VTPR (short for Volume-Translated Peng–Robinson)
is an estimation method for the calculation of phase equilibria of mixtures of chemical components. The original 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
In statistical thermodynamics, the UNIFAC method ( UNIQUAC Functional-group Activity Coefficients)Aage Fredenslund, Russell L. Jones and John M. Prausnitz, "Group-Contribution Estimation of Activity Coefficients in Nonideal Liquid Mixtures", '' ...
.
Principle
VTPR is a group contribution
equation of state
In physics, chemistry, and thermodynamics, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or internal ...
. This is class of prediction methods combine equations of state (mostly cubic) with
activity coefficient
In thermodynamics, an activity coefficient is a factor used to account for deviation of a mixture of chemical substances from ideal behaviour. In an ideal mixture, the microscopic interactions between each pair of chemical species are the same (or ...
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
Mix, mixes or mixing may refer to:
Persons & places
* Mix (surname)
** Tom Mix (1880-1940), American film star
* nickname of Mix Diskerud (born Mikkel, 1990), Norwegian-American soccer player
* Mix camp, an informal settlement in Namibia
* Mi ...
.
The usage of an equation of state introduces all thermodynamic relations defined for equations of state into the VTPR model. This allows the calculation of
densities
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek language, Greek letter Rho (letter), rho), although the Latin letter ''D'' ca ...
,
enthalpies,
heat capacities, and more.
Equations
VTPR is based on a combination of the
Peng–Robinson equation of state Cubic equations of state are a specific class of thermodynamic models for modeling the pressure of a gas as a function of temperature and density and which can be rewritten as a cubic function of the molar volume.
Equations of state are generally ...
with a mixing rule whose parameters are determined by UNIFAC.
Equation of state
The
Peng–Robinson equation of state Cubic equations of state are a specific class of thermodynamic models for modeling the pressure of a gas as a function of temperature and density and which can be rewritten as a cubic function of the molar volume.
Equations of state are generally ...
is defined as follows:
The originally used α-function has been replaced by the function of Twu, Bluck, Cunningham and Coon
.
The parameters of the Twu 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.
Mixing rule
The VTPR mixing rule calculate the parameter a and b of the equation of state by
with
and
by the parameters a
i und b
i of the pure substances, their mole fractions x
i and the residual part of excess Gibbs energy g
E. The excess Gibbs energy is calculated by a modified UNIFAC model.
Model parameters
For the equation of state VTPR needs the
critical temperature
Critical or Critically may refer to:
*Critical, or critical but stable, medical states
**Critical, or intensive care medicine
*Critical juncture, a discontinuous change studied in the social sciences.
*Critical Software, a company specializing in ...
and
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
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 Twu 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
vapor–liquid equilibrium
In thermodynamics and chemical engineering, the vapor–liquid equilibrium (VLE) describes the distribution of a chemical species between the vapor phase and a liquid phase.
The concentration of a vapor in contact with its liquid, especially at ...
and liquid–liquid equilibrium data, activity coefficients of mixtures, heats of mixing) are needed. These are normally provided by factual data banks like the
Dortmund Data Bank
The Dortmund Data Bank (short DDB) is a factual data bank for thermodynamic and thermophysical data. Its main usage is the data supply for process simulation where experimental data are the basis for the design, analysis, synthesis, and optimizati ...
which has been the base for the VTPR development.
Volume translation
VTPR implements a correction to the pure component densities resp. volume. This volume translation corrects systematic deviations of the Peng–Robinson equation of state (PR EOS).
The translation constant is obtained by determining the difference between the calculated density at T
r=0.7 and the real value of the density obtained from experimental data. T
r is close to the
normal boiling point
The boiling point of a substance is the temperature at which the vapor pressure of a liquid equals the pressure surrounding the liquid and the liquid changes into a vapor.
The boiling point of a liquid varies depending upon the surrounding envir ...
for many substances. The volume translation constant c
i
is therefore component specific.
This volume/density translation is then applied to the complete density/volume curve calculated by the PR EOS. This is sufficient because the calculated curve has the right slope and is only shifted.
The Peng–Robinson equation of state is then
Modifications to the UNIFAC model
UNIFAC use two separate parts to calculate the activity coefficients, a combinatorial part and a residual part. The combinatorial part is calculated only from group specific constants and is omitted in the VTPR model. VTPR use only the residual part calculated from interaction parameters between groups.
This has the side effect that r
i values (van der Waals volumes) are not needed and only the van der Waals surfaces
q
i are used.
In addition, the q
i values are not constant properties of the groups, instead they are adjustable parameters and fitted to experimental data together with the interaction parameters between groups.
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 T
c=304.19 K and P
c=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.
VTPR describes this binary mixture quite well, the
dew point
The dew point is the temperature to which air must be cooled to become saturated with water vapor, assuming constant air pressure and water content. When cooled below the dew point, moisture capacity is reduced and airborne water vapor will cond ...
curve as well as the bubble point curve and the critical point of the mixture.
Electrolyte systems
VTPR normally cannot handle
electrolyte
An electrolyte is a medium containing ions that is electrically conducting through the movement of those ions, but not conducting electrons. This includes most soluble salts, acids, and bases dissolved in a polar solvent, such as water. Upon dis ...
containing mixtures because the underlying UNIFAC doesn't support salts. It is however possible to exchange the UNIFAC
activity coefficient
In thermodynamics, an activity coefficient is a factor used to account for deviation of a mixture of chemical substances from ideal behaviour. In an ideal mixture, the microscopic interactions between each pair of chemical species are the same (or ...
model by a model that supports electrolytes like
LIFAC.
[Eileen Collinet, ]Jürgen Gmehling
Jürgen Gmehling (born January 13, 1946 in Duisburg) is a retired German professor of technical and industrial chemistry at the Carl von Ossietzky University of Oldenburg.
Biography
His career started with an apprenticeship as a laboratory assi ...
, "Prediction of phase equilibria with strong electrolytes with the help of the volume translated Peng–Robinson group contribution equation of state (VTPR)", ''Fluid Phase Equilibria'', 246(1–2), 111–118, 2006. ,
See also
*
PSRK PSRK (short for Predictive Soave–Redlich–Kwong)
is an estimation method for the calculation of phase equilibria of mixtures of chemical components. The original goal for the development of this method was to enable the estimation of properties o ...
(Predictive Soave–Redlich–Kwong), VTPRs predecessor of the same group contribution equation of state type but using a different equation of state, a different α function, and a different UNIFAC modification.
Literature
External links
*
{{DEFAULTSORT:Vtpr
Thermodynamic models