# Modelling of Boil-Off and Sloshing Relevant to Future Liquid Hydrogen Carriers

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}) equivalent emission due to propulsion than conventional oil-powered ships [9]. These ships cannot be carbon-neutral unless the CO

_{2}is captured during the voyage [10]. To make matters worse, the methane contained in the BOG can have 25 times the warming effect of CO

_{2}if released directly into the atmosphere [11]. On the other hand, an LH2 ship does not contribute any significant greenhouse gas emissions during its voyage. LH2 ships can also result in lower overall CO

_{2}equivalent emissions than LNG ships (depending on the fuel source), showing excellent carbon neutrality potential [7]. Given the compelling nature of hydrogen, the construction of LH2 carriers is of vital importance for future decarbonization strategies, and their design can benefit from existing knowledge on LNG ships. However, the boiling point of LH2 is around 90 °C less than that of LNG, raising several technical challenges since BOR effects on LH2 can be more pronounced and more insulation and reliquefaction infrastructure is required to contain the fuel [12,13,14].

^{3}of fuel [15], the only existing LH2 carrier ship (the Suiso Frontier) transports 1250 m

^{3}at a time. This ship employs one horizontal, cylindrical, vacuum-insulated tank and utilises a hybrid natural gas and hydrogen turbine for propulsion [16]. A working LH2 carrier has also recently been commissioned with a volumetric capacity of the order of an LNG ship: it will transport 160,000 m

^{3}of LH2 in 4 spherical vacuum-insulated tanks [17,18]. Other variants of conceptual LH2 ships that employ several forms of propulsion are still under investigation. For example, future LH2 ships may be propelled by hydrogen combustion using reciprocating engines [19] or combined-cycle gas turbines [19,20]. Alternatively, ships may employ electric propulsion with fuel cells using proton exchange membrane (PEM) [21,22,23,24], solid oxide [22], or molten carbide [19] technology. The possibility of including a reliquefaction unit on these ships also exists, as discussed by Ahn et al. [19].

^{3}of LH2 in a test ship subjected to manoeuvre-induced sloshing in several planes of motion. Wei et al. [32] further investigated sloshing-induced pressure changes in the specific case of a tank filled with LH2. Other cryogenic fluids have been studied more closely. For example, Behruzi et al. [33] and Grotle and Aesoy [34] modelled pressure and temperature changes in sealed LNG tanks that result from sloshing at various excitation frequencies and amplitudes. In some cases, thermodynamic changes can also be related to BOG generation during a ship voyage, as in Wu and Ju [35]. In other cases, the BOR in unsealed tanks can be directly investigated at various slosh excitations as in Yu et al. [36] and Mir et al. [37]. In addition, Ludwig et al. [38] derived a relationship between sloshing frequency, sloshing amplitude, and heat transfer rate across the liquid-vapour interface, general enough to apply to any cryogenic fuel. For modelling purposes, the sloshing-induced heat transfer is of special interest; hence it must be considered in any thermodynamic modelling approach focusing on the BOR.

^{3}capacity, for which data are available. The approach is then employed to explore the effects of (i) weather conditions, (ii) operating velocity, (iii) sloshing, and (iv) ship design on the BOR properties of a conceptual LH2 ship.

## 2. Thermodynamic Model

#### 2.1. Generalised Thermodynamic Model

#### 2.2. Thermodynamic Model for an Unsealed, Laden Cryogenic Tank

^{2}) is the surface area of heat transfer across the tank walls to the vapour.

^{3}capacity LNG carrier. ${\mathrm{U}}_{\mathrm{if}}$ is, by definition, independent of any changes caused by sloshing. Thus, for any SSF value around the design point, a corresponding temperature of vapour, hence BOR, can again be determined using Equations (8) and (9). The dependency of $\mathrm{SSF}$ on external weather conditions and ship velocity may also be defined using existing data and relationships from the literature, as will be shown later.

