Calculation of Boil-Off Rate of Liquefied Natural Gas in Mark III tanks of ship carriers by numerical analysis

Accepted Manuscript Title: Calculation of boil-off rate of liquefied natural gas in Mark III tanks of ship carriers by numerical analysis. Author: Mario Miana, Regina Legorburo, David Díez, Young Ho Hwang PII: DOI: Reference:

S1359-4311(15)01039-X http://dx.doi.org/doi: 10.1016/j.applthermaleng.2015.09.112 ATE 7109

To appear in:

Applied Thermal Engineering

Received date: Accepted date:

1-6-2015 13-9-2015

Please cite this article as: Mario Miana, Regina Legorburo, David Díez, Young Ho Hwang, Calculation of boil-off rate of liquefied natural gas in Mark III tanks of ship carriers by numerical analysis., Applied Thermal Engineering (2015), http://dx.doi.org/doi: 10.1016/j.applthermaleng.2015.09.112. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Calculation of Boil-Off Rate of Liquefied Natural Gas in Mark III Tanks of Ship Carriers by Numerical Analysis. a

a

a

Mario Miana *, Regina Legorburo , David Díez , Young Ho Hwang a

b

ITAINNOVA, Instituto Tecnológico de Aragón, Materials & Components, María de Luna 7 – 8, 50018 Zaragoza (Spain).

b

DONGSUNG FINETEC, 120, Hyeopdongdanji – gil, Miyang – myeon, Anseong – si, Gyeonggi – do, 456 – 843 (Republic of South Korea).

* Corresponding author. Tel: +34.976 01.11.57; Fax: +34 976.01.18.81; E-mail address: [email protected] (M. Miana).

Highlights -

4 numerical approaches calculate Boil-Off Rate for Mark III tanks of an LNG ship.

-

Approaches 1 and 2 define 2D & 3D models with selected temperature configurations.

-

A reduced order model (ROM) is defined by an equivalent thermal conductivity.

-

Approaches 3 and 4 apply this ROM to 2D & 3D sections of tanks.

-

BOR varies with insulation thickness: 0.0919 % for 270 mm and 0.0631 % for 400 mm.

Abstract. The heat flow from environment to LNG stored in Mark III of ship carriers is calculated in this paper by numerical simulations. Four different approaches are defined and evaluated: Approach 1 starts from simple 2D numerical computations of heat fluxes over representative sections of the insulation barriers in 10 specific temperature configurations defined by published data. Approach 2 evolves toward full 3D simulations of insulations layers under the same

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temperature configurations. A Reduced Order Model is next developed by calculating equivalent thermal conductivity for insulation barriers. This equivalent thermal conductivity is applied in the fluid flow and heat transfer simulation from the environment to the LNG in 2D and 3D models by 3

Approaches 3 and 4, respectively. For a full ship with a capacity of 165000 m with 270 mm thickness insulation barriers, the obtained BOR and the overall heat transfer coefficient vary 2

2

from 0.895 % and 0.0656 W/m ·ºC for Approach 2 to 0.0945 % and 0.0693 W/m ·ºC for Approach 3. For Approach 4, the BOR and overall heat transfer coefficient is 0.0919 % and 2

0.0674 W/m ·ºC. When the thickness of the insulation barrier is increased to 400 mm, these 2

initial values are reduced to 0.0631 % and 0.0453 W/m ·ºC.

Keywords. Liquefied Natural Gas, Boil-Off Rate, Reduced Order Model, Heat Transfer, Computational Fluid Dynamics, Finite Element Methods.

Nomenclature. CNC

natural convection correlation constant;

CEq

equivalent thermal conductivity calculation constant, dimensionless;

CP

heat capacity (J/kg ºC);

g

gravitational acceleration (9.81 m/s );

h

convective heat transfer coefficient (W/m ºC);

HVap

LNG enthalpy of vaporization (511 kJ/kg);

L

characteristic length (m);

m

mass (kg);

Nu

Nusselt number, dimensionless, Nu = h·L/;

Pr

Prandtl number, dimensionless, Pr = / ;

Q

heat flow (W);

q’’

heat flux (W/m );

R’’T

thermal resistance (m ºC/W);

2

2

2

2

2

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M. Miana et al: Calculation of Boil-Off Rate of Liquefied Natural Gas in Mark III Tanks of Ship Carriers by Numerical Analysis.

Ra

Rayleigh number, dimensionless, Ra = g··T·L /· ;

Re

Reynolds number, dimensionless, Re = W·L/ ;

S

conductive shape factor, dimensionless;

t

thickness (m);

T

temperature (ºC);

U

overall heat transfer coefficient (W/m ºC);

V

volume (m );

W

characteristic velocity (m/s);

3

2

3

Greek symbols. 2



thermal diffusivity (m /s);



thermal expansion coefficient (ºC );



wall angle with horizontal direction (rad);



thermal conductivity (W/m ºC);



kinematic viscosity (m /s);



density (kg/m );



difference.

-1

2

3

Subscripts. 0

reference conditions;

App.1, App.2, App.3, App.4 Approach 1, 2, 3 or 4; Cdam

cofferdam;

Env

environment (seawater or air);

Eq

equivalent;

Ev

evaporated;

i

insulation layer;

IH

inner hull;

Ld

loaded;

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Prw / Stn

prow or stern;

Acronyms. LNG

Liquefied Natural Gas;

ROM

Reduced Order Model;

CCS

Cargo Containment System;

R-PUF

Reinforced Polyurethane Foam;

FP

Flat Panel;

C90

90º Corner;

C135

135º Corner;

BOR

Boil-Off Rate, defined in equation 2 (% evaporated LNG mass / loaded LNG mass ·

day); NG

Natural Gas;

FEM

Finite Element Method.

1.

Introduction. The grow of Natural Gas (NG) as an energy source after the oil shock in the early 1970s led

to an ever rising consumption of Liquefied Natural Gas (LNG). By 2013, Natural Gas accounted for 25 % of global energy consumption [1]. NG is transported from reserves to consumer zones by pipelines or shipped in liquid state or as compressed gas. Liquefied Natural Gas covers the 10 % of the global demand for NG, so the global LNG fleet has grown steadily: it stood at 357 vessels in 2013 and 31 new vessels were scheduled for delivery in 2014 [1]. The advantages of shipped LNG over pipeline systems are well known: firstly, a greater adaptability to cover the growing distances from reserves to consumer zones; secondly, the limiting export capacity of pipelines systems; and thirdly, the potential problems due to unstable political situations when international pipelines entail the crossing of a number of countries and borders [2]. However, LNG is not without its own problems. Liquefied Natural Gas is carried at

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cryogenic temperatures near to its saturation temperature, typically about -162 ºC, and this implies a net heat transfer from the environment, in spite of the excellent insulation of ship tanks. This heat flow yields the generation of a boil-off gas that must be vented out of the tank to avoid large rises in pressure that could damage its mechanical structure. Moreover, evaporation is not homogeneous because LNG is a mixture of hydrocarbons and nitrogen. The most volatile components (nitrogen and methane) evaporate first, which is known as the LNG ageing or weathering. These methane losses imply worse liquid qualities when the ships arrive at the regasification terminal [3]. The evaporation rate of LNG during shipping is often characterized by the BOR parameter, which is the percentage of evaporated LNG per day with respect to the initial LNG loaded. In summary, the BOR prediction is the main figure to measure the heat transfer from the environment to LNG stored in tanks for shipping. The BOR parameter plays a major role in the economics of the LNG trading, since the evaporation of the methane contained in LNG involves significant energy and financial losses [4]. To minimize this heat transfer, the walls of Cargo Containment Systems (CCS) are composed of superb insulation layers made of perlite in NO-96 tanks or plywood, reinforced polyurethane foams (R-PUF) and membranes in Mark III tanks [5]. However, the calculation of the overall heat flow through the insulation barriers is not a trivial task, given the following difficulties: 

The large dimensions of the tanks ( 40 m long) compared to the thickness of the insulation barriers ( 0.27 m) or the thickness of the stainless steel membrane in contact with the LNG ( 1.2 mm).



