Propagation of Uncertainty

In the design of chemical processes, the importance of understanding parametric uncertainty is widely recognised. Propagation of uncertainty calculates the effects of the uncertainty in model inputs on the model outputs. This information is essential when quantifying confidence in a model’s outputs and helps to determine whether model outputs will meet requirements given the anticipated variation in inputs. Once probability distributions are available for the input parameters, the objective is to determine the probability distribution function (PDF) or error range of the output variables of interest; the process is illustrated in Figure 4.1 (Vasquez and Whiting, 2005).

Figure 4.1: Propagation of uncertainty process flow through a process model.

Whiting and co-workers provided perhaps the most notable contribution to parametric uncertainty quantification of thermodynamic models. The authors performed extensive investigations to determine the impact of physical property uncertainty on process design and demonstrated uncertainty propagation techniques through the use of MCS (Reed and Whiting, 1993; Whiting, 1996; Vasquez and Whiting, 1998; Clarke et al., 2001; Vásquez, Whiting and Meerschaert, 2010).

As an example, Whiting et al., (1999) studied the effect of the experimental data source on the uncertainty of thermodynamic models by using different data sets for the regression of NRTL binary interaction parameters. The regressed information was used to perform MCS to generate 100 sets of binary interaction parameters to represent a statistical sampling of the most probable model parameters, assuming accurate and precise measurement of experimental data. The

Input Uncertainty

Deterministic Model

Uncertainty Propagation

Output Uncertainty

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statistical analysis concluded that a dependency existed between the values of the binary interaction parameters and the data source. Furthermore, combining different experimental data sources into a combined experimental data pool did not improve the uncertainty estimates. However, performing analysis on individual data sets yielded an improved quantification of uncertainty and risk (Whiting et al., 1999). The study also observed that the uncertainties associated with different experimental data sources could not be correlated with the goodness of fit of the regressions. At the time, limited quantitative parametric uncertainty information was available and these earlier studies were performed using average uncertainties estimated for different physical properties, such as the evaluations performed by the Design Institute for Physical Properties (DIPPR).

Hajipour and Satyro (2011) expanded on these techniques through the development of an extensive database of critical parameters, acentric factors and binary interaction parameters and encoded it with parametric uncertainty information (Hajipour and Satyro, 2011; Hajipour, 2013; Hajipour et al., 2014). A unique feature of their database is that it includes fundamental physical property data available through NIST Thermo Data Engine (TDE). This feature is a notable improvement on the approach of earlier studies as it is based on actual parametric uncertainty. The authors also used MCS to evaluate the effect of the propagated physical property uncertainties on process simulation results.

Bjørner, Sin and Kontogeorgis, (2016) recently applied uncertainty quantification as a tool to evaluate and compare the performance of the cubic plus association (CPA) thermodynamic model. The authors note that in an effort to improve the performance of models such as CPA and SAFT additional terms are often added, leading to an increase in adjustable parameters. The resultant high correlation between parameters makes it difficult to estimate pure compound parameters that are unique. The field of heavy hydrocarbon supercritical fluid extraction with CO2 is particularly plagued by this problem due to the quadrupolar moment of the solvent. A

MCS was used to propagate the effect of parametric uncertainty to various model properties and binary phase equilibrium calculations by using three and five adjustable parameter model variants. The study confirmed significant parameter uncertainties because of high parameter correlation.

Mathias (2014) proposed an error propagation scheme as an approach to perform parametric uncertainty quantification. In this approach, probabilistic methods (i.e. Monte Carlo) are not utilised, but instead, a semi-empirical method perturbs the activity coefficients in proportion to

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the derived experimental data uncertainty. This is based on the simplifying assumption that the liquid mixture can be approximated by the Margules equation as a set of pseudo binaries. Mathias demonstrated the use of the method and how it relates uncertainties in K-values with variabilities in process design through several examples (Mathias, 2014; Mathias, 2016). The results were consistent with the rules of thumb published by Fair (1980). As this technique cannot provide a probabilistic view it is not considered for this work.

The majority of the literature related to chemical process design uncertainty quantification has used probabilistic methods to represent sources of uncertainty and then methods, such MCS, to propagate the sources through a deterministic model. MCS is considered the state of the art method (Villegas et al., 2012). In the event that a probabilistic description of the required input parameter uncertainties is not available, an alternative non-sampling-based method is generalised polynomial chaos expansion (gPC). This method is described as a spectral representation of a random process by the orthonormal polynomials of a random variable (Duong et al., 2016).

Although Ghanem and Spanos, (2003) reported that gPC is suitable for engineering purpose it has seen limited application in chemical process design. Villegas et al., (2012) reported a study in which polynomial chaos expansion was used for uncertainty quantification in chemical reactors. In this case, it was coupled with computational fluid dynamic calculations for modelling and simulating reactive flows. Duong et al.,(2016) applied gPC for uncertainty quantification and sensitivity analysis of a complex chemical process. The gPC method was developed in a Matlab™ environment and connected to HYSYS™. The gPC method performed in agreement with the results from a Monte Carlo method. The gPC method is expected to mature in the future and appears to be promising, but will not be considered in this thesis. The details of the MCS approach is reviewed in the following section.