## Unifying Transport Phenomena and Thermodynamic Equilibrium Models: A theoretical study of current and novel equilibrium based models in transport phenomena

##### Abstract

Transport phenomena are based on the conservation of mass, momentum, and energy, and provide equations that give insight into the motion of fluids, the driving force behind this motion, and the dissipation of such driving forces. The rigorous governing transport equations are functions of space and time. On the other hand, thermodynamics is another field of science that focuses mainly on equilibrium states, which are neither functions of space nor time. Thermodynamics can be used to describe the behavior of pressure, density, and chemical potential. Moreover, chemical and phase equilibrium are central topics within thermodynamics.
The focus of this dissertation is placed on the intersection between transport phenomena and thermodynamics. This intersection is here seen as i) the pressure field, ii) the density field, iii) the transport equation for energy, iv) chemical reaction in the transport equation for species mass, and v) mass transfer in multifluid flow in the transport equation for total and species mass. The work is twofold, in which the first part focuses primarily on point iv), while the second part focuses mainly on point v). Both parts require points i), ii), and iii) to be considered.
The first part of this work employs transport equations for mass, species mass, momentum, and energy for a reactive gas in a fixed bed of porous catalysts. In the transport equation for species mass, the source/sink term originating from the chemical reaction is computed from chemical equilibrium rather than reaction kinetics. The chemical equilibrium is computed by minimizing Gibbs or Helmholtz energy at every numerical discretization point. For the Gibbs energy approach, the minimization is performed while holding the temperature and pressure constant at the numerical discretization points. For the Helmholtz energy, the minimization is performed while holding the temperature and volume constant at the numerical discretization points. The volume to be utilized has two alternatives: i) the volume may be obtained from the iteration of the equation of state (EoS), and ii) by discretizing in space with the finite volume method where the volume may be obtained from the numerical grid. For the Helmholtz energy approach, the numerical solution obtained by utilizing the volume from alternatives i) and ii) were compared and revealed identical result. Moreover, the Gibbs and the Helmholtz energy minimization methods also gave identical results. On the other hand, the Gibbs energy method spent less computational time than the Helmholtz energy method to converge the set of equations. In the transport equation for energy, the heat capacity and the molar heat of reaction were computed from the EoS with residual functions denoting the departure from the ideal gas state.
The procedure above was performed for the Soave–Redlich–Kwong EoS, and the virial expansion truncated after the second term. Furthermore, two different chemical processes were investigated: the steam–methane reforming (SMR) and the methanol (MeOH) process. It is emphasized that the reactor concept investigated in the first part of this dissertation is not intended to replace reaction kinetics. The reaction rate expressions developed through rigorous catalytic kinetics experiments should be used when they are available, and if no reaction rate expression is available, the equilibrium-based reactor investigated here could serve as an alternative in process design studies.
The second part of this work derives a set of mass transfer equations that have their basis in the continuity of mass fluxes through the interphase separating two adjoining phases. Contrary to the common practice, which employs Henry's law, the mass transfer equations developed here feature a complete phase equilibrium description that incorporates all components, including the component that is commonly referred to as the solvent (the component in excess). The phase equilibrium description at the surface allows for a flexible framework, where the thermodynamic model can be chosen to best fit the components in the process investigated. It has been shown that the new framework is a generic case of the commonly employed Henry's law, and it is illustrated for three different EoSs and two different processes. The Soave–Redlich–Kwong EoS was employed for the single-cell protein process, and the Peng–Robinson and the perturbed-chain statistical association fluid theory (PC-SAFT) EoSs were employed for the Fischer–Tropsch synthesis. For the complex Fischer–Tropsch synthesis process, the novel phase equilibrium based mass transfer equations were preferable due to Henry's law coefficients being unavailable for many species in the Fischer–Tropsch synthesis process. On the other hand, Henry's law coefficients were available for all species for the SCP process.
The mass transfer expressions based on the Henry's law and the complete phase equilibrium were qualitatively and quantitatively different for both processes. The mass transfer of the solvent was not modeled with Henry's law in either of the processes, and the differences in mass fractions were seen to vary by, at most, 150 % (relatively) in the SCP process. In the Fischer–Tropsch synthesis process, several weaknesses were described for the method based on Henry's law, e.g., i) mass transfer being a function of the chosen solvent, ii) the solvent as a concept being imprecise and ambiguous, and iii) the mass transfer of the solvent is implicitly computed since the Henry's law coefficient does not exist for this component. These weaknesses were not present for the phase equilibrium based method. However, the increase in computational cost accompanied by solving a complete phase equilibrium problem is not to be disregarded. For instance, the mass transfer expression based on the PC-SAFT EoS was found to be 260 times slower than the equivalent mass transfer expression based on Henry's law. On the other hand, the mass transfer expression based on the Peng–Robinson EoS was only 5 times slower than the equivalent mass transfer expression based on Henry's law. Hence, there are large variations in the computational requirements of the new mass transfer expression based on the complexity of the EoS employed.

##### Has parts

Paper I: Øyen, S. B., Jakobsen, H. A., Haug-Warberg, T., & Solsvik, J. (2020). Differential Gibbs and Helmholtz reactor models for ideal and nonideal gases: Applications to the SMR and methanol processes. Chemical Engineering Science, 234, 116257. https://doi.org/10.1016/j.ces.2020.116257 This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).Paper 2: Øyen, Sindre Bakke; Jakobsen, Hugo Atle; Haug-Warberg, Tore; Solsvik, Jannike. Interface Mass Transfer in Reactive Bubbly Flow: A Rigorous Phase Equilibrium-Based Approach. Industrial & Engineering Chemistry Research 2021 ;Volum 60. s. 17705-17732 https://doi.org/10.1021/acs.iecr.1c03131 This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Paper 3: Øyen, Sindre Bakke; Jakobsen, Hugo Atle; Haug-Warberg, Tore; Solsvik, Jannike. Mass transfer modeling and sensitivity study of low-temperature Fischer-Tropsch synthesis. Chemical Engineering Science (CES) 2022 ;Volum 259. https://doi.org/10.1016/j.ces.2022.117774 This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).