#### 2.3. Addition of a Reliquefaction Unit

^{−1}) represents the rate of fuel reliquefaction. Likewise, Equation (3) for the liquid is replaced with:

## 3. Ship Model

#### 3.1. Ship Sizing

#### 3.2. Power Consumption and Fuel Utilisation

^{−1}) is the lower calorific value of the fuel. The unknowns in Equations (14)–(17) are the power consumption calibration factor (${\mathrm{c}}_{\mathrm{use}}$, dimensionless), the efficiency of fuel cells (${\mathsf{\eta}}_{\mathrm{FC}}$) if present (if not, ${\mathsf{\eta}}_{\mathrm{FC}}=1$), and the efficiency of the ship IC engines or electric motors (${\mathsf{\eta}}_{\mathrm{eng}}$). For combustion ships, the unknown values can be determined by scaling them to match empirical data on power, fuel utilisation rate, and velocity. This is further discussed using data for an existing LNG carrier. For electric ships, ${\mathsf{\eta}}_{\mathrm{eng}}$ and ${\mathsf{\eta}}_{\mathrm{FC}}$ may be found from the literature (e.g., see References [52,53,54,55]).

#### 3.3. Sloshing

## 4. Investigated Fuel Carriers and Design Alterations

#### 4.1. Conventional Liquefied Natural Gas Carrier

^{2}. A value of ${\mathrm{c}}_{\mathrm{use}}=0.07121$ thus appeared to fit the data best. Equation (23) relates the resultant velocity, expressed in knots, to the efficiency of the ship engines:

**Table 3.**Fuel consumption data for the conventional LNG ship at design point [28].

Input | Unit | Quantity |
---|---|---|

Ship Velocity | kn | 16.7 |

Beaufort Number | - | 2 |

Fuel Utilisation Rate for Propulsion (Best Fit) | kg s^{−1}, %/day | 0.757, 0.0905 |

Power Consumption (Best Fit) | kW | 14,600 |

^{−1}K

^{−1}, which is within the range of values tabled by Ref. [64]. At an insulation thickness of 0.53 m, ${\mathrm{U}}_{\mathrm{ins}}$ is thus 0.092 W m

^{−2}K

^{−1}.

^{−2}K

^{−1}.

Input | Unit | Quantity | Source |
---|---|---|---|

Internal Pressure of Tank | bar | 1.01325 | [28] |

Average External Pressure | bar | 1.01325 | [67] |

Average External Sea Temperature | K | 288 | [66] |

Heat Transfer Rate to Liquid | kW | 386 | [28] |

Temperature of Liquid | K | 110 | [28] |

Specific Enthalpy of Vapour Relative to Specific Internal Energy of Liquid * | kJ kg^{−1} | 685.8 | [44] |

Total Boil-Off Rate | kg s^{−1}%/day | 0.757 0.0905 | [28] |

Surface Area of Tank in Contact with Vapour (Calculated from The Tank Dimensions) | m^{2} | 8610 | [63] |

Surface Area of Tank in Contact with Liquid (Calculated from The Tank Dimensions) | m^{2} | 23,660 | [63] |

Surface Area of Liquid-Vapour Interface (Calculated from The Tank Dimensions) | m^{2} | 8296 | [63] |

#### 4.2. Conceptual Liquid Hydrogen Carrier

^{−1}is 41.7% of that of the LNG ship at any velocity or BN (see Equation (17)).

^{−1}K

^{−1}.

#### 4.3. Addition of a Reliquefaction Unit

#### 4.4. Use of Electric Propulsion

## 5. Model Application

#### 5.1. Boil-Off Properties of Combustion Ship Models

^{2}, so its power consumption is also lower (see Equation (14)). The latter also means that higher velocities can be reached by the LH2 ship before the rated power is exceeded.

#### 5.2. Effect of Insulation Thickness

#### 5.3. Effect of Electrification

#### 5.4. Additional Fuel Tank Design Variations

#### 5.5. Sizing Considerations for Targeting a Specific Delivered Energy

^{3}and 18,600 m

^{3}of tank structure, respectively. The “Other Volumes” (Table 11) onboard the ship are assumed constant so that the resulting LH2 ships designed for 5000 km and 15,000 km voyages would be approximately 1.73 and 1.77 times the size of the LNG ship (based on internal volume). Such a ship would require major alterations to re-optimise design features such as its streamlining and stability, so its sizing and power consumption would differ significantly from those presented in this study.