The large number of involved heat transfer phenomena: LNG convection inside of the tank; conduction through insulation layers; natural convection in ballast compartments of irregular shapes; forced convection from ship to water and air environment.



The measurement of thermal properties for materials in a wide temperature range, ranging from -162ºC to +45 ºC.

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

The lack of detailed validation results for on-board LNG tanks.

To overcome these difficulties, numerical simulations are proposed in this paper as the 3

basic tool to predict the BOR parameter for a 165000 m capacity LNG ship carrier with 4 Mark III tanks. The scientific literature on numerical simulations for LNG tanks covers different aspects among which stress analysis [6, 7], studies on vibration caused by the sloshing phenomena [8], or the thermal analysis of representative sections of a tank [9, 10]. Chen et al [11] analyze the pressure and temperature changes in LNG storage tanks by means of dynamic models while references [12 – 15] estimate the BOR using computational fluid dynamics 2

assuming an overall heat transfer coefficient of 0.411 kJ/h·m ·ºC. Experimental results have been obtained only for very simplified geometries under laboratory conditions [16 – 20]. It should be noted that the solution of the energy transport equation by numerical methods in any system needs the definition of a computational grid able to capture the main geometric features, such as the different insulation layers to be considered with Mark III tanks. Although mesh sizes can be enlarged in zones with small temperature changes, smooth transitions between the different gap sizes are recommended. The previously defined typical length scales 12

ranging from 1.2 mm to 40 m can lead to creating grids of up to 7·10 cells to represent a 3

quarter of a 40000 m capacity ship tank. As the number of computational cells is increased, the time to run simulations is extended and the difficulties in analyzing such a detailed domain rise. By way of a comparison, one of the 12

largest numerical grids ever solved was composed of 4·10 cells [21]. Therefore, detailed numerical simulations of all geometric features of LNG tanks are not affordable. Moreover, a very large mesh also requires an extended computational time to be converged, and the detailed information provided by such an enormous amount of data is not of interest for the analysis of design modifications, where overall conclusions are more useful in the design stages than comprehensive information about every point. Previous numerical studies on heat transfer in LNG ship tanks [22 – 27] were based on finite difference approximations, only considering the conduction through the different insulation layers.

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To overcome all these inconveniences, this paper proposes a set of 4 efficient approaches to predict the BOR in Mark III tanks combining detailed numerical simulations of fluid flow and heat transfer and Reduced Order Models. Reduced Order Models (ROMs) are a group of wellestablished technologies to reduce the size of the computational model with a minimum loss of accuracy [28]. ROMs have been successfully applied in a wide number of applications including transient heat transfer phenomena [29, 30] and aerodynamic loads [31]. The paper is structured as follows: Section 2 describes the problem to be investigated and Section 3 defines the developed methodologies; Section 4 collates the results achieved by the different approaches and judges their advantages and disadvantages. From these results, a new design modification is proposed and evaluated consisting of an increase in the thickness of the insulation barrier. Finally, section 5 summarizes the main conclusions drawn from this investigation and outlines the planned future work.

2. Definition of the problem. 2.1. Description of Mark III Tanks on LNG ship carriers. The problem to be investigated is the heat transfer from the external environment to the 3

LNG stored inside a 165000 m capacity ship carrier. The LNG ship contains 4 Mark III tanks following the arrangement and the prismatic shapes shown in Figure 1 and having the characteristic dimensions collected in Table 1. According to IMO regulations [32], ships for carrying liquefied gases in bulk must be of the double hull type. The external hull is 50 mm thick steel and the inner hull is 18 mm thick steel. The space between the inner and the outer hull is divided into ballast compartments also composed of 18 mm thick steel. These compartments are assumed to be filled up with air. The insulation barriers are directly attached to the inner hull by a mastic layer. The prismatic shape of the tanks is fitted with insulation layers by defining four different bodies, called Flat Panels, 90º and 135º Corners and Trihedrons. These bodies are composed of a primary barrier, directly in

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contact with LNG, and a secondary barrier which is thicker than the primary barrier. Between these barriers, a 0.6 mm thick impermeable triplex layer prevents the LNG leaking through the outside. The primary barrier is composed of a 1.2 mm thick corrugated stainless steel membrane, a plywood layer and the R-PUF layer. The secondary barrier is composed of R-PUF supported by two plywood layers. The initial detailed description of these CCS can be found in [33]. Figure 2 shows the assembly of the Flat Panels and Corners. The nomenclature of the different Flat Panels and Corners is described in Figures 1b and 1c. It depends on their relative position to the seawater or the air. The vertical walls are Flat Panels 1, and they line to the cofferdam, which is the space between two consecutive tanks, or directly to environment (seawater and air) at the prow and the stern of the ship. The bottom wall is named as Flat Panel 2, and the top wall is Flat Panel 8. The sidewalls are Flat Panels 3 to 7, starting from the bottom wall. The water line divides the vertical sidewall of Tanks 2, 3 and 4 into Flat Panels 4, under the water line, and Flat Panel 5, over the water line. For Tank 1, the water line falls within FP 3, so there is no FP 4. The lateral inclined top wall is further divided into two Flat Panels: FP 6, wetted by LNG, and FP 7, wetted by NG. The name of each edge comprises two figures: the first one indicates the angle of the edge and the second one represents the name of one of the attached Flat Panels. Edges starting from 1 correspond to 90º Corners while edges starting from 3 correspond to 135º Corners which are the same length as the tank. Lastly, edges lining to the cofferdams are called CDAMS Edges, corresponding to 90º Corners, likewise.

2.2. Properties of the Materials. The insulation barriers and the external and inner hulls are composed of 7 different materials; their thermal conductivities are given in Table 2. The thermal conductivity of the plywood, the stainless steel and the R-PUF and are obtained from references [10, 34, 35]. For the rest of materials, thermal conductivity was directly provided by suppliers. For the air filling the

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compartments between the inner and the outer hull, density is modelled using equation 1, following the Boussinesq approach:

   0 1   ·T  T 0 

(1)

3

where 0 is the reference density (1.209 kg/m ),  the thermal expansion coefficient (0.00348 -1

ºC ) and T0 the reference temperature (15 ºC). The specific heat, the thermal conductivity and viscosity are obtained by linear interpolation with temperature using the values tabulated in Table 3 [36].

