Boil-Off Rate Behavior in a Double-Shell Vacuum-Insulated Cryogenic Storage Tank with Multilayer Insulation

Abstract

Cryogenic storage systems require precise management of boil-off gas (BOG) to prevent pressure buildup caused by external heat ingress. This study investigates sloshing-induced heat transfer and BOG generation in a double-shell vacuum-insulated cryogenic storage tank subjected to dynamic excitation. Experiments were conducted under dynamic operating conditions, during which BOG generation and boil-off rate (BOR) were measured over a range of vibration frequencies, acceleration levels, and fill ratios. The results reveal a strong frequency-dependent sloshing behavior that significantly influences BOG generation. Under dynamic conditions, the BOR exhibited a non-linear dependence on filling ratio, with a maximum response occurring at 50% filling, reflecting an optimal balance between sloshing-enhanced heat transfer and geometric confinement. Based on the experimental observations, an integrated numerical heat transfer model incorporating multilayer insulation (MLI) performance and vacuum level effects was developed and validated against experimental data. The validated numerical model successfully predicts pressure evolution and BOG generation under operational sloshing conditions, providing valuable insights for optimizing cryogenic tank design and minimizing boil-off losses.

1. Introduction

Fossil fuels, such as coal and oil, continue to be the main sources of energy, and the pollutants and carbon they emit when burned are one of the leading causes of climate change, which is accelerating global warming. Various countries worldwide have signed global climate agreements to address the growing threats of climate change [1]. In line with this global trend, the adoption of renewable energy sources, such as wind and solar power, has increased significantly, particularly in developed countries. Consequently, demand for environmentally friendly energy sources, including natural gas and hydrogen, has risen substantially [2]. Among environmentally friendly energy sources, liquefied natural gas (LNG) and liquid hydrogen (LH₂) have emerged as representative and highly promising alternatives. LNG is produced by cooling natural gas to approximately 110 K, which reduces its volume by up to a factor of 600, enabling efficient storage and transportation with reduced environmental impact [3,4]. Hydrogen, in contrast, exhibits an exceptionally high gravimetric energy density of approximately 142 MJ/kg. When stored in a liquefied state, its volume is reduced to nearly 1/800 of that in the gaseous phase (LH₂ density: 70.8 kg/m³), thereby substantially enhancing volumetric energy density and transportation efficiency [5,6]. In addition, liquefied hydrogen storage offers an energy storage efficiency that is four to five times higher than that of high-pressure compressed hydrogen (35–70 MPa). Therefore, to achieve high storage efficiency, natural gas and hydrogen must be stored at cryogenic temperatures, and the storage system must be properly insulated to maintain a large temperature difference between the internal cryogenic fluid and the external environment. Cryogenic storage tanks typically employ insulation materials and systems such as high-vacuum insulation, polyurethane foam, and multilayer insulation (MLI). Despite the application of these high-performance insulation technologies, heat ingress from the surrounding environment is inevitable in cryogenic liquid storage tanks [7].

Figure 1 illustrates the general mechanism of boil-off gas (BOG) generation in cryogenic storage tanks. BOG is defined as the amount of liquefied gas that vaporizes per unit time due to heat ingress [8]. As shown in the figure, heat is transferred from the external environment through the insulation system into the cryogenic tank. The ingressed heat induces buoyancy-driven convective flow within the liquid, promotes thermal stratification, and ultimately causes partial vaporization of the cryogenic fluid, resulting in the generation of BOG. Such vaporization is the primary mechanism responsible for BOG generation in cryogenic liquid storage tanks. The generation of BOG results in an increase in the internal pressure of the storage system, which can pose significant safety concerns if not properly managed. To handle continuous BOG generation, LNG storage tanks typically employ purging, venting, or re-liquefaction systems to control pressure buildup and ensure safe operation [9,10,11].

To reduce boil-off gas (BOG) generation, cryogenic storage systems employ various types of insulation systems [12]. Among these, the vacuum multilayer insulation (MLI) system is a high-performance insulation approach that consists of alternating reflective layers and spacer materials, designed to minimize radiative heat transfer [13]. Figure 2 presents a schematic of a multilayer insulation (MLI) structure composed of reflective layers separated by insulating spacers. The fundamental principle of MLI is to reduce radiative heat transfer through multiple reflections by positioning reflective shields within the vacuum space between the warm and cold surfaces of a cryostat [14]. MLI exhibits optimal performance under high-vacuum conditions (below 10⁻³ Pa) and is capable of suppressing all three heat transfer mechanisms—radiation, conduction, and convection [15,16]. Compared with conventional insulation materials such as polyurethane foam or perlite, MLI significantly enhances cryogenic storage efficiency by reducing effective thermal conductivity to the range of approximately 0.01–1 mW/m·K, depending on the layer density and vacuum level. In particular, optimization of the interlayer spacing enables effective suppression of radiative heat transfer in high-temperature regions while minimizing solid-state conduction in low-temperature regions [17,18]. Despite these advantages, MLI systems are susceptible to increased heat ingress caused by layer compression, seam treatment, and penetrations introduced during installation. Moreover, insulation performance can deteriorate rapidly when vacuum integrity is compromised. In addition, predicting the long-term thermal performance of MLI under varying operating conditions and filling levels remains a significant challenge [19,20,21].

To estimate the vaporization rate, various methods for analyzing the self-pressurization behavior of cryogenic liquefied gas storage tanks have been investigated through experimental and numerical approaches. However, existing studies exhibit several limitations that restrict their applicability. These limitations can be broadly classified into experimental constraints, limitations in the validation of numerical models, and discrepancies between idealized static conditions and actual operating environments. Experimental studies offer fundamental advantages in understanding the physical phenomena occurring in cryogenic storage tanks, as they provide reliable data for model development and validation and help elucidate the underlying mechanisms governing complex thermodynamic behavior. Previous experimental investigations by Seo et al. [22], Barsi [23], and Fesmire [24] have provided valuable insights into self-pressurization, heat transfer mechanisms, and insulation performance in cryogenic storage systems. In particular, experimental data serve as an essential benchmark for validating and calibrating numerical models, enabling clear identification of discrepancies between theoretical predictions and actual system behavior.

However, most experimental studies have been conducted using scaled-down storage tanks due to constraints related to cost, safety, and laboratory space [25]. Because the scaled tanks differ significantly from those of full-scale tanks, the resulting heat ingress characteristics may not accurately represent those of actual storage systems. Consequently, full-scale experimental investigations are necessary to capture realistic thermal and pressurization behavior [26]. Moreover, many experimental studies have focused on a limited number of parameters, and the high cost and safety concerns associated with cryogenic testing make it difficult to reproduce large-scale tanks or conduct long-term storage experiments. These limitations restrict systematic investigations under diverse operating conditions or extreme environments.

