Improved Property Predictions by Combination of Predictive Models

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Improved property predictions by combination of predictive models Sebastian Kaminski, Evagelos Kirgios, André Bardow, and Kai Leonhard Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.6b03125 • Publication Date (Web): 23 Feb 2017 Downloaded from http://pubs.acs.org on February 25, 2017

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Industrial & Engineering Chemistry Research

Improved property predictions by combination of predictive models Sebastian Kaminski, Evagelos Kirgios, André Bardow, and Kai Leonhard∗ Lehrstuhl für Technische Thermodynamik, RWTH Aachen University, 52056 Aachen, Germany E-mail: [email protected] Phone: +49 (0)241 80 98174. Fax: +49 (0)241 80 92255

Abstract Property predictions are essential when dealing with molecules that have not been investigated experimentally yet. The accuracy of current predictive models like predictive PCP-SAFT and COSMO-RS is limited. We propose a combination of predictive models in order to yield a higher accuracy. Information from both predictive models are combined in PCP-SAFT parameter space using a log-likelihood function. Experimental vapor pressures, enthalpies of vaporization and liquid densities over a wide temperature range are used to evaluate the predictions. The average error in the combined property prediction is lower than the error of the individual models. Even more, also the maximum error is considerably lowered.

1

Introduction

Accurate process design requires accurate thermodynamic property data for all molecules in the process. 1 Furthermore, the selection or even design of molecules can be a degree 1

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of freedom in process design, as is the case in the design of novel biofuels, 2 working fluids, 3,4 entrainers, 5 or solvents. 6–8 Determining the thermodynamic properties of all potential molecules experimentally requires a prohibitive effort. Alternatively, thermodynamic properties can be determined by predictive models. Most predictive models employ quantitative structure-property relationships (QSPR), group contribution (GC) models or quantum mechanics (QM). 9 QSPR models predict thermodynamic properties from the molecule structure, expressed by a set of descriptors. Often, the descriptors are chosen by data mining techniques. 10 Thus, most QSPR models are empirical. 11 QSPR models exist for many properties, e.g. thermophysical properties, 3,11,12 toxicity, 13 lubricity of biofuels, 14 and binary interaction parameters for equations of state, 15 just to mention a few. A comprehensive review is given in Ref. 11. GC models predict thermodynamic properties based on fragments or functional groups of a molecule. A prominent example of a GC model is the UNIFAC 16 excess Gibbs free energy model. A broad range of GC models have emerged for different variants of the statistical associating fluid theory (SAFT) equation of state. 17–25 GC models are parametrized by relating groups to thermodynamic properties. Consequently, GC models can be used for molecules for which all groups are parametrized only. QM-based methods can predict thermodynamic properties of a broader range of substances, in principle. A well-known QM-based model is the Conductor-like Screening Model for Real Solvents (COSMO-RS). 26,27 COSMO-RS calculates thermodynamic properties through surface charges of a molecule embedded in a conductor. Another QM-based prediction method is predictive Perturbed-Chain Polar SAFT (predictive PCP-SAFT). 28 Predictive PCP-SAFT 28 determines parameters of the PCP-SAFT 29–31 equation of state. In its current version, predictive PCP-SAFT can predict parameters for non-associating compounds containing carbon, hydrogen and oxygen. A method to obtain association parameters from highly accurate ab initio calculations was proposed recently. 32 Performing those ab initio cal-

2


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culations is, however, very demanding regarding computational and human time and therefore not applicable for routine applications today. Generally, QM-based prediction methods are often applied in screenings. 4,8 Still, accuracy of thermophysical properties calculated with QM-based methods is limited and more accurate prediction methods are desired for process design. Usually, there is a trade-off between a broad applicability of predictive methods and accuracy. While GC methods are applicable to a limited range of substances, within their core area of parametrization they are often more accurate than broadly applicable QMmethods. Recently, Kaminski et al. 33 suggested a method to improve parameters obtained by predictive PCP-SAFT with minimal experimental information. By combining predicted PCPSAFT parameters with a single experimental vapor pressure data point, the average error in vapor pressure prediction over a wide temperature range could be decreased from ∼40 % to < 5 %. We follow up on that approach by combining predictive PCP-SAFT not with experimental data but with data from an independent predictive method, thus remaining fully predictive. We choose COSMO-RS to provide these independent data due to its widespread use in industry and academia. We show that the combination of predictive PCP-SAFT and COSMO-RS yields more accurate thermodynamic properties than each individual prediction method.