## 6. Conclusions and Recommendations for Future Work

- (1)
- An LH2 carrier with the same fuel tank volume and insulation thickness as an LNG carrier can contain 16.8% of the fuel mass and 40.2% of the fuel energy. The unforced BOR of the LH2 carrier is 8.94 times higher than that of an LNG ship.
- (2)
- The heat transfer and boil-off effects of sloshing on an LH2 carrier are more significant than those on an LNG carrier. In particular, the rate of BOR increase with BN on board the LH2 ship is twice as large relative to an LNG carrier.
- (3)
- Adding a reliquefaction unit to the vessel reduces the fuel depletion rate by at least 38.7%. However, this reduction is highly dependent on the weather and ship velocity, so reliquefaction introduces a significantly higher sensitivity of the fuel depletion rate and delivered fuel to the operating conditions.
- (4)
- A parametric analysis illustrated that 1.04 to 6.62 times the insulation thickness of glass wool is required to allow the LH2 carrier to have BOR properties equivalent to the LNG ship, primarily due to the lower LH2 temperature.
- (5)
- An LH2 carrier powered by fuel cells and electric motors delivers at least 1.1% more cargo fuel than one with internal combustion engines due to the lower volume of the electric propulsion system and the higher efficiency of fuel cell and electric motor propulsion.
- (6)
- An LH2 carrier operating with fuel cells and reliquefaction must be at least 1.73 times larger by volume than the LNG carrier to deliver the same energy.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Additional Data Tables

**Table A1.**Properties of sea water [67].

Value Name | Unit | Quantity |
---|---|---|

Density of Sea Water | kg m^{−3} | 1026 |

Dynamic Viscosity | Pa-s | 0.00117 |

**Table A2.**Power and fuel utilisation rates of the conventional LNG ship at velocities of 10 to 19.5 kn [28].

Velocity (kn) | Propulsive Power (kW) | Fuel Utilisation Rate (ton/day) |
---|---|---|

10 | 3442 | 29 |

11 | 4537 | 34 |

12 | 5837 | 40 |

13 | 7334 | 47 |

14 | 9033 | 55 |

15 | 10,936 | 63 |

16 | 13,037 | 73 |

17 | 15,340 | 83 |

18 | 17,878 | 95 |

19 | 20,746 | 108 |

19.5 | 22,361 | 115 |

Input | Unit | Quantity | Source |
---|---|---|---|

Volumetric Power Density of Heat Exchangers in Reliquefaction Unit * | W m^{−3} | 75,700 | [75] |

Estimated Percentage Volume of Aluminium in Reliquefaction Unit Heat Exchangers * | % | 4 | [75] |

Density of Aluminium | kg m^{−3} | 2700 | [76] |

Gravimetric Power Density | W kg^{−1} | 1020 | [77] |

Volumetric Power Density | MW m^{−3} | 1.74 | [77] |

## Appendix B. Variation of Boil-Off with Fuel Tank Number and Shape

#### Appendix B.1. Effect of Fuel Tank Number with Cuboidal Tanks

**Figure A1.**Variation of total boil-off rate with Beaufort number (BN) and number of fuel tanks ($n$) at a ship velocity (${\mathrm{v}}_{\mathrm{ship}}$) of 10 kn (

**a**) An LNG ship; (

**b**) An LH2 ship with the same insulation thickness as the LNG carrier, without reliquefaction.

#### Appendix B.2. Conversion to Spherical Tanks

^{2}, the total volume of tank material is 13,200 m

^{3}and the ${\mathrm{U}}_{\mathrm{ins}}$ value is 0.023 W m

^{−2}K

^{−1}(based on the thermal conductivity of glass wool estimated in Section 4.1).

## Appendix C. Heat Transfer Properties of Vacuum Insulated Tanks

^{−2}K

^{−1}when considering the emissivity of aluminium (see Table A4) and a vacuum gap thickness of 0.53 m for a cuboidal tank containing hydrogen. For a spherical tank, the value is 0.083 W m

^{−2}K

^{−1}.

