Simulations of thermodynamic systems are necessary in the design of new industrial processes. It is therefore essential that the results of such simulations reflect how the system would perform in reality. At the same time, it is an advantage if the thermodynamic simulations do not require much time per simulation, as rapid results enable more options to be investigated in a given amount of time. For optimisation programs it is important that the thermodynamic models are robust and accurate in order for the program to obtain useful information when evaluating the objective function and the constraints. In this study we investigate the trade-offs and compromises between computational time, accuracy and robustness on an organic Rankine cycle (ORC) optimisation problem.
Although thermodynamic models play a major role in simulations of power cycles, little work seems to have been dedicated to investigating how they influence the accuracy, computational time and robustness of the simulation. Because it has been difficult to find such studies for power cycles, we have expanded the focus of our literature survey to include the study of thermodynamic models for other processes as well. However, of all the studies that were found that investigated equations of state, none studied the robustness of the thermodynamic model when implemented in a simulation tool, nor the time it took to obtain results. In fact, they only examined the accuracy of the various thermodynamic models.
Frutiger et al. [
1] studied how the simulation result of a power cycle can significantly change as a result of the uncertainty of a few critical parameters such as critical temperature, critical pressure and the acentric factor. Applying a Monte Carlo procedure to 40 different working fluids, they showed how ignoring the uncertainties in the parameters can lead to highly erroneous simulation results, as the 95% confidence interval may be large in cases where important parameters are very uncertain. Furthermore, switching the ranking scheme for the potential fluids between being based on mean net power and according to the lower bound of the 95% confidence interval, changed the relative standing of the fluids. Stijepovic et al. [
2] studied how measures of Rankine cycle performance, such as net power, exergy efficiency and cost of power, are influenced by a few parameters of the working fluid, qualitatively showing for example that the molar volume of saturated liquid has a negative impact. Their results highlight the need for accurate calculations on this parameter. Mazzoccoli et al. [
3] studied the use of seven different equations of state (EOSs) to predict the temperatures, pressures and densities of binary mixtures with CO2, and compared the results with experimental data. The authors comment that deviations in temperature and pressure from the experimental values are negligible, and so the study focuses on predicting the liquid density. The results show that the cubic EOSs have the highest margin of error, with Lee–Kesler–Plocker (LKP), Perturbed Chain – Statistical Associating Fluid Theory (PC-SAFT) and Groupe Européen de Recherches Gazières (GERG) EoS all offering better predictions. For LKP and PC-SAFT, the authors had to fit the binary interaction parameters, k
ij, to obtain the best results. This was unnecessary for GERG, which managed to have among the lowest deviations in this study. The use of different EOSs to predict thermodynamic parameters was also looked into by Li and Yan [
4,
5], who employed various cubic EOSs to predict the vapour–liquid equilibria (VLE) and molar volumes of liquids and gas for binary mixtures with CO2. Abdollahi-Demneh et al. [
6] studied 23 EOSs and evaluated their ability to predict thermodynamic parameters such as vapour pressure and enthalpy of vaporization of 102 pure substances, and found that LKP and the three-parameter cubic EOS from Patel–Teja (PT) were among the best EOSs, having an average absolute percentage deviation of less than 3.41% for the saturated liquid molar volume and less than 1.78% for the saturated vapor molar volume of alkanes and cycloalkanes. Liu et al. [
7] studied how well seven EOSs were able to predict the densities of five pure hydrocarbons across a large range of pressures. They also studied implementations of the Soave–Redlich–Kwong (SRK) EOS that used temperature-dependent volume translations, and a volume translation that was both temperature-and density-dependent. This differed from other studies that did not use these dependencies. Their findings show that PC-SAFT consistently has the lowest mean absolute deviation. The study also showed that temperature and density translating the SRK EOS did not much improve the result of the SRK EOS. Orbey and Sandler [
8] studied different mixing rules in combination with the Peng–Robinson (PR) EOS for estimating the excess enthalpy and VLE of four binary mixtures. The authors consider this to be a stringent test for the models. They found that fitting the parameters to data for one property, and then predicting the other, yielded poor accuracies, although fitting both simultaneously gave good results for both parameters. Brown [
9] studied three different implementations of the PR EOS, and how the accuracy of the EOS changes, using BACKONE and REFPROP 8.0 as references. The three implementations differed in how many parameters were inputs in the calculation, with the remaining parameters having to be estimated. As expected, their results show that the implementation with the fewest estimated parameters gave the most accurate results. While previous studies compared purely simulated results to certain references, Kuboth et al. [
10] showed how combining experimental data with the simulations by implementing correlations for heat exchanger performance, pressure loss and refrigerant sub cooling could significantly improve the accuracy of the results gained from the simulations of power cycles.
Other studies have focused primarily on the accuracy with regards to a single or a few parameters of the various EOSs, and while it is very important that the EOSs accurately predict thermodynamic parameters, there are additional characteristics of the EOSs that are important when they are used in simulation or optimization tools. One such aspect is how robust an EOS is in the given simulation tool; if the simulation often crashes or requires frequent restarts, it may be too cumbersome to use in spite of whatever improvements in accuracy it may achieve. Complex models that are highly accurate are often less robust than their simpler alternatives, particularly when calculating phase equilibria. The reason for poor robustness is explained in detail by Wilhelmsen et al. [
11]. The crux of the problem is that the functional relationship between two parameters (e.g., pressure and density at a constant temperature) can have several roots in these more advanced models, and therefore there are several infeasible solutions that must be eliminated. Another feature that is important in an EOS implementation is its computational time. Having an EOS that requires less computational time is an advantage, as it allows the user to obtain results rapidly and make adjustments to the model where necessary. These facets have not been discussed and compared in literature to the best knowledge of the authors. Given how little these aspects have been discussed in the literature, we set out to provide quantitative data on all three categories. The EOSs are subsequently ordered based on overall performance. Furthermore, where the published studies have primarily revolved around how accurately EOSs estimate individual parameters, this work focuses on how well the EOSs performed when optimizing an ORC and shows how the use of different models find different sets of optimal operating conditions and different maxima for power production.