Thermodynamic Consistency-Based Validation Approach for Equation of State Methods in Process Simulators

1.Objectives

Development of physical property estimation module on process simulators requires proper validation approaches to confirm that the entire process of derivation and programing of a number of equations has been correctly performed. Conventional approaches, such as comparison between analytical solutions and numerical solutions, can be applicable to each step in the process. However, recent equations of state (EOS) method involve huge numbers of complex equations and steps in the process, and eventually require increased efforts for validation. So, this work seeks for a new validation approach which can check consistency between input and output of the EOS methods to detect any error in the entire process.

2. Methodology

Extensive variables such as Gibbs energy show 1^st order homogeneousness. For example residual Gibbs energy at constant temperature and pressure can be expressed by the following equation using its composition derivatives.

           ...(1)

Applying similar concept to excess Gibbs energy yields the well-known Gibbs-Duhem equation for activity coefficient, which has been applied to variation of experimental data.  In this work, we discuss validation approach based on Eq.(1) for EOS methods in process simulators.

3.Proposed Approach

Many of recent EOSs are described with residual Helmholtz energy at constant temperature and volume (A_res,T,V). Prediction of physical properties often get through density calculation at first, followed by that for fugacity coefficients. This work proposed a validation approach which checked that the following two procedures gave exactly the same value for residual Gibbs energy at constant temperature and pressure (G_res,T,P).

1)Direct calculation from A_res,T,V

The following equation was directly obtained from the definition of G_res,T,P and A_res,T,V.

                      ...(2)

2)Calculation from fugacity coefficient.

Just substituting fugacity coefficient into Eq.(1) yielded;

                            ...(3)

The dimensionless value of (G_res,T,P /nRT) by Eq.(2) must match that of Eq.(3), if all the derivation and programing of fugacity coefficient and density from EOS is perfect, and if the convergence tolerance for density is enough small.

4.Example Case

For the purpose of testing the approach, a standalone program for predicting vapor-liquid equilibrium (VLE) using Huron-Vidal method¹⁾ was created to confirm the ability of the above validation approach. When all the derivation and the programing had been perfectly done, (G_res,T,P /nRT)  calculated by Eq.(2) perfectly matched that of Eq.(3) in a precision limit of computation. The deviation was less than a value typically like 1.0x10^-15[-] (depending on calculation environment).

When we have intentionally bought an error into the derivation, or have intentionally eased the convergence tolerance for density, significant deviation in a magnitude of 1.0x10^-3[-](depending on calculation environment) was found between the values of (G_res,T,P /nRT), even when the calculated VLE did not alter a lot by the introduction of the error. Eventually the test results indicated that the above validation approach was enough sensitive to detect errors.   

5.Conclusions

We have established a software validation approach based on thermodynamic consistency to confirm that the entire process of the derivation and the programing was perfectly conducted in prediction modules for phase equilibrium. The approach turned out to be enough sensitive to detect error, and eventually contribute a lot to ensuring the software quality.

[Nomenclature]

(Alphabet) A: Helmholtz energy, G: Gibbs energy, n: quantity, R: Gas constant, T: Temperature, V: volume, x: mole fraction, Z: Compressibility factor, ∅: Fugacity coefficient

(Subscript)  i: component identification number, ig: ideal gas, res: residual

[Literature Cited]

1) Marie-Jose Huron and Jean Vidal; Fluid Phase Equil.3,255-271(1979)