Solution and Parameter Identification of a Fixed-Bed Reactor Model for Catalytic CO2 Methanation Using Physics-Informed Neural Networks

In this study, we develop physics-informed neural networks (PINNs) to solve an isothermal fixed-bed (IFB) model for catalytic CO2 methanation. The PINN includes a feed-forward artificial neural network (FF-ANN) and physics-informed constraints, such as governing equations, boundary conditions, and reaction kinetics. The most effective PINN structure consists of 5–7 hidden layers, 256 neurons per layer, and a hyperbolic tangent (tanh) activation function. The forward PINN model solves the plug-flow reactor model of the IFB, whereas the inverse PINN model reveals an unknown effectiveness factor involved in the reaction kinetics. The forward PINN shows excellent extrapolation performance with an accuracy of 88.1% when concentrations outside the training domain are predicted using only one-sixth of the entire domain. The inverse PINN model identifies an unknown effectiveness factor with an error of 0.3%, even for a small number of observation datasets (e.g., 20 sets). These results suggest that forward and inverse PINNs can be used in the solution and system identification of fixed-bed models with chemical reaction kinetics.

Approximations for the Concentration and Effectiveness Factor in Porous Catalysts of Arbitrary Shape: Taylor Series and Akbari-Ganji’s Methods

A mathematical model of reaction-diffusion problem with Michaelis-Menten kinetics in catalyst particles of arbitrary shape is investigated. Analytical expressions of the concentration of substrates are derived as functions of the Thiele modulus, the modified Sherwood number, and the Michaelis constant. A Taylor series approach and the Akbari-Ganji's method are utilized to determine the substrate concentration and the effectiveness factor. The effects of the shape factor on the concentration profiles and the effectiveness factor are discussed. In addition to their simple implementations, the proposed analytical approaches are reliable and highly accurate, as it will be shown when compared with numerical simulations.

Oxygen Deficient (OD) Combustion and Metabolism: Allometric Laws of Organs and Kleiber’s Law from OD Metabolism?

The biology literature presents allometric relations for the specific metabolic rate (SMRk) of an organ k of mass mk within the body of mass mB: SMR_k ∝ mBfk (body mass allometry, BMA). Wang et al. used BMA, summed-up energy from all organs and validated Kleiber’s law of the whole body: SMRM ∝ mBb’, b’ = −0.25. The issues raised in biology are: (i) why fk and b’ < 0, (ii) how do the organs adjust fk to yield b’? The current paper presents a “system” approach involving the field of oxygen deficient combustion (ODC) of a cloud of carbon particles and oxygen deficient metabolism (ODM), and provides partial answers by treating each vital organ as a cell cloud. The methodology yields the following: (i) a dimensionless “group” number GOD to indicate extent of ODM, (ii) SMRk of an organ in terms of the effectiveness factor; (iii) curve fitting of the effectiveness factor to yield the allometric exponents for the organ mass-based allometric laws (OMA); (iv) validation of the results with data from 111 biological species (BS) with mB ranging from 0.0075 to 6500 kg. The “hypoxic” condition at organ level, particularly for Covid-19 patients, and the onset of cancer and virus multiplication are interpreted in terms of ODM and glycolysis.

Diffusion and reaction in pore hierarchies by the two-region model

The two-region (“Kärger”) model of diffusion in complex pore spaces is exploited for quantitating mass transfer in hierarchically organized nanoporous materials, consisting of a continuous microporous bulk phase permeated by a network of transport pores. With the implications that the diffusivity in the transport pores significantly exceeds the diffusivity in the micropores and that the relative population of the transport pores is far below that of the micropores, overall transport depends on only three independent parameters. Depending on their interrelation, enhancement of the overall mass transfer is found to be ensured by two fundamentally different mechanisms. They are referred to as the limiting cases of fast and slow exchange, with the respective time constants of molecular uptake being controlled by different parameters. Complemented with reaction terms, the two-region model may equally successfully be applied to the quantitation of the combined effect of diffusion and reaction in terms of the effectiveness factor. Generalization of the classical Thiele concept is shown to provide an excellent estimate of the effectiveness factor of a chemical reaction in hierarchically porous materials, solely based on the intrinsic reaction rate and the time constant of molecular uptake relevant to the given conditions.