2

Theory

In the following, we introduce predictive PCP-SAFT in Sec. 2.1 and COSMO-RS in Sec. 2.2 and finally, in Sec. 2.3, we discuss methods for combining predictive PCP-SAFT and COSMORS for more accurate predictions. The expected decrease in uncertainty is discussed with an example in Sec. 2.4 and computational details are given in Sec. 2.5.

3



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2.1

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The PCP-SAFT equation of state and parameter prediction

The Perturbed-Chain SAFT (PC-SAFT) 29 equation of state uses a chain consisting of hard spheres as a reference for a perturbation theory treatment of dispersion interactions. With additional terms for dipole-dipole 31 and quadrupole-quadrupole 30 interactions, the equation is termed Perturbed-Chain Polar SAFT (PCP-SAFT). Hard chains consist of m hard spheres with a diameter σ. The spheres interact with a dispersion interaction strength ε/k. Thus, the 3 PCP-SAFT parameters m, σ and ε/k have physical meaning. However, they do not relate directly to any observable property. 34 The polar interactions in PCP-SAFT are described by the dipole moment µ and the quadrupole moment Θ as parameters. Both µ and Θ can be taken from experiments or from quantum mechanical calculations and are not considered as adjustable pure component parameters by the authors of the multipole terms. 30,31 Following Ref. 28, we use multipoles calculated with quantum mechanics and, as in Ref. 33, we do not alter the multipole moments in our combination approach. The parameters m, σ and ε/k can be predicted for non-associating compounds with predictive PCP-SAFT. 28 Predictive PCP-SAFT is based on molecular descriptors obtained from quantum mechanical calculations. Those descriptors are then used in a multilinear correlation to obtain the predicted PCP-SAFT parameters. 28 All predicted PCP-SAFT parameters used in this study are calculated as described in Ref. 33. In the following, we summarize the 3 adjustable parameters ε/k, σ and m in the parameter vector θ





ε/k     θ=  σ    m

4

.


(1)

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2.2

COSMO-RS

The Conductor-like Screening Model (COSMO) 35 is a solvation model based on quantum mechanics. The considered molecule is placed in a cavity within an ideal conductor. The surface charge of the cavity is calculated to screen the electrostatic charges of the molecule. The Conductor-like Screening Model for Real Solvents (COSMO-RS) 26 considers closely-packed ideally-screened molecules. In reality, there is no conductor between molecules. The energy necessary to remove the conductor between the molecules can be calculated by means of the screening charges. That energy is the basis for the pressure-independent calculation of the chemical potential of the liquid phase as a function of temperature. 27 The chemical potential of the gas phase is calculated with an empirical equation based on quantum mechanically obtained gas phase energy and COSMO conductor energy. The chemical potentials can be used for further thermodynamic calculations like vapor pressure pvap and enthalpy of vaporization ∆H vap . For further information the reader is referred to Ref. 26. The liquid density at 298 K can be calculated using a quantitative structure property relationship (QSPR) model with descriptors from COSMO-RS. 36 The temperature at which the properties pvap and ∆H vap are used in our approach might influence the result. Not only might there be systematic errors, but also the variance might change over temperature. To investigate the influence of different temperatures, we choose temperatures of 0.5 and 0.7 of the critical temperature Tcrit for the predicted vapor pressure pvap and the enthalpy of vaporization ∆H vap . We also considered the liquid density ρliq which can be calculated at 298 K only. Those data are calculated with the commercially available COSMOthermX15 program, version 3.0, 37 using the TZVP parametrization level. The liquid density calculated with COSMO-RS is pressure-independent, but a pressure has to be specified for the calculation of the density with PCP-SAFT. When using density data for our approach, we thus assumed a pressure of 1 MPa, because most molecules in our set are in a liquid state at 298 K and 1 MPa.

