Insulation Performance of Vacuum-MLI Cavity Under Varying Residual Gas Pressure: Analytical Study and Application to Liquid Hydrogen System

Abstract

In this study, the influence of residual gas pressure within a vacuum insulation cavity on the insulation performance of a multi-layer insulation (MLI) system was investigated through thermal analysis. Based on an electrical analogy, a thermal resistance network was constructed, considering heat transfer through the insulation system by gas conduction, solid conduction, and surface radiation. Lees’ four-moment model was employed to calculate the gas conduction across a wide range of vacuum conditions, including medium-to-low vacuum situations. The analysis shows that total heat flux and effective thermal conductivity exhibited non-linear increases as the pressure approached atmospheric level. This trend was successfully validated by comparisons with experimental data from the literature, thereby confirming the rationality of the proposed analytical model. Furthermore, the contributions of individual heat-transfer modes to the total heat flux within the insulation system were scrutinized, thereby revealing their redistribution patterns. Under high-vacuum conditions, solid conduction and radiation were the primary modes of heat transfer. However, with increasing pressure, the proportion of gas conduction rose markedly, becoming the primary heat-transfer mode under medium-vacuum and low-vacuum conditions. Finally, a validated analytical technique was utilized to predict heat-transfer characteristics under cryogenic boundary conditions associated with liquid hydrogen storage.

1. Introduction

Cryogenic fluids, such as liquefied natural gas (LNG), liquid hydrogen (LH₂), and liquid nitrogen (LN₂), are characterized by extremely low boiling points, making them highly susceptible to significant vaporization losses even with small amounts of heat intake [1,2]. These losses are responsible for economic disadvantages as well as severe safety risks like structural damage from rapid internal pressure escalation, and in extreme cases, catastrophic events like explosions [3]. Effective thermal management through the use of high-performance insulation technologies is essential for the storage and transport of these fluids [4].

Among the various materials used for insulating tanks and pipelines containing cryogenic fluids, multi-layer insulation (MLI) material consists of spacers with low thermal conductivity and reflectors, as illustrated in Figure 1. This configuration offers excellent insulation performance by minimizing solid conduction and radiative heat transfer from the external environment to the cryogenic fluid [5]. When combined with a vacuum insulation system, MLI material also reduces heat transfer by gas, thereby further enhancing the thermal barrier capability [6,7].

The insulation characteristics of a vacuum-MLI system depend on the pressure conditions produced by the residual gases within the vacuum regime [8]. Therefore, to apply this system in storage vessels and transport pipelines, its characteristics under different residual gas pressures must be evaluated. However, experimental evaluations are cost- and time-intensive. In contrast, thermal analysis can be performed relatively economically and rapidly, and furthermore, it enables the circumstantial prediction of heat-transfer characteristics within the insulation system, which are challenging to observe through testing.

The heat-transfer characteristics of MLI under vacuum insulation conditions have been examined in many previous studies using thermal analyses [9,10,11,12,13,14,15,16,17]. However, most of these studies were conducted under high-vacuum conditions, where the absolute residual gas pressure was very low; there are very few similar studies under medium-to-low vacuum conditions. Maintaining high-vacuum conditions poses significant technical challenges, and entails substantial economic and safety concerns [18]. Thus, MLI application should be considered not only under high-vacuum conditions, but also under medium-to-low vacuum conditions that may arise due to vacuum degradation or breakage. This requirement necessitates research focused on predicting heat-transfer characteristics within these pressure ranges.

In this study, one-dimensional steady-state thermal analysis of a vacuum-MLI system was conducted on the basis of an electrical analogy to determine the effect of residual gas pressure on heat-transfer characteristics. The heat-transfer modes considered in the calculations were surface radiation, solid conduction, and gas conduction. An appropriate model for gas conduction was selected to predict heat transfer under medium-to-low vacuum conditions. The insulation performance was evaluated based on the calculated effective thermal conductivity, which was successfully validated through the experimental data obtained by Fesmire and Johnson using LN₂ [19]. In addition, the contribution of each heat-transfer mode to the total heat transfer within the insulation system was investigated in detail. Finally, thermal insulation performance under the temperature boundary of LH₂ was predicted using the validated analytical method.

2. Calculation Methods

2.1. Thermal Resistance Circuit

The physical model for the calculation, representing the boundary exposure (cryogenic fluid and ambient air) and the vacuum-MLI cavity, is schematically shown in Figure 2. The thermal resistance circuit employed for the thermal analysis of the insulation system is shown in Figure 3; radiative and conductive thermal resistances (R) were connected in parallel for each layer, with each resistance calculated as the reciprocal of the corresponding thermal conductance [20].

