## 1 Introduction  
High-vacuum multilayer insulation (MLI) is widely used in the thermal insulation systems of space cryogenic propellants. In the outer space environment, the residual gas in MLIs can easily escape and the interlayer pressure can be as low as \(10^{-6}\) Pa. The ideal interlayer vacuum is the basis of the super thermal insulation for MLIs. However, deviations in the interlayer pressure from the ideal vacuum always occur in large cryogen liquid containers and lead to a remarkable decline in the thermal insulation performance of MLIs. Numerous studies have shown that when the gas pressure is less than \(1\times 10^{-4}\) torr, the residual gas conductivity in MLI can be significantly reduced, and the radiation conductivity and solid thermal conductivity are dominant. Hence, MLIs have a good effect on the shielding of these two kinds of heat flux. When high-vacuum insulation is applied to containers of liquid hydrogen, the vacuuming process of the insulation structures should be optimized to build an ideal interlayer pressure distribution. Therefore, monitoring the vacuum of MLIs in cryogen containers is a very important issue [1]. Directly measuring the interlayer pressure is the best way to monitor MLI vacuums. Many researchers have conducted experimental tests of the interlayer pressure. [PERSON] [2] used thermocouple gauges monitor the MLI interlayer vacuum in a liquid hydrogen container. It was found that the MLI interlayer vacuum did not reach the desired level. However, the MLI vacuum was improved when pretreatment measures such as perforation were used. [PERSON] [3] placed thermocouples between interlayers to measure the vacuum of MLIs in a liquid nitrogen calorimeter. The results proved that the combination of aluminized Mylar and glass fibers had good insulating properties and found that the interlayer pressure might be 10-50-times higher than the vacuum chamber pressure. It was also shown that the interlayer pressure was higher than the pressure of the vacuum chamber, and the effect of the residual gas conduction cannot be neglected in theoretical calculations. [PERSON] [4] placed the capillary tubes in the MLI interlayer, and external micro-pressure gauges were used to measure the MLI vacuum for dynamic and static vacuuming processes. The results showed that dynamic vacuum and baking had a beneficial effect on the MLI vacuum, but the measuring error of the capillary tube was significant, and the interlayer pressure distribution results were similar to a \"\(\cap\)\" shape. The interlayer pressure distribution was closer to an \"L\" shape based on the static evacuation. According to the published literature mentioned above, the interlayer pressure may vary in the heat leakage direction, and the maximum pressure may occur in the middle layers. Small calorimeters have been used to study the thermal insulation performance of MLIs, and the basic parameters of these test sets are summarized in Table 1.  
Due to the small volume and lower usage of MLIs, a small cryogenic container can be vacuumed more easily than a large one. Hence, the thermal insulation performance of MLIs in large cryogen containers is unsatisfactory, although it is excellent in small-scale experimental studies with a small calorimeter. So far, the above measurement methods for MLI vacuums have required additional equipment such as vacuum gauges and capillaries, and only local interlayer pressure can be obtained. On the other hand, theoretical prediction of the interlayer pressure of MLIs may be a good method to predict their thermal insulation performance. In fact, the interlayer pressure distribution of MLIs is influenced by many factors, for example the outgassing of the materials, the types of adsorbents, the properties of the reflectors and spacers, and the detailed structures of MLIs (e.g., layer density and the perforation). As far as we know, it is difficult to develop a high-precision predicting model to determine the interlayer pressure of MLIs for large-scale cryogen storage.