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2.3

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Combination of predictive models

In this article, we suggest a method to combine predictive methods in order to yield more accurate predictions of vapor pressure, enthalpy of vaporization, liquid density and all other equilibrium properties accessible by PCP-SAFT. The combination of different methods requires some sort of averaging of information. The probably most intuitive approach is calculating the same property with both methods and averaging them (combination in property space). Another combination possibility is performed in PCP-SAFT parameter space, adapting the approach of Kaminski et al. 33 Both approaches are presented in the following. 2.3.1

Combination in property space

The combination in property space for the models A and B can be calculated easily analytically by minimizing an expression derived from the log-likelihood function 38

Φ (y) =

(y − yA )2 (y − yB )2 + V yA V yB

,

(2)

with y being the property calculated by the combined approach, yA and yB being the property calculated by the individual prediction models and VyA and VyB being the variances of the predictions. The minimum of Eq. 2 can be found by setting the first derivative with respect to y equal to zero and solving for y. The resulting solution is

y=

yA /VyA + yB /VyB 1/VyA + 1/VyB

.

(3)

If the variances of COSMO-RS and predictive PCP-SAFT are comparable, Eq. 3 simplifies to the arithmetic mean y=

yA + yB 2

.

(4)

At least for vapor pressure, this assumption is valid for this study. In the following, we use Eq. 4 when referring to the combination in property space. 6


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A disadvantage of a combination in property space is that the property has to be calculated with both prediction methods for every state point of interest. As a consequence, this approach is not applicable if a property cannot be calculated by one of the prediction methods. 2.3.2

Combination in PCP-SAFT parameter space

Combining information from predicted PCP-SAFT parameters (Sec. 2.1) and from COSMORS (Sec. 2.2) in PCP-SAFT parameter space can be done using standard statistical methods like log-likelihood. 38 The combination yields new parameters for PCP-SAFT. The objective function for parameter combination is 33 Φ (θ) = ∆y T V yˆ −1 ∆y + ∆θ T V θ0 −1 ∆θ {z } {z } | |

(5)

,

stay close to predicted PCP-SAFT parameters

fit COSMO-RS prediction

ˆ , yî being the datum i from COSMO-RS, V yˆ the variance matrix of y ˆ, with ∆y = y (θ) − y ∆θ = θ − θ 0 , θ 0 the predicted PCP-SAFT parameter set, V θ0 the variance matrix of θ 0 . The second term of Eq. 5 is strictly convex and helps counteracting the occurrence of local minima. The inverted variance matrix V θ0 for predicted PCP-SAFT parameters is 33 ε/k

V −1 θ0



ε/k  0.00444 K  = σ  0   m 0

σ

m

0

−2

0

255.52 Å

−2

147.75 Å

−1



  −1 147.75 Å   .  140.79

(6)

The variance of data from COSMO-RS V yˆ is Vyˆ,i,j = yî yˆj Vyˆrel ,i,j

7

,


(7)


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with Vyˆrel ,i,j

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Ncomp

X yî,k − yi (θ exp ) yˆj,k − yj (θ exp ) k k = Ncomp − 1 k=1 yî,k yˆj,k 1

,

(8)

and Ncomp being the number of components used for calculating the variance. As in Ref. 33, we use a constant relative variance for properties. Properties like the vapor pressure can differ in orders of magnitude for different molecules. Therefore, the characterization of the vapor pressure variance seems more appropriate in a relative way. The relative variance Vyˆrel is calculated using properties yi calculated with PCP-SAFT parameters which are fitted to experimental data θ exp from Ref. 28 as reference, instead of using the experimental data directly. This reference is consistent with θ exp being the reference for the calculation of V θ0 . 33 Hence, V yˆ can be seen as an effective variance, combining the prediction-error of COSMO-RS and the model-error of PCP-SAFT. The same diverse set of 49 molecules that Kaminski et al. 33 used to calculate V θ0 were used for calculating V rel yˆ : pvap (0.5 Tcrit ) pvap (0.7 Tcrit ) ∆H vap (0.5 Tcrit ) ∆H vap (0.7 Tcrit ) ρliq (298 K) p

V rel yˆ

vap

(0.5 Tcrit )