2.3. Sailing conditions. IMO regulations [32] set the sailing conditions considering that the temperature of the seawater is 32 ºC and that of the air is 45 ºC. The velocity of the ship is defined as 19 knots. The LNG tanks are considered to be filled up to 98 % of capacity. The lower part is occupied by LNG at -162ºC, while the upper area is filled with NG.

2.4. BOR definition. BOR is defined as the percentage of the evaporated LNG mass per day with respect to the initial loaded LNG mass [3], so the heat flow (Q) received by LNG is translated into BOR by equation 2:

BOR 

Q ·24 ·3600 ·10 V · ·H vap

3

·100

where  is the density of LNG density and Hvap the enthalpy of vaporization of LNG (425 kg/m

(2)

3

and 511 kJ/kg) at -162 ºC and 1 bar. If BOR is calculated for each tank, Q and V are the total heat flow received by the LNG stored inside of each tank and the total volume of this tank.

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However, BOR can be calculated for the full ship too, so Q and V are the heat flow received by the total LNG transported by the ship and the full ship capacity.

3. Numerical approaches for BOR calculation. Four successive approaches are developed to obtain the heat flow transferred from the environment to the LNG based on simplified numerical simulations and Reduced Order Models of heat transfer through the insulation barriers of LNG tanks. The proposed approaches can be classified into two groups depending on their working procedure and the applied geometrical description, as shown in Figure 3. Approaches 1 and 2 use the same compounding method: both approaches obtain the heat flux transferred through Flat Panels and 90º and 135º Corners for some selected configurations through numerical simulations. These heat fluxes are then multiplied by the total heat transfer area giving the net heat flow transferred through global locations, for example, Flat Panel 1 or Edge 3-6. The sum of all heat flows transferred through the walls of the tanks will provide the input heat flow that will be used to calculate BOR. The difference between Approaches 1 and 2 is that Approach 1 runs 2D numerical simulations of simplified geometries of Flat Panels and Corners, while Approach 2 considers the full 3D geometries of Flat Panels and Corners. The second method applies the general framework of Reduced Order Models (ROMs) to derive Approaches 3 and 4. Specifically, the ROMs technology applied to this system searches for an equivalent thermal resistance while only considering a single material along the insulation layers. This material will be defined by equivalent thermal conductivities to yield the same heat transfer simulations as the above detailed numerical solutions. To achieve this, the running of comprehensive numerical simulations of individual components like Flat Panels and 90º and 135º Corners is proposed. These comprehensive numerical simulations are the same as those calculated in Approach 2.

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From these highly detailed results, an effective or integration parameter is extracted. For the insulation walls of Mark III tanks, the integration parameter is the equivalent thermal conductivity of the materials that comprise the walls. An effective thermal conductivity will be defined for Flat Panels and Corners to condense out all heat transfer resistances at the different layers that compose the different bodies of the insulation barriers. Once more, the difference between Approaches 3 and 4 is that Approach 3 runs 2D numerical simulations of fluid flow and heat transfer in representative 2D slices of ship arrangement, while Approach 4 considers full 3D sections of the tanks and the ballast compartments.

3.1. Approach 1: 2D Numerical Simulations of FP and Corners in Specific Configurations. The heat transfer from the environment to the LNG stored in MARK III tanks can be modeled as the heat flux through a planar wall under different temperatures on either side. The different layers that compose the tank walls act as like resistors in series for the heat transfer, and therefore, applying Fourier’s law, the heat flux transferred through the wall is simply the temperature difference divided by the total thermal resistance [37]:

q  

T Env  T LNG R T

(3)

The total thermal resistance is the sum of the resistance of each layer that composes the insulation barrier, and this resistance is a function of the thermal conductivity of the material that composes the layer and the thickness and the conductive shape factor of said layer. The dependence of the thermal conductivity of R-PUF on temperature makes a direct calculation of the thermal resistance by analytical formula like, for example, the planar wall resistance [37] complicated. To solve this problem, Approach 1 is divided into three steps. First, the heat fluxes through Flat Panels and Corners are numerically calculated considering the properties of materials defined in section 2.2 and the temperature for LNG and for the inner hull obtained from

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the literature [25]. Trihedrons are not considered due to their small heat transfer area by comparison with Flat Panels or Corners. The second stage is the calculation of the heat transfer areas of Flat Panels and Edges defined in Figures 1b and 1c. The third step calculates the heat flows transferred through the different walls by multiplying the previously obtained heat fluxes and heat transfer areas, yielding the total heat flow which is finally translated into BOR using equation 2. The temperatures of the inner hull and the membrane reported by [25] are split in Table 4 into 10 specific configurations described in Figure 4. LNG is considered to be at -162ºC while natural gas presented at the top of the tank is at -158 ºC. The temperatures in Table 4 are then imposed as boundary conditions for the numerical model. The simplicity of this approach is applied to the definition of the numerical domain too. Only the 2D representative sections of Flat Panels and 90º and 135º Corners shown in Figure 5 are investigated. The numerical calculations of heat fluxes are performed using the Finite Element Method by the ABAQUS 6.12 software [38]. The numerical grids are composed of about 200,000 cells. The obtained values are presented and analysed in section 4.1.

3.2. Approach 2: 3D Numerical Simulations of FP and Corners in Specific Configurations. Approach 2 follows the same method as above, in other words, the calculation of heat fluxes through Flat Panels and Corners in the 10 specific configurations determined by [25]. However, Approach 2 increases the detail level of the obtained results because the heat fluxes are achieved through 3D numerical simulations of the complete geometries of Flat Panels and Corners, including the steel corrugations in contact with LNG, instead of the characteristic 2D slices as in the previous section. The selected grids are composed of 2.2 millions cells for Flat Panels and 700,000 cells for Corners. The obtained results are summarized in Section 4.1.

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3.3. Definition of Reduced Order Models for Heat Transfer in Mark III Tanks. A realistic prediction of BOR can be obtained through detailed numerical simulations of the fluid flow and heat transfer considering the external hulls, the ballast compartments, the inner hull, the metallic sheets that form the ballast compartments and the insulation walls. The heat transfer from the environment to the inner hull will be simulated without any significant simplification, defining a numerical grid to solve the continuity and the momentum and energy transport equations in these zones. The Reduced Order Model is defined for the insulation walls, which are discretized through a numerical grid of 10 cells in the normal direction, as shown in Figure 6. Such coarse mesh is not able to capture all geometrical features of the Flat Panels and Corners. Thus, the Reduced Order Model proposes the definition of an equivalent material for each insulation body (Flat Panels and 90º and 135º Corners), characterized by the density and the heat capacity of the R-PUF as this accounts for the highest percentage of the total volume of the barriers. The thermal conductivity of this equivalent material is determined to yield the same thermal resistance as is obtained for Flat Panels and Corners in Approach 2, following equation 4:

 Eq   R  PUF

 q App.2  Mod q App.2,

  R  PUF ·C Eq

(4)

where R-PUF is the thermal conductivity of R-PUF displayed in Table 2, q’’App.2 the heat flux obtained in Approach 2 and q’’App.2,Mod the heat flux obtained in a “modified” Approach 2 where all layers but the inner hull are only composed of R-PUF. The configuration with the largest difference of temperature from the inner hull to LNG is selected, in other words, configuration D for Flat Panels and 90º Corners and Configuration E for 135º Corners. As a result, the equivalent parameter CEq integrates the exhaustive information from the 3D numerical models into a single parameter to characterize the thermal resistance from inner hull to LNG.

