# Development of a predictive equation of state

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An equation of state (EOS) is important for modelling fuel properties for injection in engine modelling in the cluster of excellence. An EOS can also be used in the design of separation and reaction processes as well as in the design of energy technology applications e.g. refrigeration. The objective is to develop a predictive equation of state based on PCP-SAFT (Perturbed-Chain Polar Statistical Associating Fluid Theory) and quantum mechanically obtained molecular information. Even though the prediction method is developed for and validated by the PCP-SAFT EOS, the aim is to develop a framework that is generally applicable to segment based equations of state, such as PCP-SAFT or SAFT-VR Mie. It is used to predict thermodynamic properties of pure compounds and mixtures.

## Parameter prediction

Initially, a scheme has been developed to determine EOS parameters from quantum mechanical (QM) calculations for non-hydrogen bonding compounds (1, 4) and applied to predict the solubility of drug compounds (2). These EOS parameters were used to predict the surface tension of new potential fuel candidates along with other model compounds with a fully predictive model and with a 1-parameter model adjusted to the easily obtainable normal boiling point (3). The scheme for EOS parameter prediction is currently constrained to molecules consisting of hydrogen, carbon, nitrogen and oxygen only. However, the molecules can be arbitrarily assembled and therefore the field of application is significantly larger than with group contribution methods. Since the molecular descriptors of the prediction method are determined by ab-initio calculations, only the structural information of a molecule is required as starting information.

## Improvement strategies to the predictive equation of state models

Copyright: © Institute of Technical Thermodynamics, RWTH Aachen UniversityIn order to improve the accuracy of the predictive equation of state models, we developed some strategies, those are: combining it with a minimal set of experimental property data (5); or combining it with a set of property data predicted from another predictive model, which is the Conductor-like Screening Model – Real Solvent (COSMO-RS) (6).

We combined previous advances to build the Segment-based Equation of State Parameter Prediction (SEPP) (7) framework. SEPP determines the geometry, dispersion and polar parameters analogously as in previous models. The workflow is shown in Figure 1. SEPP uses QM calculations (blue) to obtain five molecular descriptors. The electron density serves as input to calculate additional structural information in a geometry intermediate model (orange). Finally, a multilinear model (green) combines the previously obtained information to derive three of the five PCP-SAFT parameters. The parameter for polar contributions is taken directly from the QM calculations. For the association parameter, we employed a novel approach to model hydrogen bonds. Based on the surface charge density profile of a molecule, hydrogen donor and acceptor sites are identified, and a parameter is determined for each possible pair.

## Current Work

Our present focus is to benchmark SEPP for different systems and molecular groups. So far, we investigated the applicability of SEPP for pure compounds with no or a single functional group. Currently we evaluate its applicability on pure compounds with multiple functional groups and the influence of conformers.

The next steps will also include testing the applicability on mixtures with self-associating compounds and mixtures with induced association. This way, we expect to illuminate the strengths and weaknesses of the framework and identify further improvements.

## Publications on this topic:

(1) Van Nhu Nguyen, Mahendra Singh und Kai Leonhard, Feelly Ruether, Kai Leonhard und Gabriele Sadowski

Quantum Mechanically Based Estimation of Perturbed-Chain Polar Statistical Associating Fluid Theory Parameters for Analyzing Their Physical Significance and Predicting Properties.

*The Journal of Physical Chemistry B*, **112(18):5693-5701**, 2008.

(2) Jan Cassens, Feelly Ruether, Kai Leonhard und Gabriele Sadowski

Solubility calculation of pharmaceutical compounds – A priori parameter estimation using quantum-chemistry

*Fluid Phase Equilibria*, **299(1): 161 - 170**, 2010.

(3) Alexander von Müller and Kai Leonhard

Surface Tension Calculations by Means of a PCP-SAFT-DFT Formalism Using Equation of State Parameters from Quantum Mechanics

*Fluid Phase Equilibria*, **356: 96-101**, 2013.

(4) Muhammad Umer, Katja Albers, Gabriele Sadowski and Kai Leonhard

PC-SAFT parameters from ab-initio calculations

*Fluid Phase Equilibria*, **362(41-50)** 2014.

(5) Sebastian Kaminski, André Bardow, and Kai Leonhard

The trade-off between experimental effort and accuracy for determination of PCP-SAFT parameters

*Fluid Phase Equilibria*, **428:182-189**, 2016

(6) Sebastian Kaminski, Evagelos Kirgios, André Bardow, and Kai Leonhard

Improved Property Predictions by Combination of Predictive Models

*Industrial Engineering Chemistry Research*, **56(11):3098-3106**, 2017

(7) Sebastian Kaminski and Kai Leonhard

SEPP: Segment-Based Equation of State Parameter Prediction

*Journal of Chemical & Engineering Data,* **65 (12), 5830-5843**, 2020