known to move heavy loads on Earth. With the available submarine waste heat of 3800 W, the heated vapor system can reach 170 meters depth in any composition, a gain of only 50 meters over the unheated version. Due to lower storage pressure, the masses of any of the nitrogen concepts are likely to be less than the noncondensible concept. In summary, the LN2 actuator concept appears to be the best of the nitrogen concepts, and is used in the concept comparison of Section 6.10.

6.9 Nitrogen Pump System

If the fluid occupying the ballast tank is denser than the Titan Sea, then the tank holding it can be made smaller. The most obvious alternative to the pump concept with sea liquid is to use liquid nitrogen, which can be taken from the atmosphere and has higher liquid density than ethane. A ballast system using it would have to attach a gas purification unit to the tank inlet in place of a valve to the sea (Figure 6-29).

Figure 6-29: Conceptual layout of a liquid nitrogen pump system.

157

The ballast tank must be large enough, as in the pump concept, that it covers the ~450 kg of buoyancy difference between a methane sea and an ethane sea (at 2.77 m3 submarine volume and 500-650 kg/m3 density range). Since the saturated liquid density of nitrogen at 93 K is 730 kg/m3, the maximum required ballast liquid volume is 0.65 m3.

With some margin of error, the tank volume can be taken as 0.85 m3. The radius of such a tank, constructed as a cylinder 5 meters long with volume 0.425 m3 (there are 2

tanks, one on each side of the vehicle), is R 16.4 cm and the surface area is Atank  5.34 m2.

The operating concept is to flood the tank through the gas purification unit (with the option of venting though the valve). To return to the surface, the pump removes liquid nitrogen to the Titan Sea.

6.9.1 Assessment

Since nitrogen is a gas at Titan surface conditions, storing it as a liquid implies

significant heat leak. The saturation temperature of nitrogen at 0.15 MPa is Tsat  81 K, and the thermal conductivity of a titanium tank is k  6.7 W/m-K. The heat leak by conduction, from the Titan Sea to the tank interior, based on thickness, surface area and a

sea temperature Tsea  93 K, is

kA T T  Q tank sea sat 430 kW (Equation 6-45) w

Pressurizing the nitrogen during liquefaction raises the saturation temperature and thereby reduces heat transfer. A fixed pressure of 0.462 MPa is required to raise the saturation temperature to 93 K, to eliminate heat leak. Even a small deviation from this

158 standard will accumulate over an 8-hour dive period, which risks boiling the liquid ballast and bursting the tank.

Another disadvantage of this concept is the very high gas liquefaction rates required to fill the ballast tank. Let the 450 kg of nitrogen be liquefied in 30 minutes, to take occupy a small portion of the dive time. Compared to 25 kg in 17 hours for the initial charging of the pressurant gas system, the required flow rate is over 600 times higher. The hardware to deliver such a flow rate at the desired purity of nitrogen would be far too bulky to reasonably fit on the submarine. Therefore, this concept is not included in the comparison.

6.10 Sea Boiling System

The removal of liquid from the ballast tank does not necessarily require a pump or a supply of pressurant gas. One alternative, after the tanks are flooded to sink, is to apply heat and boil the ballast liquid. The pressure of the newly created vapor can be made adequate to expel the liquid and create buoyancy. A conceptual layout of such a scheme is shown in Figure 6-30.

Figure 6-30: Conceptual layout of methane/ethane gas system

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6.10.1 Assessment

The initial state considered in the enthalpy change, for both composition extremes, is 93 K and 0.15 MPa. The final state is saturated vapor at 1 MPa, which is needed to expel ballast liquid under any design conditions.

For a 5% methane sea, the mass is based on the volume taken in (0.72 m3) and the sea density of 650 kg/m3, for a mass of 450 kg. The value of enthalpy rises from -136 kJ/kg at the initial state to 542 kJ/kg at the final state. Thus the total energy requirement for boiling the ballast tank is 305 MJ.

For an 85% methane sea (with ~14% dissolved nitrogen), the mass is based on the volume taken in (0.15 m3) and the sea density of 520 kg/m3, for a mass of 78 kg. The value of enthalpy rises from -89 kJ/kg at the initial state to 529 kJ/kg at the final state.

Thus the total energy requirement for boiling the ballast tank is 48 MJ.

At the submarine waste heat of 3800 W, it would take 22 hours to boil off the liquid ballast in the worst case, 5% methane. Compared to a total dive time of 8 hours, this is unacceptably long. Therefore, the liquid boiling concept is not used for the trade study.

6.11 Comparison

The total mass and power consumption of each concept is shown in Table 6-10.

The power cited is the maximum total power during any stage of operation, out of an estimated available power of 860 W (5).

Concept Total Mass (kg) Maximum Total Power (W) Complexity* Pump 180 320 1 Bladder only 210 800 2 Noncondensible, no 664 (neon) 740 2 separator 362 (helium) Noncondensible with 560 (neon) 300 3 separator 600 (helium)

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Liquid Nitrogen 180 500 3 Actuator Table 6-10: Mass and power estimates of the options in this trade study.

*1 is simplest and 3 is most complex.

Table 6-11 summarizes the overall pros and cons of the concepts described in this paper. The bladder system can bring the vehicle to the surface quickly, but is slow to sink and cryogenic bladder materials have no heritage. The pump system would only need to expand the bladder once, but limits the allowable composition change in any single dive.