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3.4. Approach 3: ROM and 2D Numerical Simulations of LNG Tanks. Approach 3 is based on 2D numerical simulations of fluid flow and heat transfer from the environment to the LNG stored inside the Mark III tanks. Four different domains are built as shown in Figure 7: two symmetrical transversal slices of the Tanks 1 and 2, one longitudinal slice of the cofferdam between Tanks 1 and 2, and one longitudinal slice of Tank 1. This conjugated fluid flow-heat transfer analysis is governed by the continuity, momentum and energy transport equations [39]. These equations are solved by the ANSYS FLUENT software, Release 15.0, using the 2D double precision solver [40]. The fluid flow is assumed to be non-isothermal incompressible turbulent flow, following the Navier - Stokes equations. Turbulence is modeled using the k -  Realizable model [41] while the turbulence generation by walls is calculated through Enhanced Wall Functions [40]. Pressure and velocity fields are coupled by the SIMPLE algorithm [42]. Second order spatial discretization schemes are applied to the momentum, energy, turbulent kinetic energy and turbulent dissipation rate, while pressure is discretized using the Body Force Weighted scheme [40]. In all domains, the shell conduction from the inner hull to the external hull through the different solid walls that form the ballast compartments is considered by meshing of these slim walls with two cells along their thickness. The detailed geometries of the insulation barriers have been replaced by equivalent thermal conductivities for Flat Panels and 90º and 135º Corners. The shell conduction of the inner and outer hulls is directly modeled using 18 and 50 mm thick steel meshed layers, respectively. The shell conduction of the stainless steel membrane is not simulated horizontally; just a tiny thermal resistance is considered vertically, defined by 1.2 mm thickness of stainless steel. The cell height attached to all walls is 7 mm to allow an accurate capture of the thermal and viscous boundary layers. The structured mesh type is selected for most of the regions except in zones with sharp angles. The selected meshes vary from 1.7 millions of quadrilateral cells for Tank 2 to 4.4 millions of cells for the cofferdam between Tanks 1 and 2. The thermal boundary condition at the cofferdam walls imposes a temperature of 5 ºC,

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while convective boundary conditions are applied at the external hull and at the membrane walls. For the external hull walls, the air or the seawater temperature defined by the IMO sailing conditions [32] is imposed and the convective heat transfer coefficient (h) is obtained from correlation no. 5, which corresponds to the average Nusselt number for turbulent parallel flow over a flat plate [39]: Nu  0 . 037 ·Re

4

1 5

·P r

3

(5)

where Nu is the Nusselt number, Re is the Reynolds number and Pr is the Prandtl number. Considering the length of the ship as the typical length scale ( 200 m) and the air and water 2

properties obtained from [36] and [39], the convective heat transfer coefficient is 12.40 W/m ·ºC 2

for the walls in contact with air and 6,529 W/m ·ºC for the walls in contact with water. These values are of the same order of magnitude as the typical values of the convective heat transfer coefficient collected in [39]. For the membrane walls in contact with LNG, is first assumed that LNG is at -162ºC for the entire tank. The convective heat transfer coefficient is estimated from correlation 6: Nu  C NC · Ra ·cos 

 n

(6)

Correlation 6 assumes that convection is the result of buoyancy forces alone, and ignores the influence of sloshing LNG motion inside the tanks. The CNC constant varies from 0.1 to 0.54; the exponent n varies from 1/4 to 1/3 depending on the orientation of the wall and  is the wall angle compared to the horizontal [39]. The convective heat transfer varies in Tank 2 from 68.4 2

2

W/m ·ºC for FP 8 to 1067.8 W/m ·ºC for FP 3.

3.5. Approach 4: ROM and 3D Numerical Simulations of LNG Tanks. Once the numerical model has been defined and applied in 2D domains for Approach 3, it can be easily extended to 3D domains for the development of the Approach 4. This section only describes significant differences with respect to Approach 3. First, the ship is divided into three

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different domains, as shown in Figure 8. Tanks 2 and 3 are directly next to the cofferdams, so a single simulation is enough to get the heat flow transferred from the environment to the LNG. Meanwhile, Tanks 1 and 4 sit against the cofferdams or lie against the prow or the stern, respectively. Considering the symmetrical behavior of the heat transfer in Tank 1 (see the analysis of Figure 14 in section 4.2), just a single mesh of a quarter of Tank 1 is built. Since Tank 4 is larger than Tank 1, the symmetrical behavior of the heat transfer can be ensured too for Tank 4, so just a single mesh is built too for Tank 4. The thermal boundary condition imposed at the vertical limiting wall will define which the case is: if the convective heat transfer is applied to seawater or the air, the simulation obtains the heat transfer from Tank 1 or 4 to the prow or the stern; if a constant temperature of 5 ºC is set, the simulation represents the heat transfer from the cofferdams. The building of the computational grids is mainly based on the sweep of the 2D built for Approach 3, including the meshing of the metal sheets that form the ballast compartments. Mapped conformed meshes are selected for the most of the domain; zones as corner compartments are meshed with pave schemes and swept along the normal direction to reduce the high cell skewness. The selected meshes comprise 10.2 million cells for Tank 1 to 13.65 million cells for Tank 2. The external and the inner hulls are also included, together with the Flat Panels and Corners. Finally, the PISO algorithm [43, 44] with skewness correction has been selected for the pressure-velocity coupling.

4.

Results and Discussions.

4.1. Results of Approaches 1 and 2. A grid independence study is performed first to ensure the quality of the results. For Approach 1, the number of cells from LNG to inner hull is doubled without any significant difference in the heat fluxes for Configurations D and E. For Approach 2, several mesh resolutions and domain extents are considered, resulting in the selected domains displayed in

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the temperature contours shown in Figure 9. The uniform thickness of all insulation layers for Flat Panels are highlighted by constant temperature profiles for height. However, the different hardwood pieces and the angle geometry yield a non-uniform evolution of temperature from the top to the bottom of Corners. Table 5 summarizes the heat transfer areas for the different Flat Panels and Corners in Tanks 1 to 4 and Table 6 displays the heat fluxes obtained through the computational simulations performed in Approaches 1 and 2. The heat flow transferred through each insulation barrier can be obtained by multiplying the heat fluxes by the corresponding heat transfer areas. Once the heat flows transferred from Flat Panels and Corners are obtained, the sum of all heat flows are applied to calculate the BOR following equation 2. It can be seen that 3D heat fluxes for Corners are 13 % to 16 % lower than 2D results. The 2D model considers that the hardwood is continuous, while the 3D geometry correctly describes the distribution across the different blocks of hardwood and R-PUF. Thus, a larger R-PUF content increases the thermal resistance of the 3D corner and yields lower heat fluxes. The difference for Flat Panels is about 2 % and it is caused by the simplification of the stainless steel corrugation. The 2D model only considers a 1.2 mm thick stainless steel layer, but the 3D model takes into account the real shape of the corrugations, the wooden pieces that support them and the air that fill the gaps between the membrane and the supporting pieces. These differences are not directly translated in the BOR calculation. Figure 10 compares the BOR obtained by the different approaches and for the different tanks. It can be seen that the contribution of Corners to the total BOR varies from 4.9 % to 6.5 %, because the heat transfer area of Corners is about 6 % of the total heat transfer area. In addition, it can be observed that larger tanks yield a lower BOR, as the surface area density (the Surface to Volume ratio) is reduced [45].