The thermal conductance for solid conduction at each node ($G_{s,n}$) was determined using the following equation, validated for materials like Dacron, silk, and glass paper [13]:

$$G_{s,n}={\displaystyle \frac{Cf{k}_n}{x}}$$

(1)

where $C$ is an empirical constant used to fit the conductivity relation to experimental data, $f$ is the relative density defined as the ratio of the bulk density of the spacer to the density of the material in its solid form, $k$ is the thermal conductivity, and $x$ is the thickness of a single layer. In addition, the subscript $n$ denotes the node number.

The following assumptions were made regarding the media used in the radiative heat-transfer analysis:

• Considering the actual operating environment, nitrogen, a radiatively non-participating medium, was chosen as the residual gas. Therefore, thermal radiation of the gas was neglected.
• In spacers with very high porosity, the insignificant radiative heat transfer was neglected [21,22].
• The spacing between reflectors was sufficiently narrow to approximate them as parallel plates. Consequently, the view factor was fixed at 1.

Thus, only the surface radiation between adjacent reflectors was considered; the radiative thermal conductance ($G_{r,n}$) is expressed as

$$G_{r,n}={\sigma \left({\displaystyle \frac{1}{{\epsilon }_n}}+{\displaystyle \frac{1}{{\epsilon }_{n-1}}}-1\right)}^{-1}\left(T_n^2+T_{n-1}^2\right)\left(T_n+T_{n-1}\right)$$

(2)

where $\epsilon$ is the emissivity, $\sigma$ is the Stefan-Boltzmann constant ($5.6705\times {10}^{-8}\mathrm{W}/{\mathrm{m}}^2{\mathrm{K}}^4$), and $T$ is the absolute temperature.

Finally, Lees’ four-moment model [23] was used to calculate the thermal conductance for gas conduction ($G_{g,n}$):

$$G_{g,n}={\displaystyle \frac{\omega {Kn}_n\left(\frac{2}{\alpha }-1\right)}{1+\omega {Kn}_n\left(\frac{2}{\alpha }-1\right)}}{G}_{g,n}^{FM}$$

(3)

where $\omega$ is the fitting parameter, ${Kn}_n$ is the Knudsen number, and $\alpha$ is the accommodation coefficient. Here, ${Kn}_n$ is expressed as

$${Kn}_n={\displaystyle \frac{k_B{T}_{m,n}}{\sqrt{2}\pi d^{2}Px}}$$

(4)

where $k_B$ is the Boltzmann constant, $d$ is the molecular diameter, $P$ is the absolute pressure [Pa] (1 Pa ≈ 7.501 mTorr), and $T_{m,n}$ is the mean temperature of nodes $n$ and $n-1$.

Lees’ four-moment model is used to estimate gas thermal conductance by extending its value from the free-molecule flow regime to the continuum regime. Therefore, the determination of the thermal conductance in the free-molecule flow regime ($G_{g,n}^{FM}$) is essential and is calculated using the following relation, which is derived from kinetic theory [24,25]:

$$G_{g,n}^{FM}=P\alpha {\displaystyle \frac{\gamma +1}{\gamma -1}}\sqrt{{\displaystyle \frac{R_u}{8\pi M{T}_w}}}$$

(5)

where $\gamma$ is the specific heat ratio, $R_u$ is the universal gas constant ($8.31441\mathrm{J}/\mathrm{m}\mathrm{o}\mathrm{l}\mathrm{K}$), $M$ is the molecular weight, and $T_w$ is the temperature at the warm boundary.

2.2. Calculation Conditions and Variables

The analytical conditions are listed in

Table 1. Calculation conditions employed in the LN₂ case.

MLI composition and properties	Reflector	Type	Double-aluminized Mylar
		Emissivity, ${\epsilon }_n$	Equation (6)
	Spacer	Type	Dacron net
		Conductivity, $k_n$	Equation (7)
	Total number of layers, $N$		40
	Total thickness, $X$		15.5 mm
Boundary conditions	T0 (cold)		78 K
	TN (warm)		293 K
Residual pressures of vacuum-MLI system	$P[\mathrm{m}\mathrm{T}\mathrm{o}\mathrm{r}\mathrm{r}$]		0.002, 0.003, 0.004, 0.01, 0.02, 0.07, 0.1, 0.3, 1, 3, 10, 100, 1040, 10,036, 99,102, 768,585