Input | Unit | Quantity | Source |
---|---|---|---|

Failure Stress of Aluminium | MPa | 438 | [81] |

Acceptable Ratio of Maximum Tensile Stress to Failure Stress (mid-range value) | - | 1.7 | [79] |

Emissivity of Aluminium | - | 0.1 | [67] |

Input | Unit | Tank Shape | Insulation Type | Quantity | Source |
---|---|---|---|---|---|

Heat Transfer Coefficient, ${\mathrm{U}}_{\mathrm{ins}}$ | W m^{−2} K^{−1} | Cuboid | Glass Wool | 0.092 | Section 4.1 |

Cuboid | Vacuum Casing | 0.080 | This Section | ||

Sphere | Glass Wool | 0.023 | Appendix B.2 | ||

Sphere | Vacuum Casing | 0.083 | This Section | ||

Surface Area for Heat Ingress Across Tank Wall, ${\mathrm{A}}_{\mathrm{in}}$ | m^{2} | Cuboid | Any | 32,200 | Section 4.1 |

Sphere | Any | 24,300 | This Section | ||

Area Dependent Heat Transfer Coefficient ${\mathrm{U}}_{\mathrm{ins}}\xb7{\mathrm{A}}_{\mathrm{in}}$ | W K^{−1} | Cuboid | Glass Wool | 2960 | Above Rows |

Vacuum Casing | 2590 | Above Rows | |||

Sphere | Glass Wool | 570 | Above Rows | ||

Vacuum Casing | 2010 | Above Rows | |||

Volume of Tank Material | m^{3} | Cuboid | Any | 17,600 | Table 1 |

Sphere | Any | 13,200 | Appendix B.2 |

**Figure A2.**Variation of area-dependent heat transfer coefficient (${\mathrm{U}}_{\mathrm{ins}}\xb7{\mathrm{A}}_{\mathrm{in}}$) and volume with insulation thickness (${\mathrm{t}}_{\mathrm{ins}}$) for four spherical and cuboidal tanks insulated by glass wool and vacuum casing onboard an LH2 ship. Values are displayed for 0.01 to 5 m.

## Appendix D. Comparison of Additional Designs

Ship Variable | Identifier Type | Categories | Identifier | Demonstration |
---|---|---|---|---|

Fuel and Propulsion Type | Colour | Methane Internal Combustion Engine | Pink | |

Hydrogen Internal Combustion Engine | Green | |||

Hydrogen Electric | Light Blue | |||

Tank Shape | Icon Shape | Cuboid | Rectangle | |

Sphere | Circle | |||

Insulation Type | Outline Type | Glass Wool | Single Line | |

Vacuum Insulation | Double Line | |||

Presence of Reliquefaction | Presence of Reliquefaction Symbol | Reliquefaction Present | Symbol Present | |

Reliquefaction Absent | Symbol Absent |

(a) | |||||
---|---|---|---|---|---|

Fuel and Propulsion Type | Ship Type | Thickness of Insulation (m) | Initial Mass of Fuel (10^{3} ton) | Mass of Tank (10^{3} ton) | Other Masses (10^{3} ton) |

Methane Internal Combustion Engine | 0.53 | 72.29 | 0.85 | 23.62 | |

0.01 | 71.67 | 1.46 | 23.62 | ||

0.07 | 73.05 | 0.09 | 23.62 | ||

0.01 | 72.60 | 0.53 | 23.62 | ||

0.31 | 72.64 | 0.49 | 23.62 | ||

0.01 | 71.67 | 1.46 | 23.62 | ||

0.07 | 73.05 | 0.09 | 23.62 | ||

0.01 | 72.60 | 0.53 | 23.62 | ||

Hydrogen Internal Combustion Engine | 0.72 | 11.65 | 1.14 | 23.87 | |

0.01 | 13.24 | 1.60 | 23.87 | ||

0.26 | 12.85 | 0.31 | 23.87 | ||

0.01 | 13.27 | 0.53 | 23.87 | ||

0.38 | 12.42 | 0.61 | 23.87 | ||

0.01 | 13.24 | 1.60 | 23.87 | ||

0.17 | 13.00 | 0.21 | 23.87 | ||

0.01 | 13.27 | 0.53 | 23.87 | ||

Hydrogen Fuel Cell | 0.71 | 11.78 | 1.13 | 23.18 | |

0.01 | 13.35 | 1.62 | 23.18 | ||

0.26 | 12.96 | 0.32 | 23.18 | ||

0.01 | 13.39 | 0.54 | 23.18 | ||

0.32 | 12.67 | 0.52 | 23.18 | ||

0.01 | 13.35 | 1.62 | 23.18 | ||

0.15 | 13.15 | 0.18 | 23.18 | ||

0.01 | 13.39 | 0.54 | 23.18 | ||

(b) | |||||

Fuel andPropulsion Type | Ship Type | Volume of Tank(10^{3} m^{3}) | OtherVolumes(10^{3} m^{3}) | FuelDepletion Rate(%/day) | Revenue Loss Rate(10^{3} $/day) |

