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In this work, an artificial neural network (ANN) aided vapor–liquid equilibrium (VLE) model is developed and coupled with a fully compressible computational fluid dynamics (CFD) solver to simulate the transcritical processes occurring in high-pressure liquid-fueled propulsion systems. The ANN is trained in Python using TensorFlow, optimized for inference using Open Neural Network Exchange Runtime, and coupled with a C++ based CFD solver. This plug-and-play model/methodology can be used to convert any multi-component CFD solver to simulate transcritical processes using only open-source packages, without the need of in-house VLE model development. The solver is then used to study high-pressure transcritical shock-droplet interaction in both two- and four-component systems and a turbulent temporal mixing layer (TML), where both qualitative and quantitative agreement (maximum relative error less than 5%) is shown with respect to results based on both direct evaluation and the state-of-the-art in situ adaptive tabulation (ISAT) method. The ANN method showed a 6 times speed-up over the direct evaluation and a 2.2-time speed-up over the ISAT method for the two-component shock-droplet interaction case. The ANN method is faster than the ISAT method by 12 times for the four-component shock-droplet interaction. A 7 times speed-up is observed for the TML case for the ANN method compared to the ISAT method while achieving a data compression factor of 2881. The ANN method also shows intrinsic load balancing, unlike traditional VLE solvers. A strong parallel scalability of this ANN method with the number of processors was observed for all the three test cases. Code repository for 0D VLE solvers, and C++ ANN interface—https://github.com/UMN-CRFEL/ANN_VLE.git. 

Vapor-liquid equilibrium (VLE) is a family of first-principled thermodynamic models for transcritical multiphase flows, which can accurately capture the phase transitions at high-pressure conditions that are difficult to deal with using other models. However, VLE-based computational fluid dynamics (CFD) simulation is computationally very expensive for multi-component systems, which severely limits its applications to real-world systems. In this work, we developed a new ISAT-VLE method based on the in situ adaptive tabulation (ISAT) method to improve the computational efficiency of VLE-based CFD simulation with reduced memory usage. We developed several ISAT-VLE solvers for both fully conservative (FC) and double flux (DF) schemes. New methods are proposed to delete redundant records in the ISAT-VLE table and the ISAT-VLE method performance is further improved. To improve the convergence of the VLE solvers, a modified initial guess for equilibrium constant is also introduced. Simulations of high-pressure transcritical two-phase temporal mixing layers and shock-droplet interaction were conducted using the ISAT-VLE CFD solvers. The simulation results show that the new method obtains a speed-up factor approximately from 10 to 60 and the ISAT errors can be controlled within 1%. The shock-droplet interaction results show that the DF scheme can achieve a higher speed-up factor than the FC scheme. The two sets of simulations exhibit the phase separation at high-pressure conditions. It was found that even at supercritical pressures with respect to each component, the droplet surface could still be in a subcritical two-phase state, because the mixture critical pressure is often significantly higher than each component and hence triggers phase separation. In addition, a shock wave could partially or completely convert the droplet surface from a subcritical two-phase state to a single-phase state by raising temperature and pressure. 

The requirement of high power outputs and high efficiencies of combustion engines such as rocket engines, diesel engines, and gas turbines has resulted in the incremented of the system pressure close to the thermodynamically critical point. This increase in pressure often leads to the fluids becoming either transcritical or supercritical in state. This has led to increased interest in both the multi-component phase change phenomena as well as their chemical reactions. In this work, an artificial neural network (ANN) aided VLE model is coupled with a fully compressible computational fluid dynamics (CFD) solver to simulate the transcritical processes occurring in high-pressure liquid-fueled propulsion systems. The ANN is trained on Python using the TensorFlow library, optimized for inference (i.e., prediction) using ONNX Run-time (a cross-platform inference and training machine-learning accelerator), and coupled with a C++ based fully compressible CFD solver. This plug-and-play model/methodology can be used to convert any fully compressible and conservative CFD solver to simulate transcritical processes using only open-source packages, without the need of in-house VLE-based CFD development. The solver is then used to study high-pressure shock-droplet interaction in both two- and four-component systems where qualitative and quantitative agreement is shown with results based on both direct evaluation and the state-of-the-art in-situ adaptive tabulation (ISAT) method. The ANN model is faster than the direct evaluation method and the ISAT model by 4 times for the four-component shock-droplet interaction. The ANN model also shows implicit load balancing as long as the MPI decomposition is performed uniformly amongst the number of cores chosen, as the inference time for ANN predict does not change with the change in thermodynamic state, unlike traditional VLE solvers. Regarding the parallel scalability of this model, good strong scaling characteristics with number of processors is also observed. 

Supercritical fluids, often present in modern high-performance propulsion systems, result from elevated operating pressures. When these systems utilize fluid mixtures as fuel or oxidizers, a transcritical effect often occurs. This effect can lead to misjudgments, as mixture critical points exceed those of individual components. Fluid mixing may induce phase separation, creating liquid and vapor phases due to the transcritical multi-component effect. Consequently, two-phase modeling is essential for transcritical and supercritical fluids. Traditional interface capturing methods, like Volume of Fluid (VOF) and Level Set (LS), present challenges such as computational expense and lack of conservatism. The Phase Field (PF) method, or the Diffuse Interface (DI) method which uses a phase fraction transport equation, emerges as a conservative alternative. Despite the absence of an initial interface in transcritical fluids, phase separation from mixing may form liquid droplets, necessitating multiphase modeling. To address these complexities, a Vapor-Liquid Equilibrium (VLE) model, coupled with the PR equation of state, is introduced. This model estimates phase fractions, liquid and vapor compositions, densities, and enthalpies through a flash problem solution. The conventional PF model is enhanced by replacing the phase fraction transport equation with VLE-derived values. The resulting VLE-based PF method is implemented into an OpenFOAM compressible solver, ensuring numerical stability with explicit phase field terms and a new CFL criterion. Test cases involve 1D interface convection and 2D droplet convection. In the 1D test, the VLE-based PF model adeptly captures interfaces, adjusting thickness as needed. The 2D droplet case, challenging due to a non-aligned Cartesian grid, exhibits uniform interface thickness and preserves droplet shape. The VLE-based PF model demonstrates versatility and reliability in capturing complex fluid behaviors, offering promising prospects for future research. 

This paper numerically investigates the evaporation characteristics of a single n-dodecane fuel droplet in high-pressure nitrogen environment relevant to rotating detonation engines. A validated computational fluid dynamics solver coupled with real-fluid thermophysical models is utilized. The effects of pressure, droplet temperature, and ambient gas temperature on the evaporation rate are analyzed by tracking the droplet diameter evolution. Two interface tracking techniques, namely a mean density-based method and a novel vapor-liquid equilibrium-based method, are implemented and compared. The results show appreciable deviations from the classical d2-law for droplet evaporation. Increasing the ambient temperature and droplet temperature (toward critical point) substantially accelerate the evaporation process. Meanwhile, higher pressures decrease the evaporation rate owing to slower species/thermal diffusions. At certain conditions, discernible differences are observed between the two interface tracking methods indicating deficiencies in the simple mean density approach. The paper demonstrates an effective computational framework for transcritical droplet evaporation simulations. And the generated high-pressure droplet evaporation datasets can inform sub-model development for spray combustion modeling.