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The design of cryogenic liquid storage solutions requires accurate methods for estimating heat ingress, from the material level to the tank level. For insulation materials, thermal performance is usually measured using ambient conditions and liquid nitrogen at 77 K as boundary temperatures. A key question is how much heat ingress increases when storing liquid hydrogen LH$_2$ at 20 K. We derive theoretical bounds on the increased heat ingress, and show that it remains below 26%. Additionally, we demonstrate that heat ingress is much more sensitive to the warm boundary temperature than the cold boundary temperature. At the tank level, we compare two methods for assessing the steady-state thermal performance of cryogenic tanks: thermal network models and the heat equation solved with the finite element method. The latter offers high accuracy and adaptability for complex geometries, while thermal network models benefit from simplicity, speed and robustness. We apply both approaches to a self-supported LH$_2$ tank concept for maritime transport and analyze sensitivity to structural support thickness, warm boundary temperature, and choice of insulation material. The thermal network model can estimate heat ingress with $\sim$1% error and the cold-spot temperature with error less than 1 K.

Future use of liquid hydrogen (LH$_2$) as an effective energy carrier will require elimination or minimization of hydrogen boil-off that is not utilized by demands in the value chain. The present work promotes local area cooling (LAC) as a promising boil-off reduction technology. In contrast to the more conventional broad area cooling (BAC), LAC targets local, concentrated heat flows e.g. through tank support structures. This yields important practical benefits, especially for large-scale tanks, due to the order-of-magnitude reduction in the size of the cooling system. Such benefits include lower capital costs and simpler installation, maintenance and coolant management. LAC applied outside the outer tank wall is particularly attractive for tanks with evacuated insulation. In a series of numerical studies, we use the finite element method to evaluate the thermal performance of LAC and BAC in the context of ship-borne LH$_2$ transport. The studies concern 40 000 m$^3$-capacity, skirt-supported tanks insulated using evacuated perlite or helium-filled polyurethane (HePUR) foam. For the perlite-insulated tank, LAC and BAC with liquid nitrogen coolant can reduce the daily boiloff rate from 0.04%/day to, respectively, 0.011%/day and 0.004%/day. The corresponding numbers for CO$_2$-based refrigeration are 0.031%/day and 0.028%/day. For the HePUR-insulated tank, which has a higher baseline boil-off rate of 0.24%/day, reduced boil-off rates down to 0.17%/day and 0.04%/day are achievable using LAC and BAC, respectively. LAC and BAC both offer increased power efficiency in comparison to reliquefaction only.

With the recent wave of digitalization, specifically in the context of safety-critical applications, there has been a growing need for computationally efficient, accurate, generalizable, and trustworthy models. Physics-based models have traditionally been used extensively for simulating and understanding complex phenomena. However, these models though trustworthy and generalizable to a wide array of problems, are not ideal for real-time. To address this issue, the physics-based models are simplified. Unfortunately, these simplifications, like reducing the dimension of the problem (3D to 2D) or linearizing the highly non-linear characteristics of the problem, can degrade model accuracy. Data-driven models, on the other hand, can exhibit better computational efficiency and accuracy. However, they fail to generalize and operate as blackbox, limiting their acceptability in safety-critical applications. In the current article, we demonstrate how we can use a data-driven approach to correct for the two kinds of simplifications in a physics-based model. To demonstrate the methodology's effectiveness, we apply the method to model several elasticity problems. The results show that the hybrid approach, which we call the corrective source term approach, can make erroneous physics-based models more accurate and certain. The hybrid model also exhibits superior performance in terms of accuracy, model uncertainty, and generalizability when compared to its end-to-end data-driven modeling counterpart.

Upcoming technologies like digital twins, autonomous, and artificial intelligent systems involving safety-critical applications require models which are accurate, interpretable, computationally efficient, and generalizable. Unfortunately, the two most commonly used modeling approaches, physics-based modeling (PBM) and data-driven modeling (DDM) fail to satisfy all these requirements. In the current work, we demonstrate how a hybrid approach combining the best of PBM and DDM can result in models which can outperform them both. We do so by combining partial differential equations based on first principles describing partially known physics with a black box DDM, in this case, a deep neural network model compensating for the unknown physics. First, we present a mathematical argument for why this approach should work and then apply the hybrid approach to model two dimensional heat diffusion problem with an unknown source term. The result demonstrates the method's superior performance in terms of accuracy, and generalizability. Additionally, it is shown how the DDM part can be interpreted within the hybrid framework to make the overall approach reliable.

In this work, we introduce, justify and demonstrate the Corrective Source Term Approach (CoSTA) -- a novel approach to Hybrid Analysis and Modeling (HAM). The objective of HAM is to combine physics-based modeling (PBM) and data-driven modeling (DDM) to create generalizable, trustworthy, accurate, computationally efficient and self-evolving models. CoSTA achieves this objective by augmenting the governing equation of a PBM model with a corrective source term generated using a deep neural network. In a series of numerical experiments on one-dimensional heat diffusion, CoSTA is found to outperform comparable DDM and PBM models in terms of accuracy -- often reducing predictive errors by several orders of magnitude -- while also generalizing better than pure DDM. Due to its flexible but solid theoretical foundation, CoSTA provides a modular framework for leveraging novel developments within both PBM and DDM. Its theoretical foundation also ensures that CoSTA can be used to model any system governed by (deterministic) partial differential equations. Moreover, CoSTA facilitates interpretation of the DNN-generated source term within the context of PBM, which results in improved explainability of the DNN. These factors make CoSTA a potential door-opener for data-driven techniques to enter high-stakes applications previously reserved for pure PBM.