. The reflector and spacer of the MLI were composed of a double-aluminized Mylar and Dacron net, respectively. A total of 40 layers were stacked with an overall thickness of 15.5 mm, in accordance with previously reported experimental conditions [19]. The emissivity (${\epsilon }_n$) of the reflector and the thermal conductivity ($k_n$) of the spacer are expressed as temperature-dependent functions [13,26]:

$${\epsilon }_n=1.1823\times {10}^{-2}+6.17562\times {10}^{-5}{T}_n$$

(6)

$$k_n=1.7\times {10}^{-2}+7\times {10}^{-6}\left(800-T_{m,n}\right)+2.28\times {10}^{-2}\ln T_{m,n}$$

(7)

The cold and warm boundary temperatures were fixed at 78 and 293 K, respectively. The convergence criterion was defined as a temperature residual below 10⁻⁶. Analyses were conducted for various residual gas pressures.

The properties of the residual gas (GN₂) and the empirical parameters required for analysis are summarized in

Table 2. Residual gas properties and empirical parameters used in calculation.

Variable		Value
Residual Gas	Type	GN2
	Molecular diameter, $d$	3.15 × 10−10 m
	Molecular weight, $M$	28.0 kg/kmol
	Specific heat ratio, $\gamma$	1.4
Accommodation coefficient, $\alpha$		0.76 [28]
Empirical constant, $C$		0.008 [20]
Relative density, $f$		0.02
Fitting parameter, $\omega$		1.4

. The fitting parameters in Lees’ four-moment model, which were introduced to extrapolate the gas conduction from the free-molecule regime through the transition up to the continuum regimes, were proposed to be 3.75 by Lees [27], 3.166 by Teagan [28], and 1.8 by Green [29]. However, in this study, optimization was performed to determine the most suitable value, leading to the selection of ω = 1.4 as the fitting parameter for the analysis.

2.3. Calculation Procedure

Figure 4 illustrates the procedure for calculating the temperature distribution, heat flux, and effective thermal conductivity in the vacuum-MLI system. The process starts with the initialization of the temperature distribution, followed by the computation of the emissivity of the reflectors, thermal conductivity of the spacers, and the Knudsen number. From these values, the thermal conductivities of the gas conduction, solid conduction, and radiation were determined. The total thermal resistance ($R_t$) was calculated as follows:

$$R_{t,n}={\left(G_{s,n}+G_{r,n}+G_{g,n}\right)}^{-1}$$

(8)

The new temperature distribution ($T_n^{new}$) was calculated by

$$T_n=T_0+{\displaystyle \frac{{\sum }_{i=1}^n{R}_{t,i}}{{\sum }_{i=1}^N{R}_{t,i}}}\left(T_N-T_0\right)$$

(9)

The analysis was performed iteratively until the temperature residual summed at all points converged to a predefined value (10⁻⁶). Once the convergence criterion was satisfied, the total heat flux was calculated using Equation (10):

$$q_t={\displaystyle \frac{1}{{\sum }_{i=1}^N{R}_{t,i}}}\left(T_N-T_0\right)$$

(10)

Finally, the effective thermal conductivity across the total thickness was determined through Equation (11):

$$k_{eff}={\displaystyle \frac{q_{t}X}{\left(T_N-T_0\right)}}$$

(11)

3. Results and Discussion

3.1. Total Heat Flux and Effective Thermal Conductivity

Figure 5 shows the calculated total heat flux ($q_{t,cal}$) and effective thermal conductivity ($k_{eff,cal}$) as functions of the residual gas pressure. These calculated values were compared with the corresponding experimental data ($q_{t,exp}$ and $k_{eff,exp}$) obtained in a previous study [19] using a boil-off calorimeter. The effective thermal conductivity computed using the conventional method ($k_{eff}^{CM}$), which has been widely adopted in relevant studies, was also included in the figure. The conventional method employs only $G_{g,n}^{FM}$ from Equation (5) for the $G_{g,n}$ calculation rather than Equation (3).

Both $q_{t,cal}$ and $k_{eff,cal}$ increased non-linearly with increasing absolute pressure. The proposed analytical methodology also demonstrated excellent agreement with the experimental data. In contrast, $k_{eff}^{CM}$ closely matched the experimental values in the high-vacuum regime. However, as the absolute pressure increased, $k_{eff}^{CM}$ gradually deviated from the experimental data and the discrepancies increased. These findings clearly demonstrate that for a thermal analysis conducted over a broad range of residual gas pressures, including medium-to-low vacuum conditions, the use of Lees’ four-moment model, as employed in this study, offers a much more accurate and reliable prediction.

