Simple Perturbation Scheme to Consider Uncertainty in Equations of

Article pubs.acs.org/jced

Simple Perturbation Scheme to Consider Uncertainty in Equations of State for the Use in Process Simulation J. Burger,*,† N. Asprion,‡ S. Blagov,‡ and M. Bortz§ †

Laboratory of Engineering Thermodynamics, University of Kaiserslautern, Erwin-Schroedinger-Strasse 44, 67663 Kaiserslautern, Germany ‡ BASF SE, Carl-Bosch-Straße 38, 67063 Ludwigshafen, Germany § Fraunhofer Institute for Industrial Mathematics ITWM, Fraunhofer-Platz 1, 676663 Kaiserslautern, Germany ABSTRACT: For use in sensitivity studies and robust optimization, uncertainties in thermodynamic models have to be considered in simulations. Instead of using less vivid confidence intervals of the models’ fitting parameters, this work proposes a perturbation scheme for the fugacity coefficients in the liquid phase which can be applied to any thermodynamic model including equations of state. The perturbation is tuned via physically vivid and comprehensible parameters which have a direct relation to experimental uncertainties. Perturbations of pure component properties and mixture properties are separately accessible. After giving the theoretical framework of the perturbation scheme, the benefits of the approach are demonstrated using process simulation examples.

1. INTRODUCTION For the simulation of chemical processes, models of physicochemical properties are required. The accuracy of these models has a large impact on the simulation results.1,2 Whatever effort is undertaken to provide precise models, the models will always contain uncertainties which have their source in the model structure, the model parameters, or both. Consequently, the quantification of uncertainties and their consideration in process simulation has attained much attention in engineering practice and research. Studying the sensitivities of (uncertain) model parameters on the process performance has become a major topic in process simulation. The models in chemical process engineering are typically complex, and thus, nonlinear relationships between the changes in model parameters and output functions have to be generally expected.3 Thus, Whiting and co-workers used Monte Carlo sampling for their sensitivity studies of several chemical engineering applications, e.g. refs 4−7. Despite the fact that uncertainties in physical models and appropriate consideration in sensitivity studies and process simulation have now been discussed for over 20 years, the topic remains up-to-date in research. Sensitivity studies, in which model parameters are varied, appear constantly in various papers. Taking out CO2 absorption, for example, references 8−10 study the impact of uncertain model parameters on process characteristics. While the motivation of the studies is quite similar, different methods to vary the uncertain parameters are used (without going into detail on this point here). One point that hampers the practical use of the above methods in engineering practice is that the methods are challenging to implement in process simulation. This is © XXXX American Chemical Society

illustrated using the following example, assuming that the vapor pressure of a component is known only within some uncertainties and that the designer is interested in how much the uncertainty of this vapor pressure will influence the attained product purity in a distillation column. Performing a sensitivity analysis would require to vary the vapor pressure in simulations and to compare the resulting product purity with the nominal one. The problem in this approach is of practical nature: most process simulators do not have a feature for “changing the vapor pressure by 5%”. Typically, the vapor pressure is entered via a parametrized equation such as, for example, the Antoine equation. A desired change in the vapor pressure has to be evoked by changing the parameters of the Antoine equation in a suitable way. While, in the simple example of a specified vapor pressure perturbation, the underlying equation can be analyzed to determine which of the model parameters has to be changed to achieve the desired variation, this method is no longer practicable for physical quantities relying on more complex models. Consider the liquid phase activity coefficient resulting from a model for the molar excess Gibbs energy (gE-model), e.g., the non-random-two-liquid (NRTL) model. The model includes binary interaction parameters which might be varied in a sensitivity analysis, but their values and uncertainties are rather abstract. It is highly desirable to discuss uncertainties and sensitivities in more intuitive, physical interpretable quantities. In his seminal work, Mathias11 presents a practical solution to this problem. Uncertainties in process simulation are not Received: July 14, 2016 Accepted: November 1, 2016

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DOI: 10.1021/acs.jced.6b00633 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

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Using the pure ideal gas at T and p as an alternative standard state, μi is related to the fugacity coefficient φi as follows:

expressed in terms of uncertainty of abstract model parameters (e.g. confidence intervals), but in terms of uncertainty in physical quantities, Mathias used the activity coefficients in the liquid phase. They are accessible in a quite direct way from experiments, which also means that experimental data can be used to quantify the uncertainty in the activity coefficients. The implementation in the process simulation is achieved by perturbation of the activity coefficients γi.