Due to these experimental limitations, numerical studies offer significant advantages that complement experimental approaches. Numerical methods enable detailed analysis of complex geometries and thermofluid phenomena under extreme conditions that are difficult to investigate experimentally, while parametric studies allow systematic evaluation of the effects of various design and operating variables. Numerous researchers have conducted numerical investigations on cryogenic storage tanks, including the development of finite element method (FEM)-based three-dimensional heat transfer models and boil-off rate (BOR) prediction models by Ludwig and Dreyer [27], Jeong [28], Lin et al. [29], and Kartuzova [30]. In addition, recent computational fluid dynamics (CFD) models have effectively simulated two-phase flow and phase-change phenomena by combining the Volume of Fluid (VOF) method with the Lee evaporation–condensation model. Advanced interface-capturing techniques based on the VOF framework have further improved the prediction of vapor–liquid interactions in cryogenic systems. Nevertheless, validation studies are essential for verifying the reliability of numerical models.

Many previous numerical studies on cryogenic storage tanks have relied on simplified assumptions, such as static boundary conditions or ideal-gas equations of state. To overcome these limitations, recent research has increasingly employed advanced numerical methods to investigate boil-off gas (BOG) generation induced by sloshing phenomena. For example, Wu and Ju [31] analyzed the BOG characteristics of a Type-C LNG tank under sloshing excitation and demonstrated the critical roles of excitation frequency and amplitude in governing interfacial heat and mass transfer processes. Despite these advances, most numerical studies still depend heavily on experimental data for validation and are often restricted to short-term simulations under limited operating conditions. In particular, many existing models continue to adopt static boundary conditions and ideal-gas assumptions, which restrict their applicability to realistic long-term operation. Consequently, such approaches remain insufficient for accurately predicting the coupled thermo-fluid behavior required for reliable design and performance assessment of cryogenic storage tanks under practical dynamic operating conditions.

Regarding the fundamental thermodynamic behavior and self-pressurization mechanisms of cryogenic storage tanks, most previous studies have been experimentally investigated under limited and static laboratory conditions. Experimental investigations by Jeon [32], Rossetti [33], Choi [34], and Kartuzova [35] have provided valuable insights into thermal stratification, boil-off gas (BOG) formation mechanisms, and insulation performance. However, most of these studies have been conducted under static laboratory conditions that differ substantially from real operational environments. Jeon [36] recently investigated the influence of sloshing on boil-off gas (BOG) generation and thermal response in liquid fuel tanks through combined experimental and numerical approaches, highlighting the complex interactions between liquid motion and thermodynamic state. Several studies conducted under maritime transport conditions have demonstrated that ship motions, including rolling and pitching, can significantly influence the pressure rise and temperature increase rates within cryogenic tanks.

As discussed above, experimental investigations of cryogenic storage tanks have often been constrained to single-variable analyses and limited operating conditions, primarily due to cost and safety considerations. Likewise, many numerical studies remain limited by insufficient validation data, short prediction horizons, reliance on static boundary conditions, and simplifying assumptions such as ideal-gas equations of state. Under such static modeling frameworks, complex phenomena encountered in real operating environments—including sloshing-induced flow, pressure fluctuations, and strongly coupled thermo-fluid interactions—cannot be adequately captured, thereby limiting the practical applicability of existing models. Recent studies, such as that by Chen et al. [37], have further shown that, in multiscale systems, external vibrations can significantly alter the coupling between convection and evaporation. These findings highlight the need for comprehensive investigations that more faithfully represent realistic operating conditions to enable reliable prediction and optimized design of cryogenic storage tanks.

The purpose of this study is to evaluate and verify the effects of fill ratio (FR) and dynamic operating conditions on the boil-off rate (BOR) of a cryogenic storage tank through an integrated approach combining experimental investigation and numerical analysis. Experiments were conducted using a 350 L double-shell cryogenic storage tank equipped with an insulation system under both static and dynamic conditions. To experimentally investigate tank behavior under dynamic conditions, boil-off gas (BOG) generation and BOR were measured over a range of vibration frequencies, acceleration levels, and fill ratios. Based on the experimental results, an integrated numerical heat transfer model was developed to account for multilayer insulation (MLI) performance and vacuum level effects. The numerical model was validated through comparison with experimental data and subsequently applied to predict pressure behavior under various insulation and operating conditions. Finally, combined experimental and numerical analyses were performed to evaluate BOR and elucidate the coupled effects of fill ratio and dynamic conditions.

2. Theory

2.1. Optimization of the Layer Density in MLI Applications

In cylindrical cryogenic storage tanks, heat ingress occurs primarily through conduction and radiative heat transfer. These heat transfer mechanisms can be effectively suppressed by maintaining a high-vacuum environment in the annular space between the inner and outer tank walls. Compared with conventional foam insulation, vacuum insulation systems offer significantly superior thermal performance. In particular, the selection of an appropriate number of multilayer insulation (MLI) layers is a critical design parameter, as it strongly influences the overall efficiency of vacuum insulation systems. Proper design of the MLI configuration is essential because excessive packing of MLI layers within a fixed insulation thickness reduces the interlayer spacing, which can introduce additional solid conduction paths and increase overall heat transfer. As a result, MLI systems exhibit an optimal layer density, typically expressed as the number of layers per unit thickness (layers/cm), at which the total heat transfer through the insulation is minimized. This optimal layer density is determined by balancing the competing effects of reduced radiative heat transfer with increased solid-state conduction as layer density increases. Accordingly, optimization of the layer density can be achieved by adjusting both the spacing between adjacent layers and the total number of layers within the insulation blanket thickness.

Several analytical approaches have been proposed to determine the optimal layer density of MLI systems. Among these, the Modified Lockheed equation and the layer-by-layer modeling approach are widely used. The Modified Lockheed equation extends the original Lockheed formulation by incorporating an additional thermal conductivity term associated with the spacer material, thereby accounting for all three heat transfer modes—radiation, solid conduction, and residual gas conduction.

$$q_{total}=q\left(radiation\right)+q\left(solid conduction\right)+q (gas conduction)$$

(1)

The generalized Lockheed equation can be calculated as [38]:

$$Q_{total}= {\displaystyle \frac{C_R\epsilon ({T_h}^{4.67}-{T_c}^{4.67})}{N}}+{\displaystyle \frac{C_S{\overline{N}}^{2.63}(T_h-T_c)(T_h+T_c)}{2(N+1)}}+{\displaystyle \frac{C_{G}P({T_h}^{0.52}-{T_c}^{0.52})}{N}}$$

(2)

where ${\overline{N}}^{2.63}$ denotes the layer density [layers/m] and $N$ represents the total number of layers. $C_S$ is the solid conduction coefficient [W/m·K], which is a function of the spacer material, while $C_R$ is the radiation coefficient, determined by the reflector material. $C_G$ denotes the gas conduction coefficient and depends on the residual gas pressure between the layers. $T_h$, $T_c$ are the outer and inner layer temperatures [K], respectively, and $P$ is gas pressure [Pa].

