High pressure modifications improve the accuracy of the van der Waals equation of state
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Peer-reviewed literature indicates that various temperature and pressure-based modifications to the van der Waals and related cubic equations of state enhance their accuracy in modeling phase behavior and fluid properties.
Abstract
The well-known Maxwell construction1 (the equal-area rule, EAR) was devised for vapor liquid equilibrium (VLE) calculation with the van der Waals (vdW) equation of state (EoS)2. The EAR generates an intermediate volume between the saturated liquid and vapor volumes. The trajectory of the intermediate volume over the coexistence region is defined here as the Maxwell crossover, denoted as the M-line, which is independent of EoS. For the vdW or any cubic3 EoS, the intermediate volume corresponds to the “unphysical” root, while other two corresponding to the saturated volumes of vapor and liquid phases, respectively. Due to it’s “unphysical” nature, the intermediate volume has always been discarded. Here we show that the M-line, which turns out to be strictly related to the diameter4 of the coexistence curve, holds the key to solving several major issues. Traditionally the coexistence curve with two branches is considered as the extension of the Widom line5,6-9. This assertion causes an inconsistency in three planes of temperature, pressure and volume. It is found that the M-line is the natural extension of the Widom line into the vapor-liquid coexistence region. As a result, the united single line coherently divides the entire phase space, including the coexistence and supercritical fluid regions, into gas-like and liquid-like regimes in all the planes. Moreover, along the M-line the vdW EoS finds a new perspective to access the second-order transition in a way better aligning with observations and modern theory10. Lastly, by using the feature of the M-line, we are able to derive a highly accurate and analytical proximate solution to the VLE problem with the vdW EoS.
Modification to the Van der Waals Equation of State
In this paper, we modify the VDW equation of state by adding a temperature factor to it. As a result, we give out a good phase diagram and the correlation of the reduced pressure and the reduced temperature when a balanced liquid-gas coexistence canonical argon-like system is considered.
Published as: Journal of Phase Equilibria, 2003, December,Vol.24, No.6, Page 533-541
arXiv categories: cond-mat.stat-mech
particularly in the modeling of vapor–liquid equilibrium and chemical engineering process design. The van der Waals equation of state may be written as ( p +
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 applied in the fields of physical chemistry and chemical engineering, particularly in the modeling of vapor–liquid equilibrium and chemical engineering
whe…
For
P
r
<
1
{\displaystyle P_{\text{r}}<1}
and
T
r
<
1
{\displaystyle T_{\text{r}}<1}
, the system is in a state of vapor–liquid equilibrium. In that situation, the reduced cubic equation of state…
The attractive force parameter ‘a’ was considered to be a constant with respect to pressure in the Peng–Robinson equation of state. The modification, in which parameter ‘a’ was treated as a variable with respect to pressure for multicomponent multi-phase high density reservoir systems was to improve accuracy in the prediction of properties of complex reservoir fluids for PVT modeling. The variation was represented with a linear equation where a1 and a2 were the slope and the intercept respectively of the straight line obtained when values of parameter ‘a’ are plotted against pressure.
This modification increases the accuracy of the Peng–Robinson equation of state for heavier fluids particularly at high pressure ranges (>30MPa) and eliminates the need for tuning the original Peng–Robinson equation of state. Tunning was captured inherently during the modification of the Peng-Robinson Equation.
The Peng-Robinson-Babalola-Susu (PRBS) Equation of State (EoS) was developed in 2005 and for about two decades now has been applied to numerous reservoir field data at varied temperature (T) and pressure (P) conditions and shown to rank among the few promising EoS for accurate prediction of reservoir fluid properties especially for more challenging ultra-deep reservoirs at High-Temperature High-Pressure (HTHP) conditions. These works have been published in reputable journals.
While the widely used Peng-Robinson (PR) EoS of 1976 can predict fluid properties of conventional reservoirs with good accuracy up to pressures of about 27 MPa (4,000 psi) but fail with pressure increase, the new Peng-Robinson-Babalola-Susu (PRBS) EoS can accurately model PVT behavior of ultra-deep reservoir complex fluid systems at very high pressures of up to 120 MPa (17,500 psi).
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