Equations
of state (EoS) are essential in the modeling of a wide
range of industrial and natural processes. Desired qualities of EoS
are accuracy, consistency, computational speed, robustness, and predictive
ability outside of the domain where they have been fitted. In this
work, we review present challenges associated with established models,
and give suggestions on how to overcome them in the future. The most
accurate EoS available, multiparameter EoS, have a second artificial
Maxwell loop in the two-phase region that gives problems in phase-equilibrium
calculations and excludes them from important applications such as
treatment of interfacial phenomena with mass-based density functional
theory. Suggestions are provided on how this can be improved. Cubic
EoS are among the most computationally efficient EoS, but they often
lack sufficient accuracy. We show that extended corresponding state
EoS are capable of providing significantly more accurate single-phase
predictions than cubic EoS with only a doubling of the computational
time. In comparison, the computational time of multiparameter EoS
can be orders of magnitude larger. For mixtures in the two-phase region,
however, the accuracy of extended corresponding state EoS has a large
potential for improvement. The molecular-based SAFT family of EoS
is preferred when predictive ability is important, for example, for
systems with strongly associating fluids or polymers where few experimental
data are available. We discuss some of their benefits and present
challenges. A discussion is presented on why predictive thermodynamic
models for reactive mixtures such as CO2–NH3 and CO2–H2O–H2S must be developed in close combination with phase- and reaction
equilibrium theory, regardless of the choice of EoS. After overcoming
present challenges, a next-generation thermodynamic modeling framework
holds the potential to improve the accuracy and predictive ability
in a wide range of applications such as process optimization, computational
fluid dynamics, treatment of interfacial phenomena, and processes
with reactive mixtures.
Abstract. Large time step explicit schemes in the form originally proposed by LeVeque [Comm. Pure Appl. Math., 37 (1984), pp. 463-477] have seen a significant revival in recent years. In this paper we consider a general framework of local 2k + 1 point schemes containing LeVeque's scheme (denoted as LTS-Godunov) as a member. A modified equation analysis allows us to interpret each numerical cell interface coefficient of the framework as a partial numerical viscosity coefficient.We identify the least and most diffusive TVD schemes in this framework. The most diffusive scheme is the 2k + 1-point Lax-Friedrichs scheme (LTS-LxF). The least diffusive scheme is the Large Time Step scheme of LeVeque based on Roe upwinding (LTS-Roe). Herein, we prove a generalization of Harten's lemma: all partial numerical viscosity coefficients of any local unconditionally TVD scheme are bounded by the values of the corresponding coefficients of the LTS-Roe and LTS-LxF schemes.We discuss the nature of entropy violations associated with the LTS-Roe scheme, in particular we extend the notion of transonic rarefactions to the LTS framework. We provide explicit inequalities relating the numerical viscosities of LTS-Roe and LTS-Godunov across such generalized transonic rarefactions, and discuss numerical entropy fixes.Finally, we propose a one-parameter family of Large TimeStep TVD schemes spanning the entire range of the admissible total numerical viscosity. Extensions to nonlinear systems are obtained through the Roe linearization. The 1D Burgers equation and the Euler system are used as numerical illustrations.
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