ln γi* = ln γim + ln γi p

μi (T , p , x) = μi pure,id.g. (T , p) + RT ln(xiφi(T , p , x)) (3)

In the above equations, x represents the composition vector. The standard state of eq 2 can be rewritten using eq 3 resulting in μi pure,l (T , p) = μi pure,id.g. (T , p) + RT ln(φi pure,l(T , p))

(1)

(4)

Therein, the superscript m denotes the unperturbed or nominal model quantity, p the perturbation, and * the perturbed quantity. To calculate the perturbation term γpi , the mixture is represented as a set of pseudo-binaries in which γpi is calculated with a one-parameter Margules equation. For every component of the mixture, there is one parameter δi to adjust the magnitude of the perturbation γpi . This parameter can be used in sensitivity studies like any other model parameter. The perturbation term is constructed in a way that it does not produce thermodynamic inconsistencies, i.e., the perturbation becomes negligible for pure components. Mathias implemented the perturbation into ASPEN PLUS using standard features of the software and demonstrated the practical benefits in a distillation example.12 In the current work, a simple scheme is presented that enables the perturbation of the fugacity coefficients in a similar way. The scheme provides a way to perturb properties of the fluid phase equilibrium (such as the vapor pressure or phase distribution coefficients) when equations of state are used to model the fluid phases. The construction of the perturbation is chosen in a way such that the perturbation parameters reflect changes in the resulting physical quantities in the most direct fashion. As part of a bigger package to support data analysis and sensitivity studies, we have implemented the scheme into BASF’s in-house simulator CHEMASIM,13 also making optimization under uncertainties possible.14 Its practical use and the meaning of its parameters are discussed here along example calculations of vapor−liquid equilibria (VLE) in mixtures with and without supercritical components.

Inserting eq 4 into eq 2, comparing with eq 3 and applying to a liquid phase yields for the fugacity coefficient φli of a liquid phase, we get φil(T , p , x) = φi pure,l(T , p)γi(T , p , x)

(5)

The perturbed fugacity coefficient of the liquid phase is φil , *(T , p , x) = φi pure,l, *(T , p)γi*(T , p , x)

(6)

In the sequel, the perturbation is constructed: The perturbed fugacity coefficient of the pure liquid is obtained from the unperturbed one by multiplying with a perturbation factor Fi. φi pure,l, *(T , p) = Fi ·φi pure,l,m(T , p)

(7)

The superscript m marks the quantities of the nominal, unperturbed models. The perturbed activity coefficient γi* is calculated according to ref 11, cf. eq 1, leading to γi*(T , p , x) = γi p(Aij , x)γim(T , p , x)

(8)

γpi ,

For the perturbation we propose a term based on the twosuffix Margules equation for the multicomponent case, which is slightly different from the term given in ref 11. g E ,p 1 = RT 2

N

(9)

j=1 k=1 N

ln(γi p) =

N

∑ ∑ xjxk(Akjxj + Ajk xk) N

∑ ∑ xjxk[2xiAji + xj(Aij − Akj) − xkAjk ] j=1 k=1

2. PERTURBATION SCHEME The perturbation scheme of this work can be applied to any equation of state (EOS), which is used to model the nominal phase behavior. The phase equilibrium is determined by the iso-fugacity criterion, i.e., for each component i the fugacity f (j) i is identical in all coexisting phases j in equilibrium. The fugacity (j) f (j) i is calculated from the fugacity coefficient φi which, in turn, is determined from the EOS. The key idea of the present work is to perturb the fugacity coefficients in order to obtain derived physical properties (e.g., the vapor pressure) that deviate from the ones obtained with the nominal model. Only the fugacity coefficients of the compounds in the liquid phase are perturbed, while the respective fugacity coefficients in the vapor phase remain untouched. By this, cancellation of the perturbation in vapor−liquid equilibrium (VLE) calculation is avoided. Before the perturbation is explained, a look on the nominal, unperturbed case is taken. Using the standard state according to Raoult’s law (pure liquid at temperature T and pressure p of the actual state), the chemical potential μi of component i is related to the activity coefficient γi in the following way: μi (T , p , x) = μi

pure,l

(T , p) + RT ln(xiγi(T , p , x))

(10)