One another important approach is the physics—based expression developed by McIntosh for the theoretical calculation of an MLI system performance. McIntosh calculation method assumes a uniform heat flux throughout the MLI system and includes all three heat transfer modes. While radiation, gas, and solid conduction heat transfer can vary from layer to layer, the overall heat flux is assumed to be constant.

In McIntosh approach, the heat transfer by radiation can be calculated as [39]:

$$q_{radiation}= {\displaystyle \frac{\sigma \left({T_h}^4-{T_c}^4\right)}{\left(\frac{1}{{\epsilon }_h}+\frac{1}{{\epsilon }_c}-1\right)}}$$

(3)

where ${\epsilon }_h, {\epsilon }_c$ is emissivity of the hot and cold surfaces; the heat transfer by gas conduction can be calculated as:

$$q_{gas conduction}= C_{G}P·\alpha \left(T_h-T_c\right)$$

(4)

where $C_G= K_G/P_{\alpha }$, $K_G$ is gas conduction,$P$ is the gas pressure, $\alpha$ is the accommodation coefficient. The heat transfer by solid conduction can be calculated as:

$$q_{solid conduction}= K_S\left(T_h-T_c\right)$$

(5)

where $K_S= C_{S}fk/∆x$, in which $C_S$ is an empirical constant, $f$ is the relative density of the separator compared to solid material, $k$ is the separator material conductivity [$\mathrm{W}/\mathrm{m}·\mathrm{K}$] and $∆x$ is the actual thickness separator between reflectors [m].

Therefore, the total heat transfer in McIntosh’s approach comes out to be the sum of Equations (3)–(5):

$$q_{total}= {\displaystyle \frac{\sigma \left({T_h}^4-{T_c}^4\right)}{\left(\frac{1}{{\epsilon }_h}+\frac{1}{{\epsilon }_c}-1\right)}}+C_{S}fk\left({\displaystyle \frac{T_h-T_c}{∆x}}\right)+C_G{P}_{\alpha }\left(T_h-T_c\right)$$

(6)

Reflector material is responsible for the radiation between shields and radiant heat transfer is proportional to the $\epsilon$ of shields.

In the present study, heat transfer by conduction was incorporated into the model by considering two dominant mechanisms. The first is solid conduction through spacers that maintain the separation between adjacent reflective layers in the multilayer insulation (MLI) system. The spacers used in this study exhibit a high porosity exceeding 90%, which significantly reduces the effective solid–solid contact area between layers and thereby suppresses conductive heat transfer. Such highly porous spacers play a critical role in preserving the thermal performance of vacuum insulation systems. The second mechanism is free molecular gas conduction caused by residual gas molecules within the evacuated annular space. As the vacuum level improves, the mean free path of gas molecules becomes larger than the characteristic spacing between the bounding surfaces, and the heat transfer regime transitions into the free molecular flow regime. In this regime, the conductive heat transfer is approximately proportional to the absolute gas pressure.

2.2. Governing Equations for Cryogenic Fluid Flow and Phase Change Modeling

Under the unsteady state assumption, the governing equations are considered with the Reynolds density-averaged variables in Cartesian tensor notation [40]. The conservation equations of mass and momentum for the phase change analysis at the interface are described as follows:

$${\displaystyle \frac{\partial \rho }{\partial t}}+{\displaystyle \frac{\partial }{\partial x_i}}\left(\rho u_i\right)=0$$

(7)

$${\displaystyle \frac{\partial }{\partial t}}\left(\rho u_i\right)+{\displaystyle \frac{\partial }{\partial x_j}}\left(\rho u_i{u}_j\right)=-{\displaystyle \frac{\partial \rho }{\partial x_i}}+{\displaystyle \frac{\partial }{\partial x_j}}\left[\mu \left({\displaystyle \frac{\partial u_i}{\partial x_j}}+{\displaystyle \frac{\partial u_j}{\partial x_i}}-{\displaystyle \frac{2}{3}}{\delta }_{ij}{\displaystyle \frac{\partial u_l}{\partial x_l}}\right)\right]+{\displaystyle \frac{\partial (-\rho \overline{u_i^{\prime }{u}_j^{\prime }})}{\partial x_j}}$$

(8)

$${\displaystyle \frac{\partial }{\partial t}}\left(\rho E\right)+{\displaystyle \frac{\partial }{\partial x_i}}\left[u_i\left(\rho E+p\right)\right]={\displaystyle \frac{\partial }{\partial x_j}}\left[\left(k_h+{\displaystyle \frac{{C_p\mu }_t}{{Pr}_t}}\right){\displaystyle \frac{\partial T}{\partial x_j}}+u_i{\left({\tau }_{ij}\right)}_{eff}\right]$$

(9)

where $\rho , p, t, {\delta }_{ij}, E, k_h, C_p, {\mu }_t, {Pr}_t, T,$ and ${\left({\tau }_{ij}\right)}_{eff}$ are the density, pressure, time, Kronecker delta tensor, total energy, thermal conductivity, specific heat, turbulence viscosity, turbulent Prandtl number, and deviatoric stress tensor, respectively. The total energy and deviatoric stress tensor can be expressed as follows:

$$E=h- {\displaystyle \frac{p}{\rho }}+ {\displaystyle \frac{u^2}{2}}$$

(10)

$${\left({\tau }_{ij}\right)}_{eff}={\mu }_{eff}\left({\displaystyle \frac{\partial u_j}{\partial x_i}}+{\displaystyle \frac{\partial u_i}{\partial x_j}}\right)-{\displaystyle \frac{2}{3}}{\mu }_{eff}{\displaystyle \frac{\partial u_i}{\partial x_i}}{\delta }_{ij}$$

(11)

Here, ${\mu }_{eff}$ represents the effective viscosity. The energy equation is coupled to the conservation of mass and momentum equations. The superscripts and denote the turbulent fluctuating component and Reynolds time-averaged component, respectively.