Methane Internal Combustion Engine | 17.62 | 171.97 | 0.0905 | 21.6 | |

0.84 | 190.24 | 0.1043 | 24.7 | ||

1.78 | 171.97 | 0.0895 | 21.6 | ||

0.44 | 171.97 | 0.0901 | 21.6 | ||

10.20 | 178.54 | 0.1048 | 25.1 | ||

0.84 | 190.24 | 0.0944 | 22.3 | ||

1.78 | 171.97 | 0.0895 | 21.6 | ||

0.44 | 171.97 | 0.0901 | 21.6 | ||

Hydrogen Internal Combustion Engine | 23.79 | 171.97 | 0.6224 | 57.7 | |

0.90 | 171.97 | 0.5945 | 62.6 | ||

6.55 | 171.97 | 0.1681 | 17.2 | ||

0.44 | 171.97 | 0.2686 | 28.4 | ||

12.70 | 171.97 | 0.3816 | 37.7 | ||

0.90 | 171.97 | 0.2463 | 25.9 | ||

4.30 | 171.97 | 0.1702 | 17.6 | ||

0.44 | 171.97 | 0.1758 | 18.6 | ||

Hydrogen Fuel Cell | 23.57 | 170.31 | 0.6258 | 58.6 | |

0.91 | 170.31 | 0.5907 | 62.8 | ||

6.59 | 170.31 | 0.1676 | 17.3 | ||

0.44 | 170.31 | 0.2678 | 28.5 | ||

10.76 | 170.31 | 0.3061 | 30.9 | ||

0.91 | 170.31 | 0.1800 | 19.1 | ||

3.82 | 170.31 | 0.1440 | 15.1 | ||

0.44 | 170.31 | 0.1377 | 14.7 |

**Figure A3.**Variation of remaining fuel mass and energy with voyage duration at BN = 2 and a resultant velocity of 16.7 knots. All investigated LH2 fuel cell ships are shown at the insulation thickness that maximises the fuel delivered after a voyage of 20 days.

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**Figure 1.**Control volume for liquid and vapour phases of cryogenic fuel in an unsealed container and control surface for the liquid-vapour interface. The cryogenic tank is assumed to operate without reliquefaction. Thermodynamic properties are shown, including the temperature of liquid (${\mathrm{T}}_{\mathrm{L}}$), vapour (${\mathrm{T}}_{\mathrm{V}}$), and external environment (${\mathrm{T}}_{\mathrm{out}}$); the mass of liquid (${\mathrm{M}}_{\mathrm{L}}$) and vapour (${\mathrm{M}}_{\mathrm{V}}$); the heat transfer rate into the liquid (${\dot{\mathrm{Q}}}_{\mathrm{L}}$), into the vapour (${\dot{\mathrm{Q}}}_{\mathrm{V}}$) and across the liquid-vapour interface (${\dot{\mathrm{Q}}}_{\mathrm{if}}$); the rate of change of the internal energy of the liquid (${\dot{\mathrm{U}}}_{\mathrm{L}}$) and vapour (${\dot{\mathrm{U}}}_{\mathrm{V}}$); the specific enthalpy flow out of the liquid (${\mathrm{h}}_{\mathrm{L}}$) and through the vapour (${\mathrm{h}}_{\mathrm{V}}$); the mass flow rate out of the liquid (${\dot{\mathrm{m}}}_{\mathrm{L}}$) and vapour (${\dot{\mathrm{m}}}_{\mathrm{V}}$) the kinetic energy transfer rate due to sloshing (${\dot{\mathrm{W}}}_{\mathrm{slosh}}$).