Inserting eqs 7 and 8 into eq 6, one finally obtains the perturbation of the fugacity coefficient φli in the liquid phase. φil , *(T , p , x) = γi p(Aij , x) ·Fi ·φil ,m(T , p , x)

(11)

Equation 11 is central to this work as it shows how the fugacity coefficients φli in the liquid phase are perturbed. The nominal values φl,m may be calculated from any EOS. The i respective calculation routines are present in most process simulators or property packages. Before the fugacity coefficient is, however, inserted into phase equilibrium conditions, it is perturbed, and the perturbed fugacity coefficient φl,i * is used in the calculations. To understand the physical consequences of the perturbations, consider for now a pure component in VLE. The phase equilibrium condition is given by

f il = f iv

(12)

p ·φil = p ·φiv

(13)

or

(2) B



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where f li and f vi are the fugacities in liquid and vapor phase, respectively. Let us consider the nominal case at constant temperature; then, for the nominal saturation vapor pressure ps,m, it holds that φil , m(ps , m ) = φiv , m(ps , m )

(14)

From the pressure dependency of the fugacity coefficient, ⎛ ∂lnφi ⎞ v − v pure,id.g. ⎜ ⎟ = RT ⎝ ∂p ⎠T

with v

pure,id,g.

(15)

= RT/p and eq 14, one obtains for the liquid phase

φil , m(p) = φiv , m(ps , m )

ps , m ⎡ exp⎢ p ⎣

∫p

p s,m

vl ⎤ dp⎥ RT ⎦

(16)

vv ⎤ dp⎥ RT ⎦

(17)

and for the vapor phase φi v, m(p) = φiv , m(ps , m )

ps , m ⎡ exp⎢ ⎣ p

∫p

p s,m

Let us now perturb the liquid fugacity coefficient as shown in eq 11. For pure components, γpi is equal to one. Thus, the properties are influenced solely by the perturbation factor Fi. For the perturbed liquid phase follows: φil , *(p) = Fiφil , m(p) = Fiφiv , m(ps , m )

ps , m ⎡ exp⎢ p ⎣

∫p

p s,m

Figure 1. Top: Nominal (FCUM = 1.0) and perturbed (FCUM = 0.8/1.2) vapor pressures of cumene. Bottom: Ratios of perturbed to nominal vapor pressures.

vl ⎤ dp⎥ RT ⎦

yields an increase in the vapor pressure. In Figure 1 (bottom), the ratio of perturbed vapor pressure ps,* to nominal vapor pressure ps,m is plotted for both values of Fi. It becomes clear that the ratios are approximately equal to the perturbation factors. The ratios approach the exact values of Fi for low temperatures and thus low pressures. The lower the pressures are, the closer the vapor phase to ideal behavior, and the lower the error in eq 21. The analytical relationship between the factor Fi and the pure component vapor pressure for a close to ideal vapor phase is the basis for its physical interpretation. Generally, Fi can also be used to approximately scale partial pressures in mixture VLEs, also for nonideal systems. This is shown in the example section. The factor γpi in eq 11 reflects additional perturbations on the mixture properties. For the calculation of γpi via eq 10, the factors Aij have to be specified. Because of the construction of the perturbation and the form of eq 9, these factors are directly related to the perturbation of the binary infinite dilution activity coefficients.

(18)

φv,i *(p)

φv,m i (p).

The vapor phase is not perturbed, i.e., = The phase equilibrium condition in the perturbed case is obtained by inserting the derived equations into eq 13. For the pressure p, the perturbed vapor pressure ps,* has to be inserted. ⎡ Fiφiv , m(ps , m )ps , m exp⎢ ⎣

∫p

ps , * s,m

⎡ = φiv , m(ps , m )ps , m exp⎢ ⎣

∫p

vl ⎤ dp⎥ RT ⎦ ps , * s,m

vv ⎤ dp⎥ RT ⎦

(19)

The last term on the left-hand side, the Poynting correction, is neglected for moderate pressures. One obtains after symbolic simplifications: ⎡ Fi = exp⎢ ⎣

∫p

ps , * s,m

vv ⎤ dp⎥ RT ⎦

Aij = ln γijp, ∞

(20)

This can be verified by considering the binary system i − j and inserting xi = 0 into eq 10. Using eq 1, yields

The latter equation gives an analytical relationship between Fi and the perturbation of the vapor pressure under the sole assumption of a negligible Poynting correction. The equation is further analyzed for the special case of an ideal vapor phase. Inserting vv = RT/p and integrating yields ps , * Fi = s , m p

(22)

γij*, ∞ = exp(Aij )γijm, ∞

(23)

This means, if one would set the factor Aij to a certain value, one would change the respective infinite dilution activity coefficient by the factor exp(Aij). It is important to note that although activity coefficients are used in the perturbation scheme, no gE-model is required in the nominal model.