0.31447   pvap (0.7 Tcrit )  −   = ∆H vap (0.5 Tcrit )  −0.021175   vap ∆H (0.7 Tcrit ) 0  ρliq (298 K) 0

−

−0.021175

0

0.20285

−0.012758

0

−0.012758

0.0028186

−

0

−

0.012075

0

0

0.0025478

0

   0    . 0    0.0025478   0.0010843 (9)

Molecules for which COSMOtherm issued a warning (out of core temperature range of COSMOtherm / below melting temperature) are omitted from the calculation of V rel yˆ . Not all data mentioned in Eq. 9 are used simultaneously for the parameter fitting in Eq. 5. Thus, V rel yˆ needs to be trimmed by deleting all rows and columns of data which are not used. For example, if the vapor pressure at 0.5 Tcrit and the enthalpy of vaporization at

8




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0.7 Tcrit is used, the variance matrix reads pvap (0.5 Tcrit ) ∆H vap (0.7 Tcrit ) V rel yˆ =

p

vap

(0.5 Tcrit )

∆H vap (0.7 Tcrit )

  

0.31447

0

0

0.012075



  .

(10)

The according ∆y is 



vap vap  p (0.5 Tcrit , θ) − pˆ (0.5 Tcrit )  ∆y =   ˆ vap (0.7 Tcrit ) ∆H vap (0.7 Tcrit , θ) − ∆H

.

(11)

We did not consider combinations with multiple vapor pressures or enthalpies of vaporization, the respecting covariances are hence not given in Eq. 9. The covariance of data which exhibit a low coefficient of determination of r2 < 0.2 are manually set to 0. As in Ref. 33, extreme outliers were discarded from the calculation of r2 . The coefficient of determination between COSMO-RS data and predicted PCP-SAFT parameters is also below 0.2. The independence of COSMO-RS and predictive PCP-SAFT is a prerequisite for using Eq. 5 in the given form.

2.4

Expected reduction in uncertainty

Before showing results, we present an example calculation of the expected uncertainty of property yi . The variance V is a measure of uncertainty. The uncertainties of the parameters θ propagate to the uncertainties of calculated property yi 38 V yi = Q i V θ Q T i

(12)

with Qi being the sensitivity vector

Qi =

dyi (θ) dθ 9

.


(13)


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The example calculation in this chapter assumes that the errors are normally distributed and have an expectation value of 0. Since the sensitivity Qi is dependent on the parameters θ, the actual numbers depend on the molecule. For the example in this section, all numbers are given for n-butane and the property yi is the vapor pressure pvap at 0.5 Tcrit . The sensitivity Q then reads

Qpvap = butane (0.5 Tcrit ) =

dpvap butane (0.5 Tcrit ) d(ε/k)

dpvap butane (0.5 Tcrit ) dσ

dpvap butane (0.5 Tcrit ) dm

−5.792 kPa −0.4295 kPa −26.37kPa K Å

Using Eqs. 6 and 14 in Eq. 12, we get a standard deviation σ =

√

(14)

.

V = 7.159 kPa for

predictive PCP-SAFT. Using Eq. 7 and the COSMO-RS prediction pvap = 11.97 kPa, we obtain a standard deviation of σ = 6.713 kPa for COSMO-RS. The inverse of variances from all individual methods methods can be summed up to the inverse of the variance of the combination, because the inverse of the variance is a measure for information content: 38 V −1 =

X

V −1 i

(15)

.

i

If both standard deviations of both original methods are equal, a factor of

√

2 is expected

for the standard deviation of combination. Eq. 15 can be applied to the scalar variance of property y (property space) as well as to the variance matrix of parameters θ (parameter space). For the combination in property space, we obtain σ = 4.897 kPa, which is about a √ factor of 2 smaller than the standard deviation of predictive PCP-SAFT and COSMO-RS. When combining information in parameter space, the information of COSMO-RS prediction propagate into a contribution of information of parameters, analogous to Eq. 12: 38 T −1 V −1 θ,j = Qj Vyˆj Qj

10

.


(16)

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Inserting Eqs. 16 and 6 into Eqs. 15 and 12, we obtain

V yi = Q i

+ Vθ−1 0

X

−1 QT j Vyˆj Qj

j

|

{z

!−1

QT i

.

