Uncertainty analysis of the CPA and a quadrupolar CPA equation of state – With emphasis on CO2

Abstract

The parameters of thermodynamic models, such as the cubic plus association (CPA) equation of state, are subject to uncertainties due to measurement errors in the experimental data that the models are correlated to. More importantly as the number of adjustable parameters increase, the parameter estimation problem becomes more complicated due to parameter identifiability issues. In this work the uncertainties in the pure compound parameters of CO 2 are investigated using several different CPA approaches, including a new quadrupolar CPA. The uncertainties are estimated using both least squares estimation and the bootstrap method for parameter estimation. The uncertainties in the parameters estimated from the bootstrap method are propagated to physical property and vapor liquid equilibrium predictions using Monte Carlo simulations.The results indicate that both the pure compound parameter uncertainty and the propagated uncertainty is negligible for the modeling approaches which employ three adjustable parameters. For modeling approaches with more than three adjustable parameters, however, there may be significant uncertainties in the pure compound parameters, as well as a high degree of correlation between the adjustable parameters. This results in significant propagated errors * Corresponding author Email address: mgabj@kt.dtu.dk (Martin G. Bjørner)Preprint submitted to Fluid Phase Equilibria December 31, 2015for certain output properties. To reduce the uncertainty in the adjustable model parameters the heat of vaporization was included as additional correlation data.This resulted in parameter distributions which followed a normal distribution more closely, however, the correlation between the adjustable parameters remained high. Overall the results indicate, that it is important to report parameter uncertainties together with their correlation matrix when a model is developed, so that better informed decisions can be made, for instance about which model extension, or association scheme should be employed.

Modeling derivative properties and binary mixtures with CO2 using the CPA and the quadrupolar CPA equations of state

“…We demonstrate in Bjørner et al [36] that the parameter uncertainty is substantial when qCPA is employed…”

“…We have recently investigated the uncertainty of the pure compound parameters for CO 2 using several different modeling approaches including the new qCPA [36]. To estimate the uncertainty of the model parameters real experimental data are employed, rather than a correlation, and a slightly different objective function is employed, which uses the variance as the weighting factor, rather than the experimental value.…”

New association schemes for mono-ethylene glycol: Cubic-Plus-Association parameterization and uncertainty analysis

“…Another approach was taken by Bjørner et al [28], who applied a statistical method -called bootstrappingto determine the confidence intervals for the parameterization of CO 2 within different variations of CPA.…”

“…Returning to statistically based methodology, Bjørner et al [28] proposed parameterization via the bootstrapping methods first proposed by Bradley Efron in the 1970s [81]. Although it took some years for the popularization of the method (even within the statistical community), its use became much more prevalent from the mid-1980s onwards [82][83][84] with the 1986 article of Efron and Tibshirani now having…”

“…For the parameter distributions and co-variance plots we chose to examine a 0 rather than Γ, in order to decouple the effects of b in Γ. For the co-variance plots it is noticeable that there is significantly less correlation than seen for the methods used by Bjørner et al [28]. The clearest case of correlation is shown for ε/R with a 0 .…”