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4.2. Results of Approaches 3 and 4. Firstly, the equivalent thermal conductivities obtained from the procedure reported in section 3.4 are reported here. Table 7 compares the obtained heat fluxes for the selected configuration in the original and modified Approaches 2 and the equivalent constant to be applied in Equation 4. The last row shows the obtained heat flux when the equivalent thermal conductivity is applied. It can be concluded that this order reduction is suitable to capture the global thermal behaviour of the insulation barrier, because the differences when using this reduced model with respect to the detailed model are less than 0.17 %. Once the equivalent thermal conductivities of Flat Panels and 90º and 135º Corners are obtained, the numerical simulations can be performed. The grid independence study has been performed by doubling the number of cells inside of the insulation barriers, yielding evident differences in heat flows for all simulated domains in Approaches 3 and 4. Qualitative results are given in Figures 11 and 12 through temperature and velocity contours in the different analysed domains. The maximum fluid velocity is located in the bottom compartments because they show the configuration that encourages the convective heat transfer, having the cold wall high up and the bottom wall low down. The ballast compartments at the corners show significant irregular velocity fields due to the instabilities caused by the plumes generated from these cold walls [37]. The temperature contours reveal that the main heat transfer resistance is obviously generated by the insulation barriers, because the drop in temperature through the air compartments is noticeably lower than it is through the Flat Panels and Corners, as it would be expected. The thermal bridges of the steel sheets comprising the ballast compartments can be identified in Figure 13 by the significant rise in temperature in these figures. The symmetry of the thermal field in Tank 1 along its longitudinal section is demonstrated by Figure 14. The profiles of the temperature difference for the inner and outer hulls are drawn starting from the symmetry line. No significant differences are found for distances lower than 5 meters from the midpoint, so the different thermal boundary conditions imposed at cofferdam

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walls and at stern or prow walls only influence in the region nearest to these zones. Thus, Tanks 1 and 4 can be divided by symmetry planes for Approach 4. Table 8 compares the numerical temperatures obtained from Approaches 3 and 4 with results reported by [25]. Only a rough comparison should be made because there are certain differences between the simulated and the published cases. For example, the thermal conductivity of R-PUF is 0.04 W/m·ºC in [25]; the LNG temperature varies in [25] from -162ºC to -158 ºC in the vapour phase, while current numerical simulations are based on a constant temperature of -162ºC and the geometries of the compartments are different, as shown in Figure 15. Reference [25] considers just five ballast compartments, while the current numerical model divides the fluid space into 12 compartments. For this reason, Table 8 shows the areaweighted average of temperature for Approach 3 and the volume-weighted temperature for Approach 4 contemplating the different compartments identified in Figure 15. Ballast compartments 2 and 4 show small differences between the published and the obtained results, due to a good concordance between the different geometries. For the inner hull temperatures for configurations A to E and CR1 to CR3, the results shown for both Approaches 3 and 4 are the area-weighted average of the wall that separates the inner hull from the ballast compartments or the metal sheets. The temperatures in the configurations show larger differences, especially for Approach 3, and for Configuration F, the top wall. Table 9 compares the heat flow and BOR obtained from other available published data with results from Approach 4. The significant differences between the geometries of MOSS and Mark III tanks and the variations in thicknesses and material properties among these references avoid a direct validation based on the reported values. However, it should noted that Approach 4 is close to the values reported from [25], considering that only a quarter of the tank is simulated in [25]. The calculation of BOR for Approach 3 is not straightforward because Edges 1-3, 1-4, 1-5 1-6 and 1-7 are not simulated while FP 2, FP 8 and Edges 1-2 and 1-8 in Tank 1 are simulated twice. The proposed solution is to consider the missing data as the average of the nearest

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simulated corners, while the repeated data is reduced by taking the average value between the two simulated domains. Moreover, the 2D domains do not represent the exact geometry, so the heat transfer area must be determined for the different edges based on the length of the tanks. Computational results are cast into the different heat fluxes obtained at the membrane level, while the heat fluxes obtained for Approaches 1 and 2 were calculated at the inner hull side. Thus, only overall results are shown in terms of the BOR calculated for the heat transfer through Flat Panels and Corners in Figure 10 together with the overall BOR and the overall heat transfer coefficient in Table 10. The heat flow through Flat Panels is again the most important, but the relative importance of edges increases by 12 % for Approach 3 and 16 % for Approach 4 due to the rise in the equivalent thermal conductivity. However, the overall heat transfer rate does not change dramatically: Table 10 reveals that the largest difference is about 6.6 % for Approaches 3 and 4 in Tank 1. In summary, the differences achieved in the overall BOR for the full ship lie below 3 % for all approaches. The overall heat transfer coefficient is defined by equation 7:

U 

Q A LNG ·T Env  T LNG

(7)



where Q is the overall heat flow received by each tank or by the full ship, ALNG is the heat transfer area of the membrane in contact with LNG, TEnv is the area-weighted average temperature of the environment, considering the respective surfaces exposed to seawater and air and TLNG is the LNG temperature. The obtained value for the full ship in Approach 4 is 0.0674 2

2

W/m ·ºC, which can be compared with the value of 0.07 W/m ·ºC obtained from equation 8:

U [14 ] 

1 1



h LNG

t





1

(8)

h Env

Equation 8 was applied in [14] for a material with different thickness and constant thermal conductivity, so a different value of U is reported in [14]. For the current case, the thermal

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conductivity of R – PUF varies with temperature, so the value at mid temperature (-80 ºC) is applied in equation 8. Then, equation 8 represents a good starting approach for BOR calculation, since conduction through R – PUF layer is the most important thermal resistance. However, if a detailed thermal analysis is required, the proposed approaches of the current investigation yield a more comprehensive description of the thermal fields inside of LNG ship tanks. These overall heat transfer coefficients can then be cast into global thermal resistances under the terms of equation 9:

Q 

T Env  T LNG RT



T Env  T IH R Out  IH



T IH  T LNG

(9)

R IH  LNG -5

The thermal resistances for the full ship obtained for Approach 4 are 4.7197·10 ºC/W for -4

-4

RT, 4.8392·10 ºC/W for ROutIH and 5.3112·10 ºC/W for RIHLNG. It can be concluded that the insulation barriers represents the 91.11 % of the overall thermal resistance. Finally, Approach 4 yields the estimation of the heat flow required to have cofferdams at 5 ºC. The cofferdam between Tanks 1 and 2 needs 206 kW; 342 kW are required for the cofferdam between Tanks 2 and 3; while 346 kW must be provided for the cofferdam between Tanks 3 and 4, yielding an overall cooling power of 894 kW for the full ship. This overall cooling power can be compared to the heat flow reported in [24], considering that reference data is just for a quarter of a smaller tank, at colder sailing conditions with different materials and thickness.