To accurately represent these physical characteristics, both solid conduction through spacers and free molecular gas conduction were explicitly incorporated into the mathematical heat transfer model. This integrated approach enables realistic prediction of heat transfer behavior under varying vacuum insulation conditions in the double-shell cryogenic storage system. To estimate the pressure rise rate in a cryogenic liquid storage tank, it is essential to employ a physical model that accurately represents the phase-change behavior of cryogenic fluids. Among various approaches, the Lee phase-change model has been widely adopted because it effectively describes evaporation and condensation processes through source-term formulations. In addition, the Lee model is compatible with both mixture and Volume-of-Fluid (VOF) multiphase modeling frameworks, making it suitable for simulating two-phase flow with phase change in cryogenic storage systems. The governing expressions of the Lee model can be written as follows [41]:

$$\dot{m_{lv}}=L_{lv}{\alpha }_l{\rho }_l\left(T-T_{sat}\right), \left(vaporization\right)$$

(12)

$$\dot{m_{vl}}=L_{vl}{\alpha }_v{\rho }_v\left(T_{sat}-T\right), \left(liquefaction\right)$$

where $\alpha$ is the volume fraction, and $T_{sat}$ is the saturation temperature. When a cryogenic liquid vaporizes or a gaseous liquid condenses, the heat energy is either absorbed or released, which can be written as:

$$S=\dot{m}H$$

(13)

where S is the heat energy, and $H$ is the latent heat. In the Lee model, the phase change occurs based on the saturation temperature of each phase and the mass transfer rate. The mass flow rate $\dot{m}$ of a cryogenic liquid undergoing a phase change is determined by the empirical coefficient L.

All the governing equations, along with the corresponding boundary conditions, were solved using the FVM.

3. Methods

3.1. Experimental Setup

Figure 3 illustrates the experimental facility used to measure boil-off gas (BOG) generation in a Parity 350 L vacuum–MLI double-shell cryogenic storage tank (Anyang, Korea) under both static and dynamic operating conditions. The experimental system consists of the cryogenic storage tank, insulation system, measurement instrumentation, and vibration testing equipment. The vacuum multilayer insulation (MLI) system comprises 40 reflective stainless-steel layers separated by spacer materials, providing a total insulation thickness of 50 mm between the inner pressure vessel and the outer containment shell.

Vibration characteristics were monitored using a Kistler accelerometer (Winterthur, Switzerland) with a measurement range of ±250 g and a frequency response from 0.5 to 10,000 Hz (sensitivity: 20.16 mV/g). Vacuum conditions within the insulation space were monitored using a Thyracont VSM vacuum sensor (Pfaffenhofen, Germany), which provides an operating range from 750 to 5 × 10⁻⁹ Torr with an accuracy of ±10% and a repeatability of ±2%. Internal temperature measurements were performed using Lakeshore temperature sensors (Westerville, OH, USA), offering high accuracy of ±22 mK at liquid nitrogen temperature (77 K). In addition, the evaporated gas flow rate was measured using an Aalborg mass flow meter (Orangeburg, NY, USA) with a measurement range of 0–20 L/min and a calibration accuracy of ±1.0%. The vacuum level in the annular insulation space was maintained at 4.0 × 10⁻³ torr to minimize heat ingress due to residual gas conduction. A dedicated vacuum injection system, consisting of a Kodivac vacuum pump (Gyeongsan, Korea) and pressure gauge, was employed to continuously monitor and maintain the required vacuum conditions throughout the experiments. Liquid nitrogen (LN₂) was selected as the working fluid for the experiments. Cas A digital weighing scale (Yangju, Korea) with an accuracy of ±0.1 kg was used to measure the real-time mass of LN₂, enabling calculation of the boil-off rate (BOR). Pressure transducers were installed to record the internal pressure of the storage tank, and safety relief valves were configured to activate at 1.5 MPa to prevent over-pressurization. To investigate the effects of dynamic excitation, the storage tank was mounted on an electrodynamic vibration test equipment capable of applying horizontal sinusoidal excitation. The vibration system included a controller and acceleration sensors to regulate and monitor the excitation frequency and amplitude. In road transportation, heavy-duty trucks experience continuous vibrations in the 5–15 Hz range due to engine operation and road excitations. Similarly, in maritime applications, mechanical vibrations from propulsion systems and environmental interactions contribute within the 1–15 Hz frequency band, despite the lower frequencies of primary ship motions. Accordingly, excitation frequencies of 5 Hz and 10 Hz were selected as representative operating conditions to evaluate the thermal response and boil-off behavior of the storage tank. Experiments were conducted at three fill ratios (30%, 50%, and 90%) and two excitation frequencies (5 Hz and 10 Hz) to systematically assess the effect of sloshing dynamics on boil-off gas (BOG) generation.

Prior to each test, the storage tank was pre-cooled by circulating liquid nitrogen to achieve thermal stabilization and minimize thermal stresses. The tank was subsequently filled with LN₂ to the target fill ratio, and the vacuum pump was activated to establish and maintain the prescribed vacuum level in the MLI space. After the internal pressure stabilized, BOG measurements were initiated. Once steady-state conditions were reached, the vibration equipment was activated at the specified excitation frequency to induce sloshing motion within the tank. Each test was continued until the liquid level decreased below the prescribed test range or a stable BOG generation behavior was observed over an extended duration.

3.2. Calculation Conditions

Figure 4 presents a schematic of the double-shell storage tank considered in this study, with key geometric dimensions and the applied heat flux ($q_w$) directions explicitly indicated. consisting of an inner liquid vessel and an outer containment vessel separated by an MLI insulation layer. As summarized in

Table 1. Modeling parameters of double-shell storage tank.

Parameter	Value
Tank capacity	0.127 m3
Breadth	624 mm
Height	612 mm
length	1414 mm

, the numerical calculation focused on the internal fluid domain of the inner tank, which has dimensions of 1.414 mm in length, 612 mm in height, and 624 mm in breadth. The modeling dimensions listed in Table 1 were selected to represent the typical capacity and aspect ratio of cryogenic fuel tanks used in heavy-duty hydrogen commercial vehicles. In the transportation sector, liquid hydrogen storage systems for long-haul trucks and buses are commonly designed within a capacity range that includes tanks on the order of 350 L, as considered in the present study. To ensure accurate numerical predictions while maintaining reasonable computational efficiency, an appropriate mesh configuration was established. Tetrahedral elements were selected for the final mesh configuration based on mesh dependency considerations and the geometric complexity of the storage tank. These elements provide high flexibility for complex geometries and are particularly suitable for modeling curved flow regions and abrupt cross-sectional variations within the tank.

A mesh independence study was conducted to verify the numerical accuracy of the fluid flow and ullage pressure predictions. The study involved successive refinement of the computational grid, including a coarse mesh with a reduced number of nodes and a fine mesh with an increased number of nodes. The numerical results obtained using different mesh resolutions were compared, and the deviations were quantified using the root-mean-square (RMS) differences calculated according to Equation (14). where ${\delta }_i$ represents the relative deviation of the target physical parameter. $X_{coarser,n}$ and $X_{finer,n}$ denote the values computed at the $n$-th time step using the coarser grid and the finer grid, respectively. Based on this analysis, a mesh configuration that ensured mesh-independent results while minimizing computational cost was selected for all subsequent calculations.