3.2. Internal Heat-Flux Distribution

The heat-flux distribution due to gas conduction in terms of the dimensionless thickness and absolute pressure is shown in Figure 6. Here, the dimensionless thicknesses of 0 and 1 correspond to the cold and warm boundaries, respectively. The heat flux due to gas conduction ($q_g$) increased non-linearly with increasing absolute pressure. This behavior is consistent with the fact that the thermal conductance of gas, as derived from Lees’ four-moment model, is a non-linear function of pressure. In the medium-vacuum regime (<1 mTorr), $q_g$ remained below 1 $\mathrm{W}/{\mathrm{m}}^2$. However, as the pressure rose, it increased sharply, eventually saturating to the order of ${10}^2\mathrm{W}/{\mathrm{m}}^2$ near ${10}^4$ mTorr. The variation in $q_g$ according to the thickness is not discernible in Figure 6 and is revisited in subsequent figures.

Figure 7 and Figure 8 depict the heat flux distributions for solid conduction ($q_s$) and radiation ($q_r$), respectively. Although the thermal conductance values derived from these heat-transfer mechanisms are inherently pressure-independent, the results demonstrate that their corresponding heat fluxes exhibit pressure-dependent variations. This phenomenon arises from the redistribution of $q_s$ and $q_r$ driven by the pressure-induced changes in $q_g$ under fixed temperature-boundary conditions.

A detailed examination of the variation in heat-flux distribution with pressure shows that, as the absolute pressure increased to 10 mTorr, the heat flux via solid conduction increased near the warm boundary but decreased near the cold boundary. This trend was reversed when the pressure exceeded 10 mTorr, causing a decrease in $q_s$ near the warm boundary and an increase near the cold boundary, which continued up to approximately 1 atm (768,585 mTorr). These pressure-dependent variations at both boundaries led to a characteristic intersection of $q_s$ profiles at a specific position within the insulation cavity. The radiative heat flux ($q_r$) also showed a trend similar to that of $q_s$, except that its absolute value was much smaller near the cold boundary. Specifically, the gradient of $q_r$ with respect to the thickness initially increased and then decreased with increasing pressure. These trends result from the non-linear behavior of gas conduction heat flux with pressure and the redistribution of heat flux, as previously described.

The relative contributions of each heat-transfer mode to the total heat flux are shown in Figure 9. Under all pressures, the proportion of solid conduction increases closer to the cold boundary, whereas the contribution of radiation decreases. Notably, the radiation contribution approaches zero near the cold boundary, which demonstrates that the radiative heat shielding provided by the reflectors is highly effective.

In high-vacuum environments at absolute pressures of 0.01 and 0.1 mTorr, solid conduction and radiation together accounted for over 90% of the total heat transfer. As the pressure increased, the contribution of gas conduction increased sharply, reaching over 50% of the total heat transfer at 1 mTorr. Under absolute pressures of 10 mTorr or higher, more than 90% of the heat transfer resulted from gas conduction. This result indicates that in medium-to-low-vacuum regimes, gas conduction becomes the predominant mechanism driving heat intrusion through the MLI.

3.3. Temperature Distribution

Figure 10 shows temperature distributions under various residual gas pressures. In all cases, these temperature profiles exhibited a steeper gradient near the cold boundary, transitioning to a more gradual slope as they approached the warm boundary. As the absolute pressure increased from high vacuum to 10 mTorr, the high-temperature regime progressively contracted, resulting in a nearly linear temperature profile at 10 mTorr. By contrast, at pressures above 10 mTorr, the high-temperature regime expanded again. This behavior is attributed to the redistribution of the heat flux influenced by pressure, which subsequently causes redistribution of temperature.

4. Application

Additional calculations were conducted using the previously validated analytical method to predict the heat-transfer characteristics of a vacuum-MLI system applied to liquid hydrogen (LH₂) storage tanks and transport pipelines. The calculation conditions used for the analysis are listed in

Table 3. Calculation conditions for the LH₂ case.

MLI composition and properties	Reflector	Type	Double-aluminized Mylar
		Emissivity, ${\epsilon }_n$	Equation (6)
	Spacer	Type	Dacron net
		Conductivity, $k_n$	Equation (7)
	Total number of layers, $N$		40
	Total thickness, $X$		15.5 mm
Boundary conditions	T0 (cold)		20.4 K
	TN (warm)		293 K
Residual pressures of vacuum-MLI system	$P[\mathrm{m}\mathrm{T}\mathrm{o}\mathrm{r}\mathrm{r}$]		0.002, 0.003, 0.004, 0.01, 0.02, 0.07, 0.1, 0.3, 1, 3, 10, 100, 1040, 10,036, 99,102, 768,585

. The configurations of the MLI and warm boundary temperature were maintained as identical to those used in the LN₂ case, whereas the cold boundary temperature was set at 20.4 K, corresponding to the boiling point of liquid hydrogen at 1 atm. The analysis was conducted at various pressures.

