Development of thermodynamically-consistent machine-learning equations of state: application to the Mie fluid Gustavo Chaparro, Erich A. Müller Imperial College London, United Kingdom The accurate description of thermophysical properties of fluids is crucial for design and engineering purposes. From a practical perspective, these properties are often correlated (and sometimes predicted) employing analytical empirical closed-form mathematical expressions denoted as equations of state (EoS). Historically, it has been considered that the inclusion of physically-sensible depictions (e.g. molecular interactions) into the EoS will improve both the robustness and reliability. These more theoretical approaches come at the expense of a long and sometimes cumbersome development. An example worth mentioning is the Statistical Associating Fluid Theory of Variable Range employing a Mie potential [ U=C ε ( (σ/r) λr - (σ/r) λa ) ], a.k.a., SAFT-VR-Mie [1] which is currently the most reliable EoS for this fluid and is capable of mapping back to molecular simulations results. In this contribution, a change in paradigm for developing EoS [2] is explored, where an artificial neural network (ANNs) is trained to “learn” the Helmholtz free energy of the Mie fluid, providing for a thermodynamically consistent data-driven EoS. Thermophysical properties of the Mie fluid are obtained using high-throughput Molecular Dynamics (MD) simulations. The LAMMPS software [3] is used to perform all simulations. The simulations are run with a cut-off of 5σ with no tail corrections and expressed in reduced Lennard-Jones units (i.e., k b =1, ε=1 and σ=1). The MD simulations consider Mie fluids with a repulsive exponent (λ r ) within the range 7-34 and an attractive exponent (λ a ) fixed to 6. The simulated phase space considers dimensionless densities (ρ * ) from 10 -4 to 1.2 and dimensionless temperatures (T * ) from 0.6 to 10.0. The computed thermophysical properties are used to train the residual Helmholtz free energy model of the Mie fluid (FE-ANN EoS). The proposed formulation incorporates physical insights, e.g. analytically fulfilling Maxwell’s relations and the ideal gas law. This data-driven model is trained using first- and second-order derivative information, such as the compressibility (Z= P*/ ρ * T * ), internal energy (U * ), isothermal compressibility (κ T ), isochoric heat capacity (C v * ), thermal expansion coefficient (α P ), heat capacity ratio (γ= C p * /C v * ) and Joule- Thomson coefficient (μ JT * ). The FE-ANN EoS is implemented using TensorFlow [4]. The trained FE-ANN EoS accurately describes the target thermophysical properties of the Mie fluid. Moreover, the physical-inspired formulation of the FE-ANN EoS allows computing other properties for which the model has not been trained for. For example, even though the model is trained using only PVT data, the model discovered -on its own- an unstable van der Waals loop, making it capable of computing vapour-liquid equilibria. The FE-ANN EoS results are on par with those of SAFT-VR-Mie, (arguably the best model for the Mie fluid) but were obtained in a span of a few months as opposed to the decades of development needed for the SAFT-VR-Mie EoS. References
1. Lafitte, et al. (2013). J. Chem. Phys., 139(15), 154504. 2. Rosenberger, et al. (2022). Phys. Rev. E, 105(4), 045301. 3. Thompson, et al. (2022). Computer Phys. Comm., 271, 108171. 4. Tensorflow (2005).www.tensorflow.org/ (accessed 6th October 2022)
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