Figure 5 illustrates the discretization of the computational domain using a non-uniform mesh refinement strategy. Fine tetrahedral elements were applied to the liquid domain to accurately capture sloshing dynamics and flow-induced pressure variations, while relatively coarser elements were used for the solid tank wall to reduce computational cost.

Table 2. Mesh independence study.

Case No.	Total Number of Nodes	Total Number of Elements	Difference in Ullage Pressure (%)
1	13,338	15,572	1.87
2	24,127	24,287	0.35
3	33,601	32,258	0.10
4	49,709	48,782	Under 0.01

summarizes the ullage pressure values obtained under different mesh resolutions considered in the numerical study. Although variations in the predicted ullage pressure were observed among the different mesh conditions, the resulting deviations were within 0.1%. Based on these results, a mesh configuration consisting of approximately 49,700 nodes was selected as an optimal compromise between computational efficiency and solution accuracy, and this mesh was adopted for all subsequent calculations.

$${\delta }_i=\sqrt{{\left({\displaystyle \frac{X_{coarser,n}-X_{finer,n}}{X_{finer,n}}}\right)}^2}$$

(14)

The governing heat transfer mechanisms, including conduction, radiation, and solid-state conduction, were incorporated into the numerical framework using a user-defined function (UDF) implemented in ANSYS Fluent (2024 R2). Turbulence effects were modeled using the shear stress transport (SST) k − ω model with low-Reynolds-number corrections, which was selected to accurately resolve near-wall viscous sublayer behavior as well as turbulence characteristics near the liquid–wall interface and tank bottom. The pressure–velocity coupling was handled using a coupled solution algorithm, and all spatial discretization terms were approximated using second-order accurate schemes to ensure numerical accuracy.

Table 3. Boundary and initial conditions.

Parameters	Values
Material	Gas nitrogen, Liquid nitrogen
Turbulence Model	SST $k-\omega$ with low-Re correction
Multiphase Model	VOF (Volume of Fluid)
Phase Change Model	Lee Model (UDF)
Spatial Discretization	Second Order Upwind
Initial pressure/temperature	101.325 kPa/77.39 K
Initial liquefied-nitrogen filling ratio	30, 50, 90%
External Wall (Static)	Natural Convection + Radiation
External Wall (Dynamic)	Forced Convection (UDF)
Internal Wall	Conjugate Heat Transfer (Coupled)
Gravitational acceleration for vibration	a(x) = $A{\omega }^{2}sin\left(\omega t\right)$
Vibration Amplitude ($A$)	4.97 mm (5 Hz), 1.24 mm (10 Hz)
Max Acceleration	4.905 ${\mathrm{m}/\mathrm{s}}^2$

summarizes the boundary and initial conditions applied in the numerical calculation of the double-shell cryogenic storage tank. The initial liquid phase was defined at a near-saturation temperature of 77.39 K under atmospheric pressure (101.325 kPa), while the gas phase was initialized at the same saturation temperature and pressure. Filling ratios of 30%, 50%, and 90% were selected to represent low, medium, and high liquid loading conditions, respectively. The external thermal boundary conditions, including natural/forced convection and radiation, were implemented as transient heat flux values calculated via User-Defined Functions (UDF). Simultaneously, the internal heat transfer between the fluid and the tank wall was modeled using a conjugate heat transfer approach to accurately resolve the local thermal interactions in both liquid and vapor regions. Dynamic excitation was introduced as a body-force boundary condition, with frequency-dependent parameters specified in Table 3 to simulate the effects of vibration. All governing equations and boundary conditions were discretized and solved using the finite volume method (FVM) implemented in the commercial CFD software ANSYS Fluent (2024 R2).

4. Results and Discussion

4.1. Experimental Results Under Static Conditions

The boil-off gas (BOG) generation of liquid nitrogen was experimentally evaluated under static conditions. Figure 6 illustrates the measured BOG generation rate as a function of the filling ratio. The experimental results exhibit a clear monotonic increase in BOG generation with increasing filling level. Specifically, the BOG generation rate was measured to be 1.396 L/min at a 30% filling ratio, 1.665 L/min at 50%, and 1.870 L/min at 90%, corresponding to an overall increase of approximately 34% across the investigated range. The observed increasing trend with filling level can be attributed to two cumulative effects: (i) an increase in the heat transfer area in contact with the tank wall and (ii) a reduction in the evaporative cooling capacity within the ullage space. As the filling level increases, the liquid–wall contact area becomes larger, resulting in enhanced heat transfer from the tank wall to the liquid through conduction and natural convection. Consequently, a greater amount of thermal energy is supplied to the cryogenic liquid, promoting vaporization. In addition, an increase in the filling level reduces the ullage volume, leading to a more rapid rise in the ullage internal pressure. The elevated ullage pressure increases the saturation temperature at the liquid–gas interface, thereby intensifying the temperature gradient near the interface. This steeper temperature gradient creates more favorable conditions for phase change, further accelerating vapor generation. As a result of these combined mechanisms, higher filling levels lead to increased BOG generation and a more pronounced pressure rise under static conditions.

4.2. Experimental Results Under Dynamic Conditions

4.2.1. Dynamic Conditions (5 Hz)

Figure 7 presents the time history of boil-off gas (BOG) generation under 5 Hz dynamic excitation for three filling ratios (30%, 50%, and 90%) over test durations of 6–24 h. Compared with static conditions, several distinct characteristics are observed.

During the initial transient phase (0–1000 s), all filling ratios exhibit markedly elevated BOG generation rates in the range of approximately 3–5 L/min. This behavior is attributed to pressure and temperature perturbations induced by sloshing motion, which intensifies interfacial heat and mass transfer. The initial BOG rates are substantially higher than those observed under static conditions, indicating an immediate acceleration of evaporation at the liquid–vapor interface caused by dynamic excitation. This transient response decays rapidly, after which the BOG generation rate stabilizes. A second dominant feature under the 5 Hz dynamic excitation condition is the rapid decay from peak BOG values to a quasi-steady state within approximately 5000–10,000 s (1.4–2.8 h), a phenomenon observed consistently across all filling ratios. At a 30% filling ratio, the BOG rate decreases from approximately 4.6 L/min to a steady-state average of 1.596 L/min (corresponding to a BOR of 1.133%/day), representing a reduction of about 65%. At 50% filling, the BOG rate decays from approximately 3.7 L/min to 1.961 L/min (BOR: 1.392%/day), corresponding to a 47% reduction. At 90% filling, the BOG rate decreases from approximately 5.6 L/min to 1.680 L/min (BOR: 1.192%/day), representing a reduction of about 70%. This behavior contrasts sharply with static conditions, under which BOG generation increases monotonically due to continuous pressure buildup. Notably, the 50% filling case exhibits the highest steady-state BOG generation rate among the three conditions, resulting in a local maximum at the intermediate filling ratio. This deviation from monotonic behavior may partly arise from uncertainty in the actual filling ratio, as uncontrolled vapor release during the stabilization process can lead to deviations from the nominal 50% condition. Finally, once a steady state is established (after approximately 10,000 s), the BOG generation rate exhibits periodic oscillations superimposed on a relatively constant mean value. These oscillations reflect the cyclic nature of sloshing-induced interfacial disturbances. In contrast to the smooth, monotonic increase observed under static conditions, the dynamic steady state is characterized by pressure-dominated stabilization, in which rapid pressurization suppresses sustained boiling despite the continued presence of sloshing motion. Overall, under 5 Hz dynamic excitation, the cryogenic storage system undergoes an initial perturbation-dominated phase followed by stabilization governed by pressure-limited saturation conditions. The resulting steady state represents a dynamic equilibrium among heat input to the liquid, evaporative losses, and ullage pressure rise, fundamentally differing from the behavior observed under static conditions.