4.3. Evaluation of the 4 Approaches. Approach 1 is based on a published procedure [25], but devoted to a lower ship capacity of 3

137,000 m with different thermal insulation materials. Approach 1 can be calculated very quickly, requiring less than 2 hrs. in a standard computer server. Approach 2 provides more detailed information about heat transfer through Flat Panels and Corners than the previous approach, but it requires more time to run (about 1 day) and it does not solve the previous

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disadvantage regarding the definition of the temperatures on the inner hull side. Approach 3 does not assume the simplifications regarding the inner hull temperatures and it gives an initial estimation of the convective and conductive heat transfer resistances through external and inner hulls and ballast compartments. However, the determination of the heat transfer areas and the heat fluxes reduces the accuracy of this approach because it does not represent the 3D behavior of the tanks. BOR is not directly calculated from numerical simulations because it is still necessary to obtain average heat fluxes and heat transfer areas from geometry files. Approach 4 proposes the numerical simulation of heat transfer and fluid flow in 3D models of the different tanks, considering the orientation of each tank to cofferdams or to the prow and the stern of the ship. The meshing of the insulation layers cannot include all geometric features such as the plywood, the triplex layers and the first and second barriers, so a Reduced Order Model is defined to account for the different thermal conductivity of each insulation barrier. This approach does not assume any simplifications about heat transfer areas. This increased model complexity also yields a rise in the additional data provided by the numerical simulations, for example, the required power to keep cofferdams at 5 ºC together with a full description of the heat transfer and fluid flow in the ship. However, Approach 4 requires the longest times for pre processing and solving the different stages of numerical calculations. For example, 1 week is needed to run the calculations for the 5 different numerical domains included in this task.

4.4. Proposal and evaluation of a design modification: increasing the thickness of the secondary barrier. Approach 4 is applied to calculate the BOR when the thickness of the secondary R-PUF layer is increased from 160.4 mm to 290.4 mm. The objective of this modification is the reduction of the heat transfer to the LNG with a minimum decrease in the capacity of the ship.

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These thicker insulation layers are directly supported by the inner hull, so the external hull, the ballast compartments and the inner hull remain unchanged from the previous section. The procedure is as follows. The numerical grids built for Flat Panels and 90º and 135º Corners in Approach 2 are modified by increasing the thickness of the secondary barrier. The temperatures defined for the selected configuration in Table 5 are applied at the membrane and inner hull sides, yielding the heat fluxes reported in Table 7. The constants for equivalent thermal conductivities to be applied on Flat Panels and Corners are obtained as the quotient of the previously obtained heat fluxes. Next, the mesh of Flat Panels and Corners is increased in the various domains in Approach 4 and the new equivalent R-PUF materials are applied. The numerical models described in section 3.5 and 3.6 are applied again, yielding the BOR and the overall heat transfer coefficient shown in Figure 16 and Table 10. The conclusion is clear: an increase of the thickness of the insulation barriers means a reduction in the capacity of the ship by 2.2 %, but the heat flow received by LNG is significant reduced (by up to 67 %) from the initial heat flow.

5.

Conclusions and future work. 4 approaches have been developed to calculate the Boil-Off Rate of Liquefied Natural Gas

when shipped in Mark III tanks, from the simplest procedure using 2D simulations in 10 specific configurations to 3D simulations of the fluid flow and heat transfer around the LNG tanks, applying Reducing Order Models to define equivalent thermal properties that can describe the heat transfer resistance through the insulation layers. The obtained results are in concordance with the available published data, so these approaches can be used to estimate BOR or to provide a detailed description of the thermal fields inside LNG ship carriers. There is a three-fold plan for future work. First, Approach 1 can be easily incorporated into a software application that can anticipate BOR in ship carriers using analytical calculations of

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heat transfer through the different insulation layers if the shape factor for each layer has already been obtained. Second, the measurement of experimental BOR data during real ship transportation would provide very useful information to validate the current numerical approaches. These measurements must include the ship geometry, the evolution over time of LNG temperature, pressure and composition, the air and seawater temperatures, the ship speed and the thermal properties of the insulation barriers to eliminate the effects of ageing in polyurethane foams. Lastly, detailed numerical simulations of significant phenomena at the LNG site including, for example, boiling, stratification, sloshing phenomena or mixing processes if the ship carrier includes a reliquefaction plant would yield additional descriptions of the thermal behavior of LNG when shipped.

Acknowledgements. This dissemination work has been funded in part by the FEDER Operative Program for Aragon (2007 – 2013).

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%20World%20LNG%20Report%20-%202014%20Edition.pdf Last access: May, 27 , 2015. [2]

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[10] SW Choi, JU Roh, MS Kim, WI Lee: Analysis of Two Main LNG CCS (Cargo Containment System) Insulation Boxes for Leakage Safety Using Experimentally Defined Thermal Properties, Applied Ocean Research, 37 (2012), pp. 72 – 89. [11] QS Chen, J Wegrzyn, V Prasad: Analysis of Temperature and Pressure Changes in Liquefied Natural Gas (LNG) Cryogenic Tanks, Cryogenics, 44 (2004), pp. 701 – 709.

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[12] MM Faruqe Hasan, AM Zheng and IA Karimi: Minimizing Boil-Off Losses in Liquefied Natural Gas Transport, Industrial & Engineering Chemistry Research, 48 (2009), pp. 9571 – 9580. [13] MS Zakaria, K Osman, AA Yusof, MHM Hanafi, MNA Saadun, MZA Manaf: Parametric Analysis of Boil-Off Gas Rate Inside Liquefied Natural Gas Storage Tank, Journal of Mechanical Engineering and Sciences, 6 (2014), pp. 845 – 853. [14] MS Zakaria, K Osman and MN Musa: Boil-Off Gas Formation inside Large Scale Liquefied Natural Gas (LNG) Tank Based on Specific Parameters, Applied Mechanics and Materials, 229 – 231 (2012), pp. 690 – 694. [15] MS Zakaria, K Osman, MNA Saadun, MZA Manf and MHMHanafi: Computational Simulation of Boil-Off Gas Formation inside Liquefied Natrual Tank using Evaporation Model in ANSYS Fluent, Applied Mechanics and Materials, 393 (2013), pp. 839 – 844. [16] D Boukeffa, M Boumaza, MX Francois and S Pellerin: Experimental and Numerical Analysis of Heat Losses in a Liquid Nitrogen Cryostat, Applied Thermal Engineering, 21 (2001), pp. 967 – 975. [17] O Khemis, M Boumaza, M Ait Ali and MX Francois: Experimental Analysis of Heat Transfers in a Cryogenic Tank without Lateral Insulation, Applied Thermal Engineering, 23 (2003), pp. 2017 – 2117. [18] O Khemis, R Bessih, M Ait Ali and MX Francois: Measurement of Heat Transfers in Cryogenic Tank with Several Configurations, Applied Thermal Engineering, 24 (2004), pp. 2233 – 2241. [19] T Kanazawa, K Kudo, A Kuroda and M Tsui: Experimental Study on Heat and Fluid Flow in LNG Tank Heated from the Bottom and the Sidewalls, Heat Transfer-Asian Research, 33 (2004), pp. 417 – 330. [20] M Belmedany, A Belgacem and R Rebiai: Analysis of Natural Convection in Liquid Nitrogen under Storage Conditions, Journal of Applied Sciences, 8 (2008), pp. 2544 – 2552.

