User:WilfriedC/Playground/PSRK

PSRK (short for Predictive Soave-Redlich-Kwong) is a estimation method for the calculation of phase equilibria of mixtures of chemical components. The orginal goal for the development of this method was to enable the estimation of properties of mixtures which contain supercritical components. These class of substances couldn't be predicted with established models like UNIFAC.

Principle
PSRK is a group contribution equation of state. This is a class of prediction methods which combine equations of state (mostly cubic) with activity coefficient models based on group contributions like UNIFAC. The activity coefficient model is used to adapt the equation of state parameters for mixtures by a so called mixing rule.

The usage of a equation of state introduces all thermodynamic relations defined for for equations of state into the PRSK model. This allows the calculation of densities, enthalpies, heat capacities, and more.

Equations
PSRK is based on a combination of the Soave-Redlich-Kwong equation of state with a mixing rule whose parameters are determined by UNIFAC.

Equation of State
The equation of state of Soave is defined as follows:

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

The originally used α-function has been replaced by the function of Mathias-Copeman .

$$\alpha(T_r) = \left[ 1+c_1 \left(1-\sqrt T_r \right) +c_2 \left(1-\sqrt T_r \right)^2 +c_3 \left(1-\sqrt T_r \right)^3 \right]^2$$

The parameters of the Mathias-Copeman equation are fitted to experimental vapor pressure data of pure components and guarantee therefore a better description of the vapor pressure than the original relation. The equation form has been chosen because the original Soave form can be obtained by setting the parameters c2 und c3 to zero. In addition, the parameter c1 can be obtained from the acentric factor by the relation

$$c_1 = 0.48 + 1.574 \; \omega - 0.176 \; \omega^2$$

if no fitted Mathias-Copeman parameter are available.

Mixing Rule
The PSRK mixing rule calculate the parameter a and b of the equation of state by

$$ { a \over {b R T} } = \sum_i { \left( {x_i} {a_{ii} \over {b_i R T}} - { { {g^E_0} \over {R T} } + \sum { x_i ln { b \over {b_i} } } \over {0.64663} }\right) } $$

and

$$ b = \sum_i x_i \; b_i$$

by the parameters ai und bi of the pure substances, their mole fractions xi and the excess Gibbs energy gE. The excess Gibbs energy is calculated by a slightly modified UNIFAC model.

Model Parameters
For the equation of state PSRK needs the critical temperature and pressure and additionally at least the acentric factor for all pure components in the considered mixture.

A better quality can be achieved if the acentric factor is replaced by Mathias-Copeman constants which have been fitted to experimal vapor pressure data of pure components.

The mixing rule uses UNIFAC which needs a variety of UNIFAC-specific parameters. Beside some model constants the most important are group interaction parameters which are fitted to experimental vapor-liquid equilibria of mixtures.

Hence, for high-quality model parameters experimental data (pure component vapor pressures and VLE of mixtures) are needed. These are normally provided by factual data banks like the Dortmund Data Bank which has been the base for the PSRK development. In few cases additionally needed data have been determined experimentally if no data have been available from other sources.

Example Calculation
The prediction of a vapor-liquid equilibrium is successful even in mixtures containing supercritical components.



The mixture has to be subcritical though. In the given example carbon dioxide is the supercritical component with Tc=304.19 K and Pc=7475 kPa. The critical point of the mixture lies at T=411 K und P≈15000 kPa. The composition of the mixture is near 78 mole% carbon dioxide und 22 mole% cyclohexane.

PSRK describes this binary mixture quite well, the dew point curve as well as the bubble point curve and the critical point of the mixture.

Model Weaknesses
In a PSRK follow-up work some model weaknesses are quoted:


 * The gradient of the Mathias-Copeman α-function is without any thermodynamic background and, if extrapolated to higher temperatures, the described vapor pressure curve tends to diverge.
 * The Soave-Redlich-Kwong equation of state describes the vapor densities of pure components and mixtures quite well but the deviations of the liquid density prediction are high.
 * For the VLE prediction of mixtures with components which have very differing sizes (e. g. Ethanol, C2H6O, and Eicosane, C20H44) larger systematic errors are found.
 * Heats of mixing and activity coefficients at infinite dilution are predicted poorly.

Weblinks

 * Short PSRK description from the developers
 * UNIFAC Consortium at der Carl von Ossietzky University Oldenburg (develops the PSRK model since 2005)