**Figure 2.**Control volume for liquid and vapour phases of cryogenic fuel in an unsealed container, and control surface for the liquid-vapour interface. The cryogenic tank is assumed to operate with reliquefaction. Thermodynamic properties are shown, which include: the temperature of liquid (${\mathrm{T}}_{\mathrm{L}}$), vapour (${\mathrm{T}}_{\mathrm{V}}$) and external environment (${\mathrm{T}}_{\mathrm{out}}$); the mass of liquid (${\mathrm{M}}_{\mathrm{L}}$) and vapour (${\mathrm{M}}_{\mathrm{V}}$); the heat transfer rate into the liquid (${\dot{\mathrm{Q}}}_{\mathrm{L}}$), into the vapour (${\dot{\mathrm{Q}}}_{\mathrm{V}}$) and across the liquid-vapour interface (${\dot{\mathrm{Q}}}_{\mathrm{if}}$); the rate of change of the internal energy of the liquid (${\dot{\mathrm{U}}}_{\mathrm{L}}$) and vapour (${\dot{\mathrm{U}}}_{\mathrm{V}}$); the specific enthalpy flow out of the liquid (${\mathrm{h}}_{\mathrm{L}}$) and through the vapour (${\mathrm{h}}_{\mathrm{V}}$); the mass flow rate out of the liquid (${\dot{\mathrm{m}}}_{\mathrm{L}}$) and vapour (${\dot{\mathrm{m}}}_{\mathrm{V}}$); the rate of fuel reliquefaction (${\dot{\mathrm{m}}}_{\mathrm{reliq}}$); the kinetic energy transfer rate due to sloshing (${\dot{\mathrm{W}}}_{\mathrm{slosh}}$); and enthalpy extraction rate during the reliquefaction process (${\dot{\mathrm{W}}}_{\mathrm{reliq}}$).

**Figure 4.**Sea and fuel properties involved in the calculation of heat transfer enhancement due to sloshing as a function of Beaufort number (BN). The trends shown are: (

**a**) Velocity of the wind (${\mathrm{v}}_{\mathrm{wind}}$), sea (${\mathrm{v}}_{\mathrm{sea}}$), and liquefied natural gas ship at rated power (${\mathrm{v}}_{\mathrm{ship}}$); (

**b**) Frequency of sea (${\mathsf{\omega}}_{\mathrm{sea}}$) and liquefied natural gas waves at rated power (${\mathsf{\omega}}_{\mathrm{L}}$); (

**c**) Amplitude of sea (${\mathrm{D}}_{\mathrm{sea}}$) and liquefied natural gas waves at rated power (${\mathrm{D}}_{\mathrm{L}}$); (

**d**) Sloshing scaling factor (SSF) of liquefied natural gas at a ship velocity of 0 kn and at rated power.

**Figure 5.**Variation of key mass flow rates with Beaufort number (BN) and ship velocity (${\mathrm{v}}_{\mathrm{ship}}$). The trends shown are: (

**a**) fuel utilisation rate for propulsion (${\dot{\mathrm{m}}}_{\mathrm{use}}$) and BOR (${\dot{\mathrm{m}}}_{\mathrm{loss}}{+\dot{\mathrm{m}}}_{\mathrm{use}}$) for an existing LNG ship model; (

**b**) fuel utilisation rate for propulsion (${\dot{\mathrm{m}}}_{\mathrm{use}}$) and BOR (${\dot{\mathrm{m}}}_{\mathrm{loss}}{+\dot{\mathrm{m}}}_{\mathrm{use}}$) for an LH2 combustion ship with the same tank insulation thickness as the LNG ship, without reliquefaction; (

**c**) fuel utilisation rates for propulsion (${\dot{\mathrm{m}}}_{\mathrm{use}}$) and reliquefaction (${\dot{\mathrm{m}}}_{\mathrm{r}\text{-}\mathrm{use}}$) for an LH2 combustion ship with the same tank insulation thickness as the LNG ship, with reliquefaction.