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[21] A Hamon: One Million Cores, A Breakthrough in CFD Simulation, International Science Grid This Week, Available on line at http://www.isgtw.org/feature/one-million-coresth

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[30] F He, L Ma: Thermal Management of Batteries Employing Active Temperature Control and Reciprocating Cooling Flow, International Journal of Heat and Mass Transfer, 83 (2015) pp. 164 – 172. [31] WA Silva: Discrete-time Linear and Nonlinear Aerodynamic Impulse Responses for Efficient CFD Analyses, Ph.D. dissertation, College William Mary, Williamsburg, VA, 1997. [32] IGC Code: International Code for the Construction and Equipment of Ships Carrying Liquefied Gases in Bulk, International Maritime Organization, September 2003. [33] J Roni and J Chauvin: The General Electric – Technigaz Mark III Containment System, GASTECH, Monaco (1978). [34] CY Ho and TK Chu: Electrical resistivity and Thermal Conductivity of Nine Selected AISI Stainless Steels, CINDAS Report 45, Thermophysical and Electronic Properties Information Analysis Center Lafayette In. (1977). [35] CJ Tseng, M Yamaguchi and T Ohmori: Thermal Conductivity of Polyurethane Foam from Room Temperature to 20 K, Cryogenics, 37 (1997), pp. 305 – 312. [36] EW Lemmon, RT Jacobsen, SG. Penoncello and D Friend: Thermodynamic Properties of AIr and Mixtures of Nitrogen, Argon, and Oxygen from 60 to 2000 K at Pressures to 2000 MPa, Journal of Physical and Chemical Reference Data, 29 (2000), pp. 331-385. rd

[37] JH Lienhard IV and JH Lienhard V: A Heat Transfer Textbook, 3 Edition, Philogiston Press, Cambridge, Massachusetts, USA (2008). [38] SIMULIA: ABAQUS Analysis User's Manual, Dassault Systèmes Simulia Corp., Providence, RI, USA (2012). th

[39] FP Incropera, DP de Witt: Fundamentals of Heat and Mass Transfer, 5 Edition, John Wiley & Sons, New York (USA), 2002. [40] ANSYS: ANSYS Fluent Theory Guide, Release 15.0, Canonsburg, PA, USA (2014). [41] TH Shih, WW Liou, A Shabbir, Z Yang, J Zhu: A New k –  Eddy – Viscosity Model for High

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Reynolds Number Turbulent Flows – Model Development and Validation, Computers Fluids, 24 (2005), pp. 227 – 238. [42] SV Patankar, DB Spalding: A Calculation Procedure for Heat, Mass and Momentum Transfer in Three – Dimensional Parabolic Flows, International Journal of Heat and Mass Transfer, 15 (1972), pp. 1787 – 1805. [43] RI Issa: Solution of the Implicity Discretizated Fluid Flow Equation by Operator – Splitting, Journal of Computational Physics, 62 (1986), pp. 40 – 65. [44] RI Issa, AD Gosman, AP Watkins: The computation of compressible and incompressible recirculating flows by a non-iterative implicit scheme, Journal of Computational Physics, 62 (1986), pp. 66 – 82. rd

[45] WM Rohsenow, JP Harnett, YI Cho: Handbook of Heat Transfer, McGraw – Hill, 3 Ed, New York (1998).

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Figure captions Figure 1: a) Tank layout; b) Tank 1: Dimensions and names of edges; c) Tanks 2, 3 and 4: Dimensions and Names of Flat Panels (Flat Panel n. 2 is the base of the tank) Figure 2: Insulation barriers: a) Flat Panels; b) 90º Corner; c) 135º Corner and Trihedron. Figure 3: Approaches for BOR calculation. Figure 4: Temperature configurations defined by [25]. Figure 5: 2D sections analysed in Approach 1: a) Flat Panel; b) 90º Corner; c) 135º Corner. Figure 6: Mesh of the Flat Panels and Corners in the Reduced Order Model (270 mm thickness). Figure 7: 2D Numerical Domains simulated in Approach 3: a) Transversal section of Tank 1; b) Transversal section of Tank 2; c) Longitudinal section of Cofferdam between Tanks 1 and 2; d) Longitudinal section of Tank 1. Figure 8: 3D Numerical Domains simulated in Approach 4. Figure 9: Temperature Contours (ºC) of Flat Panels and 90º and 135º Corners in Configuration D: a) Approach 1; b) Approach 2. Figure 10: BOR obtained for Approaches 1 to 4: a) Tank 1; b) Tanks 2 and 3; c) Tank 4; d) Ship. Figure 11: Contours obtained for Approach 3: a) Air temperature (ºC); b) Air velocity (m/s). Figure 12: Contours obtained for Approach 4: a) Air temperature (ºC), orientation to Cofferdam; b) Air velocity (m/s), orientation to Cofferdam; c) Air temperature (ºC), orientation to Prow / Stern d) Air velocity (m/s), orientation to Prow / Stern. Figure 13: Temperature contours (ºC) at the inner wall obtained for Approach 4: a) Orientation to Cofferdam; b) Orientation to Prow / Stern. Figure 14: Temperature difference for the cofferdam and the stern or prow sides. Figure 15: Ballast compartments for temperature comparison with [25]. Figure 16: BOR obtained for 400 mm thick insulation.