(21)

Hence, it holds that the perturbation of the vapor pressure corresponds roughly to a multiplication with Fi if the vapor phase is close to an ideal vapor phase. This is demonstrated using the example of cumene (isopropyl benzene) and a PRSKEOS (model details of the model are given in the next section). In Figure 1 (top), cumene’s vapor pressure is plotted over the temperature. The nominal model (Fi = 1) and two perturbed models (Fi = 0.8/1.2) are shown. As expected, increasing Fi

3. APPLICATION EXAMPLES When discussing model performance, one distinguishes between known model error and a possible but unknown model error which is the model uncertainty. The presented scheme can be used to study the influence of both, known model error and model uncertainty, within process simulation C



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by varying the perturbation factors within certain intervals during parametric studies, sensitivity studies, or robust optimization. The intervals have to be chosen in a way so that the deviations caused by the perturbation are at least as big as the model error or the model uncertainty. For the sake of demonstration, in the application examples model errors are treated as model uncertainties, although they might be quantifiable exactly. 3.1. Vapor−Liquid Equilibrium of Cumene + pDiisopropylbenzene. In the first example, the vapor−liquid equilibrium in the binary system of cumene (CUM) and pdiisopropylbenzene (DPB) is considered. The two components shall be separated by distillation at 67 mbar. The fluid properties in the system are modeled with the Soave− Redlich−Kwong (SRK) EOS with Peneloux volume correction.15 (The choice of the nominal EOS type and flavor is not crucial for the perturbation scheme. This EOS version is chosen randomly for the sake of demonstration. The presented perturbation scheme is compatible to any nominal EOS as shown later.) The nominal parameters are given in the Appendix. The boiling diagram at 67 mbar is given in Figure 2 in which the nominal model is shown as solid curves.

Figure 3. Deviations of experimental data for the vapor pressure from the one calculated with the nominal model (nominal ps). The curves indicate the deviation of perturbed models from the nominal one. Top: cumene, □,17 Δ,18 and ○19. Bottom: p-diisopropylbenzene, □,20 Δ,21 ○,22 and ◇.23

Regarding DPB (bottom part of Figure 3), the experimental data show negative deviations from the nominal model, on average around −20%. A factor FCUM = 0.8 improves the model quality, and a variation between 0.58 and 1.05 captures the remaining uncertainties. Figure 2 shows the perturbed model with FCUM = 1.2 and FBPD = 0.8 as dashed curves. Through perturbation of the pure component vapor pressures, the description of the VLE of the binary mixture is significantly improved. Note that the mixture properties such as limiting activity coefficients have not been perturbed in this example and might yield further improvements. The influence of the uncertain vapor pressures on a process design is considered in a case study using a process simulator. The process, consisting of a single distillation column, is sketched along its specifications on the top part of Figure 4. Motivated by the discussion along Figure 3, five scenarios are evaluated: • S0 (Base case): FCUM = 1.2; FDPB = 0.8 • S1: FCUM = 1.35; FDPB = 0.58 • S2: FCUM = 1.1; FDPB = 0.58 • S3: FCUM = 1.35; FDPB = 1.05 • S4: FCUM = 1.1; FDPB = 1.05. The trade-off between the column height (number of equilibrium stages) and the reflux ratio required to meet the specifications is shown in the bottom part of Figure 4. The large uncertainty in the vapor pressure leads to a large uncertainty in the process design. When comparing the worst scenario with the best scenario (dashed curves), the minimum reflux ratio is roughly doubled as is the minimum number of stages. To avoid high production costs caused by large safety margins in

Figure 2. Vapor−liquid equilibrium curves in the system cumene + pdiisopropylbenzene at 67 mbar. Solid curves, nominal model; dashed curves, perturbed model with FCUM = 1.2 and FDPB = 0.8; markers, experimental data (BASF).