(17)

}

expected variance of combined parameters

The combination in property space can only use the information of the property that is combined. As can be seen in the sum in Eq. 17 (and in Eq. 5 when applying the method), many predicted properties can be used for the combination in parameter space. Since a new PCP-SAFT parameter set is obtained, even the prediction of properties that were not used in the combination are effected. When using the vapor pressure at 0.5 Tcrit and the enthalpy of vaporization at 0.7 Tcrit from COSMO-RS for the combination in parameter space, the expected standard deviation for vapor pressure at 0.5 Tcrit is σ = 3.763 kPa, which is significantly lower than the standard deviation from combination in property space.

2.5

Computational details

For the application of our combination method with predictive PCP-SAFT and COSMO-RS, the following steps have to be performed: 1. calculate parameters with predictive PCP-SAFT as in Refs. 28 and 33 2. calculate properties (e.g. pvap (0.5 Tcrit ) and H vap (0.7 Tcrit ), see Tab. 1) with COSMORS 3. minimize Eq. 5 to yield PCP-SAFT parameters of the combination In this study, we used in-house software based on the ThermoC framework 39 . The objective function (Eq. 5) is minimized using the Nelder-Mead simplex algorithm. If other prediction methods than predictive PCP-SAFT or COSMO-RS are to be utilized, the according variances for use in Eq. 5 need to be determined first. The variances need to be determined only once for each prediction method. The variance should be determined 11



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with a diverse set of molecules. If the prediction method is known to have systematically different errors for different chemical families, a variance for each chemical family should be determined.

3

Results

In the following, we present the predicted thermodynamic properties for different combinations of predictive PCP-SAFT and COSMO-RS. We use a set of 50 molecules to test the proposed combinations. This set of 50 molecules contains linear and branched alkanes, alkynes, esters, ethers, ketones, aldehydes and cyclic molecules. The predicted properties are compared to experimental data from Refs. 40–62. Experimental vapor pressures and liquid densities are available for all 50 compounds, experimental enthalpies of vaporization for 42 compounds. In Tab. 1, the average relative root mean square deviation (RMSD) calculated with

avg. rel. RMSDy =

1 Ncomp

v u X u t

Ncomp

i=1

1 Ndata,i

Ndata,i

X k=1

exp yi,k − yi,k exp yi,k

!2

(18)

are shown for the original prediction methods (predictive PCP-SAFT and COSMO-RS) and all studied combinations. In Eq. 18, Ncomp is the number of components, Ndata,i is the number of data points for component i, yi,k is the calculated value of data point k and exp is the experimental value of data point k. First, the root mean square of the relative yi,k

deviation is calculated for each component over a broad temperature range, which are then arithmetically averaged over all components. Temperature ranges for all components are given in the Supporting Information. For the vapor pressure, the combination in property space (fourth block in Tab. 1) shows √ the expected decrease in error of roughly factor 2, as discussed in Section 2.4. The error of enthalpy of vaporization for the combination in property space is larger than for predictive

12


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PCP-SAFT. A closer look at the errors of the original prediction methods reveals that errors of the enthalpies of vaporization predicted with COSMO-RS are systematically too high at high temperatures. Thus, the prerequisites for a successful combination in property space (cf. Sec. 2.4) are not given for enthalpy of vaporization. Predictions of liquid density cannot be combined in property space, because COSMO-RS cannot calculate densities at temperatures other than 298 K. When using one data point from COSMO-RS for the combination in parameter space (first block in Tab. 1), there is always improvement in vapor pressure and enthalpy of vaporization prediction. The fact that the combination of predicted PCP-SAFT parameters with liquid density predictions from COSMO-RS does not improve accuracy of vapor pressure prediction much is not surprising, as Kaminski et al. 33 already discovered that even with experimentally obtained liquid densities no improvement is seen. On the other hand, the combination utilizing liquid density predictions from COSMO-RS greatly improves the prediction of liquid density compared to predictive PCP-SAFT alone. In all other combinations the liquid density prediction stays on the same good level of predictive PCP-SAFT. When using two data points from COSMO-RS (second block in Tab. 1), the vapor pressure and enthalpy of vaporization predictions improve for all studied combinations. Improvement in liquid density predictions are especially pronounced for combinations which include the liquid density information from COSMO-RS. Using three data points from COSMO-RS (third block in Tab. 1) slightly improves predictions compared to only using two data points from COSMO-RS. Predictions for all considered properties improved. The biggest effect can be seen in vapor pressure predictions, for which the error almost halves in comparison to the original prediction methods. All presented combinations in PCP-SAFT parameter space are able to decrease the average error compared to the original prediction methods. It is worth mentioning that not only the average error, but also the maximum error decreases.