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Tables. Table 1. Characteristic dimensions of tanks. Insulation thickness 270 mm 3 Net Volume at 100 % loaded (m ) 400 mm Breadth (m) Height (m) Length (m) Tank

1

2, 3

4

27009 26335 34.4 27.9 33.7

49391 48353 38.5 27.9 47.0

40713 39826 38.5 27.9 41.2

Table 2. Thermal conductivity of solid materials. Material Epoxy Plywood [10] Stainless steel [34] Steel Triplex R-PUF [35]

Glass wool

Temperature (ºC)  (W/m·ºC) -2.7 10 0.12 -163 10.146 -45 -0.35 -160 0.013 -80 0.019 0 0.022 40 0.025 -160 0.014 -80 0.022 0 0.031 40 0.035

Table 3. Air properties as a function of temperature [36]. Temperature (ºC) -20 0 20 27 35 45

 (W/m·ºC) 0.02281 0.02436 0.02587 0.02640 0.02699 0.02772

31

CP (J/kg·ºC) 1005.74 1005.90 1006.36 1006.60 1006.92 1007.39

 (kg/m·s) –5 1.620·10 –5 1.722·10 –5 1.821·10 –5 1.854·10 –5 1.893·10 –5 1.940·10

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Table 4. Temperatures reported by [25]. Configuration A B C D E F G CR1 CR2 CR3

TLNG (ºC) -162.00 -162.00 -162.00 -162.00 -158.00 -158.00 -162.00 -162.00 -162.00 -162.00

TIH (ºC) 20.02 21.91 29.19 31.51 30.96 32.44 5.00 27.44 32.62 38.09

Insulation barrier FP 2, Edge 1–2 FP 3, Edge 1–3 FP 4, FP 5, Edge 1–4, Edge 1–5 FP 6, Edge 1–6 FP 7, Edge 1–7, Edge 3–8 FP 8, Edge 1–8 FP 1 CDAMS, Edge CDAMS Edge 3–3 Edge 3–4 Edge 3–6

Table 5. Heat transfer areas for Approaches 1 and 2 (inner hull side). Zone Edge 1–8 Edge 3–8 Edge 1–7 Edge 1–6 Edge 3–6 Edge 1–5 Edge 1–4 Edge 3–4 Edge 1–3 Edge 3–3 Edge 1–2 Edge CDAMS Total Edges FP 8 FP 7 FP 6 FP 5 FP 4 FP 3 FP 2 FP 1 CDAMS FP 1 Prow / Stern Total FP Total Tank

Body C90 C135 C90 C90 C135 C90 C90 C135 C90 C135 C90 C90 FP FP FP FP FP FP FP FP FP

2

2

Configuration Tank 1 (m ) Tanks 2, 3 (m ) F 19.599 0.000 E 34.939 48.680 E 2.716 0.000 D 13.566 0.000 CR3 34.939 48.680 C 4.809 0.000 C 0.000 0.000 CR2 34.939 48.680 B 36.066 0.000 CR1 34.939 48.680 A 19.706 0.000 G 96.462 306.978 332.679 501.699 F 699.880 981.203 E 94.015 528.690 D 469.657 528.690 C 160.252 1000.352 C 0.000 406.864 B 244.391 0.000 A 1041.121 479.013 G1 703.807 1390.171 C 763.110 1892.325 763.110 0.000 4939.342 7709.009

32

2

Tank 4 (m ) 19.598 42.690 2.684 18.818 42.690 9.910 4.152 42.690 10.193 42.690 27.595 107.014 370.724 858.562 462.591 462.591 876.293 356.395 0.000 419.080 1216.430 946.163 946.163 6544.266

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Table 6. Heat fluxes obtained for Approaches 1 and 2. 2

Configuration A B C D E F G CR1 CR2 CR3

q’’App.1 (W/m ) FP C90 C135 13.467 12.828 -13.643 12.994 -14.333 13.646 -14.557 13.858 -14.255 13.558 16.053 14.399 13.694 -12.122 11.555 ---15.971 --16.530 --17.134

2

q’’App.2 (W/m ) FP C90 C135 13.191 11.322 -13.364 11.470 -14.037 12.048 -14.255 12.235 -13.962 11.982 13.757 14.104 12.102 -11.872 10.195 ---13.672 --14.152 --14.670

Table 7. Heat fluxes obtained for the equivalent thermal conductivity.

2

q’’App.2 (W/m ) 2 q’’App.2,Mod (W/m ) CEq 2 q’’App.2, Eq R–PUF (W/m )

FP 14.255 13.578 1.050 14.280

270 mm C90 12.235 9.191 1.331 12.265

C135 13.757 10.725 1.283 13.771

FP 9.505 9.155 1.038 9.504

400 mm C90 8.208 5.904 1.390 8.175

C135 9.589 7.875 1.218 9.586

Table 8. Comparison of temperatures of ballast compartments and configurations of inner hull with results for Tank 2 obtained by [25]. Ballast compartment / Configuration t1 t2 t3 t4 t5 A B C D E F G CR1 CR2 CR3

T[25] (ºC)

TApp.3 (ºC)

TApp.4 (ºC)

38.67 39.12 36.67 29.00 25.84 20.02 21.91 29.19 31.51 30.96 32.44 5.00 27.44 32.62 38.09

41.20 38.69 32.93 29.87 29.84 27.62 26.92 30.48 34.92 37.74 41.03 2.76 27.86 28.20 34.90

36.35 37.65 30.55 28.62 27.55 23.09 23.15 25.67 30.24 36.40 25.90 0.12 26.70 26.32 35.22

33

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Table 9. Comparison of current study with previous investigations reported in scientific literature. Sailing Conditions TAir Env TSea Env (ºC) (ºC) 5 0 -18 0 25 18 45 32 -18 0 45 32

Results BOR Q (W) (% Ev / day) 157458 0.1552 99820 0.1188 123500 0.1470 136000 0.1619 101500 0.1171 132400 0.1527

Reference

Type of Tank

Tank 3 Volume (m )

Insulation thickness (m)

 (W/m·ºC)

[14]

Mark III

40447

0.53

0.0605

[22]

MOSS

41051

0.22

Unknown

[23]

MOSS

32960

0.22

0.030

[24]

Mark III

138000 (full ship)

0.24

0.0536

-18

0

89700

--

[25]

Mark III

7536

0.25

0.04

0 32 29

Current study

Mark III

27009 49391 40713

-18 45 28

0.27

Varying with T

45

32

16317 18972 18469 71443 104197 97224

0.0863 0.1003 0.0977 0.1074 0.0856 0.0969

Comments

Tank volume estimated from overall dimensions

Heat flow received from Cofferdam; results for 1/4 of tank Volume estimated from Q and BOR; results for 1/4 of tank Heat flow and BOR from App. 4

Table 10. Comparison of BOR and the overall heat transfer coefficient obtained using the different Approaches and the different thicknesses.

Tank 1 Tanks 2, 3 Tank 4 Ship

BOR (% mEv / mLd·day) 2 U (W/m ·ºC) BOR (% mEv / mLd·day) 2 U (W/m ·ºC) BOR (% mEv / mLd·day) 2 U (W/m ·ºC) BOR (% mEv / mLd·day) 2 U (W/m ·ºC)

App. 1 0.1091 0.0670 0.0860 0.0676 0.0955 0.0676 0.0921 0.0675

34

270 mm App. 2 App. 3 0.1061 0.1145 0.0652 0.0703 0.0836 0.0872 0.0657 0.0686 0.0929 0.0990 0.0658 0.0701 0.0895 0.0945 0.0656 0.0693

App. 4 0.1074 0.0660 0.0856 0.0673 0.0969 0.0687 0.0919 0.0674

400 mm App. 4 0.0765 0.0458 0.0583 0.0448 0.0660 0.0457 0.0631 0.0453

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