Experimental VLE data are shown for comparison as markers. There are considerable deviations of the nominal model from the experimental data. In particular, the pure component boiling temperatures deviate strongly. The nominal model has difficulties in representing the vapor pressures of the pure components. This becomes more obvious when experimental data of the pure vapor pressures are compared to the nominal model; see Figure 3 for a plot of the deviations over the temperature. There is a clear systematic error in the model which would need an updated fit. This is beyond the scope of this article. Here, the goal is to use the perturbation scheme to (a) generally improve the representation of the experimental data and (b) reflect the remaining uncertainties of the model. In the previous section, it has been shown that the vapor pressures can be directly perturbed using the factors Fi. Regarding cumene (top part of Figure 3), the experimental vapor pressure is throughout larger than the one resulting from the nominal model; that is, on average around 20% larger. Thus, a perturbed model using FCUM = 1.2 yields a vapor pressure closer to the experimental data (see solid curve in Figure 3). To reflect the remaining uncertainty (or error) of the model, the factor can be varied between 1.1 and 1.35. D



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when dividing the target value 1.013 bar by the nominal vapor pressure; for methane, the factor FME = 1.028. Assuming an ideal vapor phase, these factors should perturb the vapor pressures in a way so that the experimental boiling temperatures are in agreement with the perturbed model, cf. eq 21. Second, the mixture properties are considered keeping the previously estimated factors FCH and FME constant. The vapor− liquid distribution coefficients (K-values) of both components, exp Kexp ME and KCH, are determined (a) from the experimental data and (b) with the perturbed model (i.e., the one with the perturbation factors Fi from above), KME * and KCH * . The relative deviation between the K-values is shown with filled markers in Figure 6. The model shows a considerable deviation of about

Figure 4. Top: Sketch of the distillation column with the specification used in process simulation. Bottom: Trade-offs between the number of stages and reflux ratio in the scenarios S1, S2, S0, S3, S4, and S5 (curves from left to right).

construction, the model’s uncertainty should clearly be reduced before the design is finalized. 3.2. Vapor−Liquid Equilibrium of Methanol + Cyclohexane. In the next example, the VLE of the system methanol (ME) + cyclohexane (CH) at 1.013 bar is considered. Gross and Sadowski16 presented a PC-SAFT EOS for the system. Figure 5 compares this nominal model with experimental data

Figure 6. Relative deviations of experimental K-values24 from the Kvalues calculated with the perturbed models of the VLE of methanol (ME) + cyclohexane (CH) at 1.013 bar. Triangles, KME; squares, KCH. Filled markers: perturbed model with FME = 1.028; FCH = 1.005; γp,∞ CH,ME = 1.0. Hollow markers: perturbed model with FME = 1.028; FCH = 1.005; γp,∞ CH,ME = 1.325.

32.5% for KCH at small concentrations of cyclohexane. For a nearly ideal vapor phase, the K-value at infinite dilution is roughly proportional to the infinite dilution activity coefficient. Thus, in order to compensate the deviation in the K-values, the infinite dilution activity coefficient γ∞ CH,ME of CH is perturbed with γp,∞ CH,ME= 1.325. The resulting model is shown in Figure 5 as dashed curves. It produces pure component boiling temperatures which are closer to the experimental ones. The updated deviations in the K-values are shown in Figure 6 as hollow markers. As expected, the deviations (especially at infinite dilution of CH) are significantly reduced by the perturbation. Perturbation of the other infinite dilution activity coefficient γp,∞ ME,CH does not significantly improve the agreement of the perturbed model with the data. This can be also anticipated from Figure 6, as there are only small deviations for KME at small concentrations of methanol. To obtain the numerical values of the perturbation factors, only one VLE flash calculation per factor was required: the simulations of both pure components’ vapor pressures, and the simulation of the VLE at high dilution of cyclohexane. The determination of the factors that are required to investigate model deficits is thus comprehensible and fast. A fit of the perturbation factors to all experimental data would indeed be possible but is not necessary. If the experimental data do not systematically deviate from the nominal model but scatter around the model, the value ranges of perturbation factors can be used to represent the uncertainty as demonstrated in the next example. 3.3. Gas Solubility of Carbon Dioxide in Methanol. Consider a gas removal process in which carbon dioxide (CO2)