4.2.2. Dynamic Conditions (10 Hz)

Figure 8 illustrates the time history of BOG generation under 10 Hz dynamic excitation for filling ratios of 30%, 50%, and 90% over test durations of 6–10 h. Compared with the 5 Hz condition, the BOG generation characteristics exhibit notable differences, highlighting the strong influence of excitation frequency on cryogenic fluid behavior. At a 30% filling ratio, the BOG generation rate remains relatively stable throughout the test duration, with a mean value of 1.407 L/min (BOR: 0.999%/day). The flow rate shows minimal temporal fluctuation, indicating stabilized heat transfer conditions. Unlike the pronounced transient behavior observed at 5 Hz, the 10 Hz excitation suppresses sloshing amplitude at this filling level, resulting in reduced interfacial disturbance and more stable BOG generation. At a 50% filling ratio, the BOG generation rate exhibits the highest mean value among the three cases, reaching 1.788 L/min (BOR: 1.269%/day). The flow rate shows moderate variability with a gradual declining trend over time. As observed under 5 Hz excitation, the intermediate filling ratio yields a local maximum in steady-state BOG generation, reflecting an optimal balance between available liquid mass and ullage volume. The elevated response may also be influenced by filling-ratio uncertainty caused by uncontrolled vapor release during pressurization and stabilization. At a 90% filling ratio, the mean BOG generation rate is 1.559 L/min (BOR: 1.107%/day), representing an intermediate response between the 30% and 50% cases. The BOG flow rate exhibits sustained fluctuations with a gradually decreasing trend, indicating continued but weakening interfacial heat transfer. Although high filling levels typically suppress sloshing due to reduced ullage volume, the 10 Hz excitation appears to maintain sufficient localized interfacial perturbations to sustain BOG generation rates comparable to those observed at lower filling ratios.

4.3. Numerical Results

4.3.1. Numerical Validation

The numerical validation study was conducted by comparing the numerical predictions with experimental results reported by Vishnu and Biju [42]. The numerical simulations were performed under conditions identical to those of the experiments, namely a high-vacuum environment of 1.0 × 10⁻⁵ torr with a 40-layer multilayer insulation (MLI) system and a total insulation thickness of 50 mm. The heat transfer model was formulated based on McIntosh’s approach for total heat transfer through MLI and was implemented using a user-defined function (UDF), ensuring consistency with the experimental conditions.

Figure 9 presents a comparison between the experimental measurements and numerical predictions of the time history of internal pressure within the storage tank under MLI conditions. The numerical results exhibited deviations of approximately 2.3% during the initial stage and about 3.8% after 30 min when compared with the experimental data. This level of accuracy is considered favorable when compared with previous numerical studies on cryogenic storage tanks, such as those reported by Barsi [23] and Ludwig and Dreyer [27]. The observed discrepancies are primarily attributed to transient effects associated with the initial thermal stabilization process. Overall, the numerical model consistently reproduced the trend of pressure rise associated with phase-change phenomena over time and showed good agreement with the results reported by Vishnu and Biju [42]. These findings demonstrate that the present computational methodology is reasonable and reliable for predicting pressure behavior in cryogenic storage tanks under MLI conditions.

4.3.2. Numerical Results Under 5 Hz Dynamic Excitation

Figure 10 shows the numerical results of dynamic sloshing behavior of liquid nitrogen inside the cryogenic storage tank under 5 Hz dynamic excitation for three filling ratios (30%, 50%, and 90%). The numerical results reveal distinct sloshing patterns that strongly depend on the initial liquid volume. At a 30% filling ratio, the free surface exhibits high-frequency oscillations concentrated near the tank centerline, with relatively low wave amplitudes. The liquid–vapor interface remains largely confined to the upper liquid region, showing minimal wave penetration toward the tank walls during the transient phase. As the system evolves toward a quasi-steady state over approximately 5–10 s, wave activity gradually diminishes, and the interface stabilizes with low-amplitude oscillations persisting throughout the simulation. At a 50% filling ratio, more pronounced interfacial disturbances are observed, characterized by asymmetric wave patterns during the transient phase (1–5 s). The liquid–vapor interface exhibits larger wave amplitudes and a broader spatial distribution across the tank width compared to the 30% case. Waves propagate from the tank centerline toward both sidewalls, producing highly dynamic interfacial motion before transitioning into a steady-state regime. Among the three filling ratios examined, the 50% filling condition demonstrates the most energetic sloshing behavior. At a 90% filling ratio, the confined ullage space above the liquid significantly restricts interfacial motion. The free surface exhibits highly localized disturbances concentrated primarily near the tank walls, while the central region of the interface remains relatively flat throughout the simulation. Wave amplitudes are substantially suppressed compared with lower filling ratios, and the interface rapidly transitions into a damped oscillatory state. The non-linear relationship between filling ratio and sloshing intensity—characterized by maximum interfacial disturbance at a 50% filling ratio—reflects the interplay between liquid inertia and ullage confinement. Lower filling ratios provide sufficient vapor volume for liquid motion but limited absolute liquid mass, whereas higher filling ratios confine the ullage and restrict interfacial acceleration. The intermediate filling ratio represents an optimal condition in which sufficient liquid mass and available ullage volume combine to produce maximum sloshing intensity.