**Figure 6.**Variation of boil-off (${\dot{\mathrm{m}}}_{\mathrm{use}}{+\dot{\mathrm{m}}}_{\mathrm{loss}}$) and fuel utilisation rate (${\dot{\mathrm{m}}}_{\mathrm{use}}$) with insulation thickness, at Beaufort numbers (BN) of 0 to 12 and ship velocities (${\mathrm{v}}_{\mathrm{ship}}$) of 16.9, 16.9, 16.7, 16.4, 15.2, 11.8, 7.06, 3.50, 1.63, 0.779, 0.390, 0.205 and 0.113 knots, respectively. The ship models are: (

**a**) an LNG ship; and (

**b**) an LH2 combustion ship without reliquefaction.

**Figure 7.**Variation of remaining fuel mass and energy with voyage duration at BN = 2 and a ship velocity of 16.7 knots. All investigated ships are shown at the insulation thickness that maximises the fuel delivered after a voyage of 20 days. The trends shown are (

**a**) the remaining mass onboard all LNG ships; (

**b**) the remaining mass onboard all LH2 ships; (

**c**) the remaining energy onboard all investigated ships.

Input | Unit | Quantity | Source |
---|---|---|---|

Number of Tanks | - | 4 | [28] |

Deadweight | ton | 95,190 | [63] |

Density of Glass Wool | kg m^{−3} | 48 | [64] |

Fraction of Fuel Vapour | % | 2 | [28] |

Fuel Capacity | m^{3} | 173,600 | [28] |

Gross Tonnage | ton | 113,000 | [63] |

Rated Power | MW | 23.4 | [28] |

Ship Beam | m | 46 | [63] |

Ship Length | m | 295 | [63] |

Thickness of Glass Wool | m | 0.53 | [28] |

Reliquefaction Unit Present | True/False | False | [28] |

**Table 2.**MAN B&W 6G70ME-C engine data [65].

Input | Unit | Quantity |
---|---|---|

Rated Power | MW | 21.84 |

Mass | ton | 665 |

Estimated Volume | m^{3} | 1568 |

Input | Unit | Quantity | |||
---|---|---|---|---|---|

Natural Gas | Source | Hydrogen | Source | ||

Temperature of Liquid | K | 110 | [28] | 20.15 | [14] |

Density of Liquid | kg m^{−3} | 425 | [28] | 70.95 | [44] |

Temperature of Vapour (Evaluated) | K | 119.5 | - | 24.3 | - |

Density of Vapour | kg m^{−3} | 1.684 | [44] | 1.071 | [44] |

Lower Calorific Heating Value | MJ kg^{−1} | 50.01 | [67] | 120 | [68] |

Specific Enthalpy of Vapour Relative to Specific Internal Energy of Liquid * | kJ kg^{−1} | 685.8 | [44] | 698.1 | [44] |

**Table 6.**LNG and LH2 data on BOR and operating temperature [14].

Input | Unit | Quantity |
---|---|---|

Boil-Off Rate of Liquefied Natural Gas Ship | %/day | 0.1204 |

Boil-Off Rate of Liquid Hydrogen Ship | %/day | 1.063 |

Temperature of Liquid Natural Gas | K | 111 |

External Temperature | K | 298 |

Input | Unit | Quantity | Source |
---|---|---|---|

Electricity Requirement Per Unit of Reliquefied Natural Gas Mass Flow Rate, ${\mathrm{e}}_{\mathrm{reliq}}$ (Approximately Mid-Range) | kWh/kg | 1.25 | [45,46,47,48] |

Electricity Requirement Per Unit of Reliquefied Hydrogen Mass Flow Rate, ${\mathrm{e}}_{\mathrm{reliq}}$ | kWh/kg | 3.30 | [57] |

Specific Enthalpy of Vapour Relative to Liquid for Natural Gas * | kJ kg^{−1} | 533.1 | [44] |

Specific Enthalpy of Vapour Relative to Liquid for Hydrogen * | kJ kg^{−1} | 494.2 | [44] |

Efficiency of Generator (Independent of Engine, Approximately Mid-Range) | % | 92.5 | [56] |

Input | Unit | Quantity | Source |
---|---|---|---|

Overall Electric Engine Efficiency (Mid-Range, Conservative Estimate to Upper Bound) | % | 92.5, 90 to 95 | [52,53] |

