PSRK equation of state

The PSRK equation of state (from English predictive Soave-Redlich-Kwong equation of state ) is an estimation method for the calculation of phase equilibria of mixtures of chemical substances. The original objective was to enable the properties of mixtures that also contain supercritical components. This class of substances could not be calculated with the previously developed methods such as UNIFAC , which estimate activity coefficients.

principle

The PSRK equation is a group contribution state equation , a class of estimation methods that combine state equations (mostly cubic) with activity coefficient models . The equation of state is used to calculate the pure substance properties and the activity coefficient model is used to describe the mixture properties. The procedure for calculating mixture properties from models originally used only for pure substance properties is known as the mixture rule.

By using the PSRK equation of state, all thermodynamic relationships can be used for equations of state that enable, for example, densities, enthalpies, heat capacities and the like. a. m. to calculate.

Equations

PSRK equation of state is based on a combination of the equation of state of Soave-Redlich-Kwong (SRK equation of state) with a mixture rule, the parameters of which are determined using the UNIFAC group contribution method.

Equation of state

The Soave-Redlich-Kwong equation of state reads:

${\displaystyle P={{R\;T} \over {v-b}}-{{a\;\alpha (T)} \over {v(v+b)}}}$

The α function originally used by Giorgio Soave is replaced by the Mathias-Copeman function .

${\displaystyle \alpha (T_{r})=\left[1+c_{1}\left(1-{\sqrt {T}}_{r}\right)+c_{2}\left(1-{\sqrt {T}}_{r}\right)^{2}+c_{3}\left(1-{\sqrt {T}}_{r}\right)^{3}\right]^{2}}$

The parameters of the Mathias-Copeman equation are adapted to experimental saturation vapor pressure data of pure substances. This allows a significantly improved description of the saturation vapor pressure. The form of the equation was chosen because setting the parameters c ₂ and c ₃ to zero results in the original Soave approach. In addition, the parameter c _{1 can be} derived from the acentric factor via the relationship

${\displaystyle c_{1}=0{,}48+1{,}574\;\omega -0{,}176\;\omega ^{2}}$

can be determined if no adapted Mathias Copeman parameters are available.

Mixing rule

The PSRK mixture rule calculates the parameters a (cohesion pressure) and b (co-volume) of the equation of state for the mixture

${\displaystyle {\frac {a}{bRT}}=\sum _{i}\left(x_{i}{\frac {a_{ii}}{b_{i}RT}}-{\frac {{\frac {g_{0}^{E}}{RT}}+\sum x_{i}\ln {\frac {b}{b_{i}}}}{0{,}64663}}\right)}$

and

${\displaystyle b=\sum _{i}x_{i}\;b_{i}}$

from the parameters a _i and b _i of the pure substances, the mole fractions x _i and the excess Gibbs Gibbs g ^E . The excess enthalpy of Gibbs is calculated using a slightly modified UNIFAC model.

Model parameters

For the part of the equation of state, PSRK requires the critical temperature and pressure as well as at least the acentric factor. A higher quality can be achieved if the acentric factor is replaced by Mathias Copeman parameters. Mathias Copeman parameters are adjusted to saturation vapor pressure data.

The mixing rule uses UNIFAC, which requires a number of UNIFAC-specific parameters. In addition to some model constants, the most important ones are the group interaction parameters, which are adapted to vapor-liquid equilibria .

For a high-quality parameterization, a large amount of experimental data (pure substance saturation vapor pressures and mixture-vapor-liquid equilibria) is necessary. These are mostly made available by factual databases such as the Dortmund database . In rare cases, even directly required material data are determined experimentally, despite the associated high costs, if no phase equilibrium data can be found from other sources.

The last available parameters were published in 2005. The UNIFAC consortium has taken on the further development of the model .

Sample calculation

A vapor-liquid equilibrium can also be predicted in mixtures that contain supercritical components.

Vapor-liquid equilibrium of cyclohexane and carbon dioxide

The mixture itself, however, must be subcritical. In the example, the carbon dioxide is the supercritical component with T _c  = 304.19 K and P _c  = 7,475 kPa. The critical point of the mixture is T  = 411 K and P  ≈ 15 MPa. The composition of the mixture is approx. 78 mol percent carbon dioxide and 22 mol percent cyclohexane .

PSRK describes this binary mixture in very good quality, both boiling and thawing curves and the critical point of the mixture are met well.

Model weaknesses

Some model weaknesses are listed in a PSRK follow-up work:

• The course of the α-function according to Mathias-Copeman is when extrapolating to high temperatures without a physical basis.
• The Soave-Redlich-Kwong equation describes the vapor densities of pure substances and mixtures quite well, but that of the liquid phase only poorly.
• When predicting vapor-liquid equilibria of mixtures in which the components are of significantly different sizes ( e.g. ethanol, C ₂ H ₆ O, and eicosane, C ₂₀ H ₄₂ ), major systematic errors occur.
• Heats of mixing and activity coefficients at infinite dilution are poorly represented.

Web links

• Short PSRK description from the chair of PSRK developers
• UNIFAC industrial consortium at the Carl von Ossietzky University of Oldenburg (has also been developing PSRK since 2005)
• Structure group classification for PSRK

Individual evidence

1. ↑ Thomas Holderbaum: The advance calculation of vapor-liquid equilibria with a group contribution state equation . In: Progress Reports VDI: Series 3 . tape 243 . VDI-Verl. ,, 1991, ISBN 3-18-144303-4 , p. 1-154 .
2. ^ Paul M. Mathias, Thomas W. Copeman: Extension of the Peng-Robinson equation of state to complex mixtures: Evaluation of the various forms of the local composition concept . In: Fluid Phase Equilibria . tape 13 , no. 0 , 1983, ISSN  0378-3812 , p. 91-108 , doi : 10.1016 / 0378-3812 (83) 80084-3 .
3. ↑ Sven Horstmann, Anna Jabłoniec, Jörg Krafczyk, Kai Fischer, Jürgen Gmehling: PSRK group contribution equation of state: comprehensive revision and extension IV, including critical constants and α-function parameters for 1000 components . In: Fluid Phase Equilibria . tape 227 , no. 2 , January 25, 2005, p. 157-164 , doi : 10.1016 / j.fluid.2004.11.002 .
4. ↑ D. Ambrose: The vapor pressures and critical temperatures of acetylene and carbon dioxide . In: Transactions of the Faraday Society . tape 52 , 1956, ISSN  0014-7672 , pp. 772 , doi : 10.1039 / tf9565200772 .
5. ↑ E. Schmidt, W. Thomas: Precision determination of the critical point of carbon dioxide and ethane by measuring the refraction of light . In: Forsch. Geb. Ingenieurwes. Ed. A . tape 20 , 1954, pp. 161-170 , doi : 10.1007 / BF02558359 .
6. ^ J. Ahlers: Development of a universal group contribution state equation , doctoral thesis, Carl von Ossietzky University of Oldenburg, 2003, ISBN 3-8322-1746-0 .