Figure 11 presents the temperature distribution inside the storage tank under 5 Hz excitation for filling ratios of 30%, 50%, and 90%. The results reveal distinct thermal stratification patterns governed by the initial liquid volume and sloshing intensity. At a 30% filling ratio, the temperature field exhibits minimal spatial variation within the liquid domain during the early stages of the simulation. Elevated temperatures are primarily observed near the liquid–vapor interface, while the bulk liquid remains close to the saturation temperature. Thermal stratification along the tank axis is weak, indicating that convective mixing dominates over thermal diffusion during the initial transient phase. As time progresses, the thermally stratified layer shows negligible growth toward the bulk liquid. At a 50% filling ratio, pronounced thermal stratification develops, with distinct temperature gradients forming between the upper and lower liquid regions. Over time, the stratified layer extends progressively toward the bulk liquid. Steep and finely resolved temperature gradients are observed near the interface, where sloshing-induced convection enhances heat transfer at the liquid–wall boundary. The 50% filling case exhibits the most significant thermal response, characterized by gradual and spatially distributed temperature evolution throughout the liquid domain. At a 90% filling ratio, the temperature field displays strong localized thermal stratification despite the suppressed sloshing amplitude associated with the confined ullage space. The restricted vapor volume forces sloshing-induced motion and heat transfer to occur predominantly near the upper tank walls, resulting in steep and sharply defined temperature gradients in these regions. Although overall convective mixing is weaker than in lower filling cases, the constrained geometry concentrates liquid motion within a limited region, producing pronounced temperature discontinuities adjacent to the wall–interface boundary. Temperature elevations are highly localized near the upper walls, while the bulk liquid remains relatively cool and nearly isothermal.

Overall, the filling-ratio-dependent thermal response reflects the coupled effects of sloshing intensity, ullage confinement, and heat transfer efficiency. At a 30% filling ratio, sloshing is distributed over a large volume, resulting in weak bulk thermal stratification. At a 50% filling ratio, moderate confinement combined with vigorous sloshing produces optimal conditions for distributed thermal stratification development. At a 90% filling ratio, severe confinement leads to steep but highly localized temperature gradients concentrated near the tank walls. Consequently, both the spatial distribution and penetration depth of thermal stratification vary significantly with filling ratio, with the 50% case exhibiting the most extensive thermal evolution into the bulk liquid and the 90% case showing the sharpest but most localized temperature gradients.

4.3.3. Numerical Results Under 10 Hz Dynamic Excitation

Figure 12 illustrates the sloshing behavior of liquid nitrogen inside the cryogenic storage tank under 10 Hz dynamic excitation for three filling ratios (30%, 50%, and 90%). Compared with the 5 Hz excitation condition, the results reveal markedly suppressed interfacial disturbances, highlighting the critical influence of excitation frequency on sloshing wave formation and amplitude. At a 30% filling ratio, the free surface exhibits minimal wave activity with very low amplitudes throughout the 10 s observation period. The liquid–vapor interface remains largely flat, with only subtle undulations appearing at later times. In contrast to the pronounced ripples observed under 5 Hz excitation, the 10 Hz excitation produces strongly damped wave motion, indicating that viscous damping at higher frequency overcomes the driving forces responsible for interfacial disturbances. Consequently, the interface rapidly transitions to a damped oscillatory state with negligible wave amplitude. At a 50% filling ratio, the free surface shows more pronounced deformation than in the 30% case, although the interfacial motion remains substantially damped compared to that observed at 5 Hz. Asymmetric wave patterns appear at later times (5 s and 10 s), with localized perturbations concentrated near the upper liquid region. The waves exhibit lower amplitudes and longer wavelengths than those observed under 5 Hz excitation. Despite the overall suppression, the 50% filling condition still demonstrates moderately enhanced interfacial disturbance relative to the 30% and 90% cases, indicating that filling-ratio-dependent effects persist even at higher excitation frequencies. At a 90% filling ratio, the free surface remains nearly flat throughout the observation period. Only minor localized perturbations are observed near the tank walls at later times, while the central interface region maintains a flat profile. The combination of a confined ullage space and high-frequency excitation effectively suppresses interfacial motion, restricting surface deformation primarily to wall-adjacent regions.

A comparison between the 5 Hz and 10 Hz excitation conditions reveals clear frequency-dependent effects on sloshing dynamics. At 5 Hz, vigorous interfacial wave motion develops, with amplitudes increasing from 30% to 50% filling. In contrast, at 10 Hz, interfacial motion is significantly damped across all filling ratios, with wave amplitudes dramatically reduced. This suppression reflects enhanced viscous damping at higher excitation frequencies. Nevertheless, the filling-ratio hierarchy remains evident, with the 50% filling ratio consistently exhibiting moderately higher interfacial response than the 30% and 90% cases, confirming that geometric confinement effects remain operative regardless of excitation frequency. These trends are generally consistent with previous sloshing studies on cryogenic tanks, which reported that increasing excitation frequency in the low-frequency range (0.53–3 Hz) intensifies sloshing and enhances heat transfer, thereby increasing BOG generation [43]. In the present study, a similar tendency is observed at 5 Hz; however, at 10 Hz, interfacial motion becomes strongly damped, leading to more stable BOG behavior while the filling-ratio dependence remains evident.

Figure 13 presents the temperature distribution inside the storage tank under 10 Hz excitation for filling ratios of 30%, 50%, and 90%. Compared with the 5 Hz condition, the results show substantially reduced development of thermal stratification, reflecting the suppressed sloshing dynamics at higher excitation frequency. At a 30% filling ratio, the temperature field remains relatively uniform throughout the liquid domain at both observed time instances. Temperature variations are minimal, with slightly elevated temperatures localized near the upper walls, while the bulk liquid remains close to the saturation temperature. Thermal stratification along the tank axis is weak, indicating that convective mixing dominates over thermal diffusion, thereby limiting the development of temperature gradients. The liquid–vapor interface exhibits minimal temperature variation despite exposure to external heat flux. At a 50% filling ratio, more pronounced temperature gradients develop compared to the 30% case, particularly evident at early times (0.5 s). Elevated temperatures appear near the upper walls and interface region, producing a moderate gradient between the upper and lower liquid regions. A thermally stratified layer begins to form in the upper portion of the liquid; however, the overall degree of stratification remains modest relative to that observed under 5 Hz excitation. The spatial extent of elevated temperature remains confined to the upper region, while the bulk liquid remains comparatively cool. Among the three cases, the 50% filling ratio exhibits a moderately enhanced thermal response. At a 90% filling ratio, the temperature distribution exhibits the most pronounced gradients, particularly at early times (0.5 s). Significant temperature elevations are observed near the upper walls and interface region, with steep gradients forming in wall-adjacent zones. Despite the confined ullage space and suppressed sloshing amplitude, localized wall heating under 10 Hz excitation generates sharp temperature discontinuities near the upper walls. The interior liquid remains relatively cool, resulting in strong thermal stratification concentrated within the boundary layer. This behavior reflects the competing effects of suppressed global convection and concentrated wall heat transfer under geometrically confined conditions.