Figure 5. Vapor−liquid equilibrium curves in the system methanol + cyclohexane at 1.013 bar. Solid curves, nominal model; dashed curves, perturbed model with FME = 1.028; FCH = 1.005; γp,∞ CH,ME = 1.325. Experimental data: □;24 Δ.25

in a binary boiling diagram. The model shows deviations in the pure component vapor pressures as well as deviations in the azeotropic temperature. This is because the model was originally fitted to experimental data at other conditions and simultaneously also to liquid−liquid equilibrium data.16 Here, the goal is to perturb the model to investigate the impact of the deviation of the model from the VLE data. First, the pure component properties, i.e., the vapor pressures, are perturbed. Using the nominal model, the vapor pressures are calculated at the experimental boiling temperatures. For cyclohexane, one obtains the factor FCH = 1.005 E



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The perturbation is directly applied to process simulation of an absorption column using the equilibrium stage model. See the top part of Figure 9 for the feed specifications of the

is absorbed from nitrogen (N2) with the solvent methanol (MEOH). The VLE is modeled again with the Soave− Redlich−Kwong (SRK) EOS; see the Appendix for the model parameters. Figure 7 shows the comparison of

Figure 9. Top: Sketch of the absorption column with the specification used in process simulation. Bottom: Trade-offs between number of stages and solvent rate to remove 99% of CO2. Solid line: nominal model. Dashed lines: uncertainties due to perturbations (cf. Figure 8).

problem. It is desired that 99% of the CO2 is removed from the gas stream. In the bottom part of Figure 9, the resulting tradeoff between column height (in equilibrium stages) and the solvent rate is shown. Through the perturbation scheme, the impact of uncertainties on the results is easily obtained showing the design engineer how much safety margin to foresee in the design.

Figure 7. Top: Solubility of CO2 in methanol (total pressure vs liquid mole fraction) at several temperatures (□, 318.15 K; +, 313.73 K; Δ, 308.15 K; ○, 298.15 K; ◇, 288.15 K). Symbols: experimental data;26,27 lines, nominal model. Bottom: Percent deviations between the nominal model and experimental data.

experimental solubilities of CO2 in methanol in comparison with the nominal model. The model shows quite some deviations (= uncertainties), as indicated in the bottom part of the Figure. The nominal model is perturbed so that the uncertainties in the pressure of up to ±10% are reflected. This is done using the FCO2 perturbation factor. Varying it between 0.9 and 1.1 leads to perturbations in the gas solubility as indicated by the dashed lines in Figure 8. (To not overload the Figure, only one temperature is shown. The perturbation at the other temperatures is analogous.)

4. CONCLUDING REMARKS A novel method for the perturbation of thermo-physical properties is presented, which is particularly applicable when the properties are modeled with equations of state. Two types of perturbation parameters have been introduced which immediately perturb the liquid fugacity coefficients and thus indirectly the properties of fluid phase equilibria. The perturbations are constructed in a way so that pure component properties can be perturbed independently from mixture properties. Unlike typical model/correlation parameters, the perturbation parameters have a vivid physical meaning: one (Fi) scales the pure component vapor pressure and the other (γp,∞ i,j ) the infinite dilution activity coefficients. Caloric properties, e.g., the enthalpy, are not perturbed by the presented scheme. We suggest perturbing these properties in direct fashion with additional schemes. The proposed perturbation scheme can be combined with any thermodynamic model of the fluid phases; it is straightforward to implement and computationally cheap. The scheme proves through examples that it is useful for model adjustment and the representation of model uncertainties. The values/intervals of the perturbation parameters are determined from few comprehensible property evaluations. The perturbation scheme provides a useful basis for sensitivity studies and robust optimizations in the simulation of physical properties and chemical processes.

Figure 8. Solubility of CO2 in methanol at ○ = 298.15 K. Symbols: experimental data;27 solid line, nominal model; dashed line, perturbed models with FCO2 = 1.1 (upper line) and FCO2 = 0.9 (lower line). F