The combinations

that utilize ∆H vap (0.7 Tcrit ) predict the enthalpy of vaporization with a higher error than 13



Table 1: Average and maximal relative root mean square deviations (rel. RMSD) for vapor pressure, enthalpy of vaporization and liquid density predictions over a wide temperature range. Temperature ranges for all molecules are given in the Supporting Information. The first block shows combinations of predictive PCP-SAFT with 1 datum from COSMO-RS, the second with 2 data from COSMO-RS and the third with 3 data from COSMO-RS. The fourth block shows results for combination in property space and the last block shows results for the original predictive models.

ρliq (298 K)



pvap (0.7 Tcrit )

combination of predictive PCP-SAFT with information from COSMO-RS avg. rel. RMSD / % max. rel. RMSD / % pvap (0.5 Tcrit )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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x x x x x x x x

x x x x x x

x x x x

x x

x x

x x x x x x x

x x x x combination in property space COSMO-RS predictive PCP-SAFT

pvap 34.6 38.3 38.3 35.4 54.6 31.3 29.8 32.2 33.1 30.5 38.9 36.8 36.1 29.4 30.2 31.2 31.1

∆H vap 8.8 9.5 7.0 10.4 10.9 7.0 8.6 8.2 6.5 9.3 9.0 6.4 10.3 6.7 8.8 6.1 9.5

ρliq 4.1 4.2 4.6 4.1 2.7 4.1 3.8 2.4 4.6 3.9 2.5 2.9 2.7 2.4 2.5 2.4 2.6

pvap 135.9 223.2 148.4 115.3 330.7 105.6 88.2 156.0 117.1 69.4 333.3 142.9 118.1 104.5 88.5 115.9 70.7

∆H vap 31.4 31.6 63.6 28.3 28.0 45.5 27.7 27.8 52.4 27.8 25.3 64.4 27.7 46.1 27.1 53.4 27.3

ρliq 11.4 14.4 22.7 22.4 10.6 12.2 11.2 6.3 35.6 13.7 7.2 22.3 8.6 6.9 7.8 7.6 8.0

44.3

13.4

-

183.3

32.9

-

66.2 57.0

22.3 11.1

4.8

292.4 416.9

117.1 68.1

41.8

14


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the according combination with ∆H vap (0.5 Tcrit ). Nevertheless, those combinations utilizing ∆H vap (0.7 Tcrit ) perform better than the equivalent combination without any information about ∆H vap . The performance of all presented combinations as well as the PCP-SAFT parameters are shown for each individual molecule in the Supporting Information. Since all combinations perform comparably well, the choice for the best combination might depend on the studied molecules. In the following, we discuss detailed results for individual molecules for the combination of predictive PCP-SAFT with pvap (0.5 Tcrit ) and ∆H vap (0.7 Tcrit ) from COSMO-RS and the combination with pvap (0.5 Tcrit ), ∆H vap (0.7 Tcrit ) and ρliq (298 K). Molecules with different characteristics are shown as examples. As can be seen in Fig. 1, for γ-valerolactone, the vapor pressure calculated with predictive PCP-SAFT deviates positively and the vapor pressure calculated with COSMO-RS deviates negatively from experimental data. Since the combination is done in parameter space, the results from the combination must not necessarily lie between those from the original methods, as can be seen at lower temperatures for the vapor pressure and even at the temperature of the enthalpy of vaporization that was used for the combination. The comparison of both combinations shows that the additional density information from COSMO-RS draws the combined prediction of the liquid density towards the prediction from COSMO-RS, but shows little differences for the prediction of vapor pressure and enthalpy of vaporization. The same trend can be observed for the other molecules. In Fig. 2, the advantage of combination in parameter space can be seen for the vapor pressure of diethyl ether. Although both original predictive methods have a positive deviation from experimental data, the combination yields much lower errors. This effect is caused by the information from the enthalpy of vaporization, because the vapor pressure information alone would yield a combined vapor pressure error with higher deviations. If for vapor pressure, both predictive methods were combined in property space, the result would always lie between those of the original predictive methods and information from other properties could not be incorporated.