A comparison of thermal behavior between 5 Hz and 10 Hz excitation conditions highlights substantial frequency effects. At 5 Hz, distinct temperature gradients develop across all filling ratios, with thermally stratified layers progressively penetrating into the bulk liquid. In contrast, under 10 Hz excitation, thermal stratification becomes weaker and more localized: the 30% and 50% filling cases exhibit suppressed temperature development, while the 90% filling case shows sharp temperature gradients concentrated near the upper walls. This frequency-dependent shift—from sloshing-driven bulk thermal stratification at 5 Hz to wall-dominated localized heating at 10 Hz—reflects the transition from vigorous convective mixing to confinement-controlled heat transfer. The intensified wall-adjacent thermal gradients observed at high filling ratios under 10 Hz excitation demonstrate that heat transfer efficiency is governed by the complex interplay between sloshing suppression and geometric confinement, leading to a non-linear filling-ratio dependence of thermal response.

4.4. Discussion for the BOG Phenomenon

Figure 14 compares the effect of filling ratio on the boil-off rate (BOR) under static, 5 Hz, and 10 Hz operating conditions, using both experimental measurements and numerical predictions. The BOR was evaluated by integrating the heat flux transferred from the tank contact surfaces to the gas phase. A clear distinction is observed between static and dynamic operating conditions. Under static conditions, the BOR increases monotonically with filling ratio, rising from approximately 1.45%/day at 30% filling to 1.50%/day at 90% filling. This trend reflects a relatively simple heat transfer mechanism dominated by conductive heat inflow proportional to the liquid–wall contact area. In contrast, under dynamic conditions (5 Hz and 10 Hz), both experimental measurements and numerical calculations exhibit a pronounced non-linear dependence on filling ratio. The BOR reaches a maximum near the intermediate filling ratio of 50%—approximately 1.4%/day at 5 Hz and 1.3%/day at 10 Hz—followed by a decrease toward 90% filling. The experimentally measured BOR at 50% filling is slightly higher than the numerically predicted value, which may be attributed to uncertainty in the effective filling ratio during the stabilization process. In particular, uncontrolled vapor release during pressurization may cause deviations from the nominal 50% filling condition. This non-linear filling-ratio dependence under dynamic excitation reflects the competing influences of sloshing-induced heat transfer enhancement and geometric confinement. At lower filling ratios, increased interfacial disturbance and enhanced effective thermal conductivity promote greater heat inflow to the liquid–vapor interface. As the filling ratio increases, however, the ullage volume becomes increasingly confined, suppressing sloshing amplitude despite the larger liquid–wall contact area. The 50% filling condition represents an optimal balance, where sufficient ullage space allows vigorous sloshing-induced mixing while maintaining adequate liquid mass for efficient heat transfer. Across all filling ratios, the 5 Hz excitation consistently yields higher BOR values than the 10 Hz condition, owing to more energetic sloshing dynamics, clearly demonstrating the influence of excitation frequency on heat transfer and BOG generation. Across all filling ratios, the experiments consistently show higher boil-off rates (BORs) at an excitation frequency of 5 Hz compared to 10 Hz. Although higher excitation frequencies generally imply greater energy input, the dominance of the 5 Hz condition can be attributed to its proximity to the natural sloshing frequency of the liquid. This near-resonant condition significantly amplifies sloshing amplitude and interfacial disturbances, in contrast to the 10 Hz case, where viscous damping and wave-breaking effects suppress the overall liquid motion. While the numerical model predicts higher BORs at 10 Hz due to the direct proportionality between excitation frequency and imposed mechanical energy, the experimental results indicate that, under realistic operating conditions, resonance-driven sloshing dynamics play a more critical role in enhancing heat transfer and BOG generation. This discrepancy highlights that the influence of excitation frequency on BOG behavior cannot be described solely by energy scaling arguments; instead, it involves complex interactions between sloshing dynamics, viscous dissipation, and interfacial heat and mass transfer processes.

Comparison with previous studies further supports these observations. Jeon et al. [32] and Lee et al. [44] reported BOR values in the range of approximately 4.0–4.5%/day under different experimental configurations, whereas the present study yields a BOR of approximately 4.9%/day. Although the absolute BOR values differ due to variations in tank scale, initial pressure conditions, and insulation design, the observed filling-ratio-dependent trends are in good qualitative agreement. This consistency, together with the close agreement between experimental and numerical results obtained in the present study, validates the numerical calculation methodology and confirms that sloshing-induced BOG generation mechanisms remain robust across a wide range of operating conditions.

5. Conclusions

This study investigated the effects of fill ratio (FR) and dynamic operating conditions on the boil-off rate (BOR) of a cryogenic storage tank using an integrated experimental and numerical approach. A validated numerical model, incorporating MLI performance and the Lee phase-change model, was developed to predict pressure behavior and BOR under various excitation frequencies. The key findings are summarized as follows:

⬝

The numerical model was successfully validated against experimentally measured boil-off gas (BOG) data under all tested conditions. The transient VOF-based phase-change approach effectively captured interfacial heat and mass transfer, confirming the reliability of the model for predicting BOG generation under dynamic sloshing conditions.

⬝

Sloshing behavior exhibits a strong frequency dependence, significantly influencing heat transfer and BOG generation. At 5 Hz, dynamic sloshing motion promotes distributed thermal stratification, resulting in a higher BOR (1.4%/day at a 50% fill ratio). In contrast, at 10 Hz, increased viscous damping suppresses sloshing amplitude, leading to localized wall heating and a reduced overall BOR (1.3%/day at a 50% fill ratio).

⬝

The effect of fill ratio under dynamic conditions shows a nonlinear trend, in clear contrast to static scenarios. Unlike static conditions, the dynamic BOR reaches a maximum at a 50% fill ratio. This peak reflects an optimal balance between enhanced interfacial disturbances due to sloshing at lower fill levels and geometric constraints—such as reduced ullage volume and weakened convective mixing—at higher fill levels.

⬝

The development of thermal stratification is strongly dependent on excitation frequency. At 5 Hz, temperature gradients develop uniformly throughout the liquid domain and gradually intensify over time. At 10 Hz, thermal generation is significantly suppressed at 30% and 50% fill ratios, whereas at a 90% fill ratio, sharp and localized temperature gradients form near the tank wall due to geometric confinement and limited convective mixing.

⬝

The proposed experimental–numerical framework enables reliable prediction of thermal behavior, pressurization, and BOR in cryogenic storage tanks under realistic dynamic operating conditions. The findings provide valuable insights for the design optimization of liquid hydrogen (LH₂) storage tanks for marine transportation, as well as cryogenic fuel tanks for space propulsion systems.