Gravimetric Power Density of Electric Engine | W kg^{−1} | 5200 | [69] |

Density of Iron | kg m^{−3} | 7880 | [69] |

Proton Exchange Membrane Fuel Cell Efficiency (Mid-Range, Conservative Estimate to Upper Bound) | % | 57, 54 to 60 | [54,55] |

Gravimetric Power Density of Proton Exchange Membrane Fuel Cell | W kg^{−1} | 1980 | [70] |

Volumetric Power Density of Proton Exchange Membrane Fuel Cell | W m^{−3} | 3,120,000 | [70] |

**Table 9.**Component mass and volume estimates for the conventional liquefied natural gas ship and a liquid hydrogen ship with the same tank insulation thickness.

Quantity | Liquefied Natural Gas | Liquid Hydrogen |
---|---|---|

Mass of Fuel (ton) | 72,000 | 12,100 |

Mass of Ballast Water (ton) | 23,200 | |

Mass of Tank (ton) | 846 | |

Mass of Engines (ton) | 713 | |

Volume of Fuel (m^{3}) | 174,000 | |

Volume of Ballast Water (m^{3}) | 22,600 |

Ship Variable | Identifier Type | Categories | Identifier | Demonstration |
---|---|---|---|---|

Fuel and Propulsion Type | Colour | Methane Internal Combustion Engine | Pink | |

Hydrogen Internal Combustion Engine | Green | |||

Hydrogen Electric | Light Blue | |||

Presence of Reliquefaction | Presence of Reliquefaction Symbol | Reliquefaction Present | Symbol Present | |

Reliquefaction Absent | Symbol Absent |

(a) | ||||||
---|---|---|---|---|---|---|

Fuel and Propulsion Type | Ship Type | Thickness of Insulation (m) | Initial Mass of Fuel (10 ^{3} ton) | Mass of Tank (10 ^{3} ton) | Other Masses (10 ^{3} ton) | |

Methane Internal Combustion Engine | 0.53 | 72.29 | 0.85 | 23.62 | ||

0.31 | 72.64 | 0.49 | 23.62 | |||

Hydrogen Internal Combustion Engine | 0.72 | 11.65 | 1.14 | 23.87 | ||

0.38 | 12.42 | 0.61 | 23.87 | |||

Hydrogen Fuel Cell | 0.71 | 11.78 | 1.13 | 23.18 | ||

0.32 | 12.67 | 0.52 | 23.18 | |||

(b) | ||||||

Fuel andPropulsion Type | Ship Type | InitialVolume of Fuel(10^{3} m^{3}) | Volume of Tank(10^{3} m^{3}) | OtherVolumes (10^{3} m^{3}) | FuelDepletion Rate(%/day) | Revenue Loss Rate (10^{3} $/day) |

Methane Internal Combustion Engine | 173.63 | 17.62 | 171.97 | 0.0905 | 21.6 | |

174.48 | 10.20 | 178.54 | 0.1048 | 25.1 | ||

Hydrogen Internal Combustion Engine | 167.46 | 23.79 | 171.97 | 0.6224 | 57.7 | |

178.55 | 12.70 | 171.97 | 0.3816 | 37.7 | ||

Hydrogen Fuel Cell | 169.35 | 23.57 | 170.31 | 0.6258 | 58.6 | |

182.15 | 10.76 | 170.31 | 0.3061 | 30.9 |

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**MDPI and ACS Style**

Smith, J.R.; Gkantonas, S.; Mastorakos, E. Modelling of Boil-Off and Sloshing Relevant to Future Liquid Hydrogen Carriers. *Energies* **2022**, *15*, 2046.
https://doi.org/10.3390/en15062046

**AMA Style**

Smith JR, Gkantonas S, Mastorakos E. Modelling of Boil-Off and Sloshing Relevant to Future Liquid Hydrogen Carriers. *Energies*. 2022; 15(6):2046.
https://doi.org/10.3390/en15062046

**Chicago/Turabian Style**

Smith, Jessie R., Savvas Gkantonas, and Epaminondas Mastorakos. 2022. "Modelling of Boil-Off and Sloshing Relevant to Future Liquid Hydrogen Carriers" *Energies* 15, no. 6: 2046.
https://doi.org/10.3390/en15062046