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(11) Mathias, P. M. Sensitivity of Process Design to Phase EquilibriumA New Perturbation Method Based Upon the Margules Equation. J. Chem. Eng. Data 2014, 59 (4), 1006−1015. (12) Mathias, P. M. Effect of VLE Uncertainties on the Design of Separation Sequences by Distillation − Study of the Benzene− chloroform−acetone System. Fluid Phase Equilib. 2016, 408, 265−272. (13) Asprion, N.; Benfer, R.; Blagov, S.; Böttcher, R.; Bortz, M.; Berezhnyi, M.; Burger, J.; Harbou, E.; von Küfer, K.-H.; Hasse, H. INES − An Interface Between Experiments and Simulation to Support the Development of Robust Process Designs. Chem. Ing. Tech. 2015, 87 (12), 1810−1825. (14) Bortz, M.; Schwientek, J.; Küfer, K.-H.; Burger, J.; von Harbou, E.; Hasse, H.; Hirth, O.; Asprion, N.; Blagov, S. Minimizing the Impact of Uncertain Model Parameters on Process Design. Chem. Ing. Tech. 2015, 87 (8), 1061−1061. (15) Péneloux, A.; Rauzy, E.; Fréze, R. A Consistent Correction for Redlich-Kwong-Soave Volumes. Fluid Phase Equilib. 1982, 8 (1), 7− 23. (16) Gross, J.; Sadowski, G. Perturbed-Chain SAFT: An Equation of State Based on a Perturbation Theory for Chain Molecules. Ind. Eng. Chem. Res. 2001, 40 (4), 1244−1260. (17) Mangold, C. Die Dampfdrücke von Benzolkohlenwasserstoffen Der Homologen Reihe CnH2n-6 Und von Gemischen Aus Benzol Und Toluol. Sitzungsber Akad Wiss Wien Math Naturwiss Kl Abt 2a 1893, 102, 1071−1104. (18) Marsden, C. Solvents Guide; Cleaver-Kume: London, 1963; Vol. 2. (19) Willingham, C. B.; Taylor, W.; Pignocco, J. M.; Rossini, F. Vapor Pressures and Boiling Points of Some Paraffin, Alkylcyclopentane, Alkylcyclohexane, and Alkylbenzene Hydrocarbons. J. Res. Natl. Bur. Stand. 1945, 35 (3), 219−244. (20) Perry, J. H. Chemical Engineers’ Handbook, 3rd ed.; McGrawHill: New York, 1950. (21) Sakoguchi, A.; Ueoka, R.; Kato, Y.; Arai, Y. Vapor Pressures of Pentamethylbenzene and M-Diisopropylbenzene. Netsu Bussei 1994, 8 (8), 161−162. (22) Melpolder, F.; Woodbridge, J.; Headington, C. The Isolation and Physical Properties of the Diisopropylbenzenes. J. Am. Chem. Soc. 1948, 70 (3), 935−939. (23) Verevkin, S. P. Thermochemical Properties of Iso-Propylbenzenes. Thermochim. Acta 1998, 316 (2), 131−136. (24) Madhavan, S.; Murit, P. S. Vapourliquid Equilibria of the System Cyclohexanemethanol: Ebulliometric Method. Chem. Eng. Sci. 1966, 21 (5), 465−468. (25) Budantseva, L. S.; Lesteva, T. M.; Nemtsov, M. S. The Liquid Vapor Equilibria in Systems Comprising Methanol and C6 Hydrocarbons of Different Classes. Viniti Import. Dortm. Database 2015 1974, 1−17. (26) DDB. Dortmund Data Base (DBB); DDBST GmbH: Oldenburg, Germany, 2015. (27) Gui, X.; Tang, Z.; Fei, W. Solubility of CO2 in Alcohols, Glycols, Ethers, and Ketones at High Pressures from (288.15 to 318.15) K. J. Chem. Eng. Data 2011, 56 (5), 2420−2429.

APPENDIX

A1. Model Parameters

The system cumene (isopropylbenzene) + p-diisopropylbenzene is modeled with a Soave−Redlich−Kwong (SRK) EOS with Peneloux volume correction.15 The parameters are given in Table 1. The parameters of the PC-SAFT model of the Table 1. Parameters of the SRK-EOS for the Substances Used in the Examplesa cumene p-diisopropylbenzene carbon dioxide nitrogen methanol

Pc (bar)

Tc (K)

ω/−

32.00 24.50 73.83 34.0 80.84

638.35 684.00 304.21 126.2 512.5

0.3444 0.3587 0.27256 0.28997 0.23616

a The binary interaction parameters are kij = 0, except in the binary system (carbon dioxide + methanol) kCO2,MEOH = −0.172 + 0.0007225· (T/K).

binary system methanol + cyclohexane are taken from ref 16. The model parameters of the system carbon dioxide + nitrogen + methanol are also given in Table 1.

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AUTHOR INFORMATION

Corresponding Author

*Tel: +49 631 205 4028. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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Simple Perturbation Scheme to Consider Uncertainty in Equations of

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