The pressurant gas systems are easily the heaviest, because the gas bottles need to be built for 30 MPa but can only use less than 10 MPa on Titan; gas blowdowns, however, are well-understood in space vehicles. The nitrogen actuator concept can harvest more of the atmosphere to compensate for leakage, but that additional hardware requirement adds complexity.

Concept Pros Cons Pump Very simple. Limited composition range per dive. Pumps are familiar. Bladder only No sea interaction. Bladder materials may not be reliable (175). Can cover any density range. Gas may leak in transit or operation. Expanded vehicle has to be packed for transit. Must manage compressor losses. Noncondensible, Can cover any density range. Gas may leak in transit. no separator Gas is lost during operation. Weight penalty: tank as pressure vessel and high-pressure pump. Vulnerable to pitching. Noncondensible Can cover any density range. Gas may leak in transit. with separator Separator failure could contaminate gas bottle. Many parts. Must manage compressor losses. Liquid Nitrogen Can cover any density range. Requires nitrogen harvesting. Actuator Leak-tolerant. Pumps are familiar. Table 6-11: Overall pros and cons of the options in this trade study

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7 Phase II Submarine

The Phase II vehicle design is shown in Figure 7-1. Examination of Table 1-1 shows that the design requirements on this orbiter-supported concept are significantly less stringent relative to the stand-alone version. There are significantly fewer dives as well. The sinking procedure is the same as in Phase I, as valves are opened to flood the tank on the surface. There are two major differences between the concepts. First, GHe is treated as a consumable here; GHe lines (not shown) connect to the liquid ballast tanks on the top of the submarine and used to push out liquid to rise to the surface. The gas used for this is lost in each dive. Second, the gas bottles are kept warm internally by a radioisotope power system. Thermodynamic analysis of the warm gas blowdown operation is somewhat trivial, because such systems have flight heritage in liquid–fuelled rocket engines using hydrogen and oxygen (185). Such analysis is still used to size the ballast system. The volume of the ballast tanks is 0.19 m3, and the GHe bottle pressure is

2.4 MPa.

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Figure 7-1: Phase II submarine design, with the ballast tanks identified. The

GHe bottles are warm internal components.

There are 2 GHe tanks each of mass 3.5 kg, and the GHe plumbing (valves, etc) has a mass of 20 kg, for a total of 27 kg. This is much less than the mass of any Phase I concept. The difference is largely driven by the reduced vehicle size with an orbiter: (1) the ballast tank is smaller and the gas bottle pressure is lower, reducing the bottle mass requirement; and (2) the bottle construction is known and need not be cryogenic. There is also no need for compressors since the gas is consumable.

8 Conclusion

This thesis contributes to the development of the Titan submarine in 3 distinct ways.

In addition, it briefly describes the ballast system of the Phase II submarine concept, which is simpler, better understood and would be used for fewer dives.

First, the primary sea components (methane, ethane and nitrogen) are assessed as a ternary system. A survey and review of VLE data for this set allows the fitting of a correlation equation, with mean absolute error of 20% or less for all conditions (nitrogen as the solvent in methane, ethane or a mixture of the two). An additional liquid phase, with much higher concentrations of nitrogen, is noted as physically valid. This solubility model is then applied to the estimation of Titan sea properties using the assumption of equality of chemical potential. Several versions of a sea model are considered, but in general, properties are affected much more strongly by composition than by depth.

Second, the solubility model is used to predict the effervescence of nitrogen gas bubbles around the submarine, which may interfere with science instruments or propellers. The nucleation model is borrowed from boiling studies, and the geometry of

163 the submarine is considered in a simplified one-dimensional view. Results are given for two operating modes: quiescent, where bubbles rise up the submarine side by their own buoyancy and natural convection; and moving, where the general motion of the sea around the submarine dominates. As long as the heat flux on the submarine skin is kept relatively low (and so the temperature difference between skin and sea is a few Kelvin or less), it appears that the amount of effervescent bubbles will be manageable.

Third, the upper and lower density bounds contribute to a trade study on several reusable ballast concepts. These are compared based on estimated mass (referenced against commercially available hardware), power requirement and complexity of design.

The use of pressurant gas to expel ballast liquid is known to be reliable, but the bottle to hold such gas adds a lot of mass. Ballast systems based on a liquid pump are several times lighter, but some variations are sensitive to changes in sea composition during a dive. Nitrogen gas can be harvested from the Titan atmosphere (by removing the small methane component), but that requires installing another subsystem on the submarine.

Which concept is preferred depends on the science emphasis and future detail design.

9 Future Work

A next step in the development of the submarine is to make system-level designs in greater detail. This work informs two areas of such development.

First, Section 5 provided an estimate of the volume and area fractions occupied by bubbles. If further inquiries demonstrate that this amount is likely to impair the operation of sensors or propellers, then the thermal management system may have to be rearranged.

Waste heat can be concentrated, for example, around the top surface of the submarine. It

164 might also help to have a measurement of the contact angle at Titan sea conditions, to verify that it is small and to determine just how small.

Second, the preferred ballast concept must be developed for total vehicle integration.

This begins with a detailed adaptation of existing LNG hardware for the Titan seas. Then the requisite hardware needs to be built into the structure of both the submarine and the ballast tank for minimum drag and weight. One alteration to the separator concept that such work might consider is the removing the gas bottle from the separator concept; instead, sea liquid would be pumped in against the pressure of a GHe/GNe cylinder of variable length.

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