The empirical parameters of the LH₂ case are listed in

Table 4. Properties and parameters for the LH₂ case.

Variable		Value
Residual Gas	Type	GHe
	Molecular diameter, $d$	2.67 × 10−10 m
	Molecular weight, $M$	4.0 kg/kmol
	Specific heat ratio, $\gamma$	1.66
Accommodation coefficient, $\alpha$		0.53
Empirical constant, $C$		0.008 [20]
Relative density, $f$		0.02
Fitting parameter, $\omega$		1.4

. The residual gas within the cavity was helium, which has a lower boiling point than hydrogen; accordingly, the accommodation coefficient was appropriately modified [30]. The empirical constants for the spacer and the fitting parameters in Lees’ four-moment model remained identical to those used in the LN₂ case.

Figure 11 shows a comparison of the total heat flux and effective thermal conductivity computed for the LH₂ case with those obtained under the LN₂ boundary conditions, as presented in Figure 5. Both results ($q_t$ and $k_{eff}$) for the LH₂ case increased non-linearly with increasing pressure, similar to the tendency observed in the LN₂ case. As expected from the difference in the cold boundary temperature ($T_0$), the total heat flux in the LH₂ case remained consistently higher than that in the LN₂ case, and the discrepancy increased with increasing pressure. Both cases showed similar effective thermal conductivity values under high-vacuum conditions. However, as the pressure increased, the LH₂ case exhibited higher values, and the discrepancy between the two cases increased gradually. The $k_{eff}$ characteristics of the LH₂ case at higher pressures are attributable to the dominance of gas conduction, with lower-molecular-weight helium employed as the residual gas. Notably, this plot ($q_t$ and $k_{eff}$ in LH₂ case) can serve as a preliminary indicator of the insulation performance of a vacuum-MLI system for LH₂, just as the experimental plot reported by Fesmire and Johnson serves as a typical indicator for the LN₂ case [19]. Nevertheless, as these LH₂/GHe results represent a predictive behavior of the validated methodology, experimental verification under these conditions is required to fully confirm the applicability of the present model.

5. Conclusions

A thermal analysis of a vacuum-MLI cavity was conducted to investigate the effect of residual gas pressure on insulation performance. The insulation cavity was modeled as a thermal network based on an electrical analogy, considering solid conduction, gas conduction, and surface radiation in each layer. In particular, gas conduction was computed using a correlation based on Lees’ four-moment model, which covers a wide range of vacuum pressure conditions. The insulation performance was evaluated using the total heat flux and effective thermal conductivity, and these results were compared with the experimental data reported in a previous study.

The results show that both the total heat flux and effective thermal conductivity exhibited a non-linear increase with increasing pressure, a trend that agreed closely with the experimental data. This finding confirmed the validity of the methodology, demonstrating that Lees’ four-moment model is a suitable approach across a wide range of residual gas pressures.

Beyond this validation, the heat-transfer behavior within the insulation system was scrutinized, revealing that the heat-flux distribution across all modes varied, owing to pressure-induced redistribution. The relative contribution of each heat-transfer mode to the total heat transfer was also investigated. The results showed that the proportion of gas conduction increased significantly with increasing pressure, becoming the dominant heat-transfer mode at pressures above 10 mTorr. This result suggests that thermal management of gas conduction is essential for enhancing the performance of insulation systems in medium-vacuum through low-vacuum conditions.

Additionally, the verified analytical method was extended to predict the insulation performance of a cavity at the cryogenic boundary of liquid hydrogen. Compared with the LN₂ case, the LH₂ case yielded a higher total heat flux over the residual gas pressure range investigated. The effective thermal conductivities of LH₂ and LN₂ were similar under high-vacuum conditions. However, as the pressure increased, the value for LH₂ increased more than that of LN₂, with the discrepancy between the two cases being gradually magnified. This phenomenon could be attributed to the employment of helium, which has a greater conductivity than nitrogen, as the residual gas in the MLI-cavity insulation system for LH₂. These analytical results may serve as insightful indicators of the effect of the LH₂-GHe combination system on the insulation effectiveness of vacuum-MLI cavities.