15



deviation from experiment

vapor pressure γ-valerolactone 60% 40% 20% 0% -20% -40% -60% -80% 300 350 400 450 500 550 600 650 700 Temperature / K


enthalpy of vaporization γ-valerolactone 40% 30% 20% 10% 0% -10% -20% -30% -40% 300 350 400 450 500 550 600 650 700 Temperature / K liquid density γ-valerolactone deviation from experiment

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

10% 8% 6% 4% 2% 0% -2% -4% -6% -8% 300 350 400 450 500 550 600 650 700 Temperature / K

Figure 1: Deviations from experimental data for different properties over the considered temperature range for γ-valerolactone. Blue solid line: Predictive PCP-SAFT; red dashdotted line / red triangle: COSMO-RS; green dashed line: combination with pvap (0.5 Tcrit ) / ∆H vap (0.7 Tcrit ); yellow dotted line: combination with pvap (0.5 Tcrit ) / ∆H vap (0.7 Tcrit ) / ρliq (298 K); blue circle: information from COSMO-RS used for the combination.

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vapor pressure diethyl ether 100% 80% 60% 40% 20% 0% -20% 200

250

300 350 Temperature / K

400

450

enthalpy of vaporization diethyl ether deviation from experiment

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


140% 120% 100% 80% 60% 40% 20% 0% -20% 200

250


400

450

liquid density diethyl ether deviation from experiment

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8% 6% 4% 2% 0% -2% -4% 200

250


400

450

Figure 2: Deviations from experimental data for different properties over the considered temperature range for diethyl ether. The line types have the same meaning as in Fig. 1.

17




vapor pressure methyl ethyl ether 1% 0% -1% -2% -3% -4% -5% -6% -7% -8% -9% -10% 340

360 380 400 Temperature / K

420

440

liquid density methyl ethyl ether deviation from experiment

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

6% 4% 2% 0% -2% -4% -6% -8% 260 280 300 320 340 360 380 400 420 440 Temperature / K

Figure 3: Deviations from experimental data for different properties over the considered temperature range for methyl ethyl ether. The line types have the same meaning as in Fig. 1. The combination was done with vapor pressure at 219 K, which is not within the temperature range of our available experimental data. The enthalpy of vaporization for the combination was calculated at 307 K.

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One of the rare cases in which the combination yields a higher error in vapor pressure than both original prediction methods is shown in Fig. 3. These cases occur preferably when both original prediction methods already yield low errors. Still, the decrease in accuracy is small. Fig. 4 shows results for 2-butyne, a molecule for which predictive PCP-SAFT yields an exceptionally high error in vapor pressure (166 %) and Fig. 5 shows results for hexanal, a molecule for which COSMO-RS yields an exceptionally high error in vapor pressure (236 %). For both molecules the combination in parameter space recovers a parameter set that lies closer to the better-performing prediction method than to the prediction method with the outlier with exceptionally high errors. The combination seems not to get distracted by extreme outliers of one prediction method. These examples demonstrate how large errors in one of the prediction methods get damped by the combination. In Figs. 6 and 7 we show the root mean square relative deviations for each molecule in projections of predictive PCP-SAFT, COSMO-RS and the combination of predictive PCPSAFT with pvap (0.5 Tcrit ) and ∆H vap (0.7 Tcrit ) from COSMO-RS. While predictive PCPSAFT and COSMO-RS yield comparable scatter (top left panel of Figs. 6 and 7), the errors are much lowered for the combination in parameter space. It can be seen that especially molecules with high error in one of the original prediction methods perform well in the combination.

4

Discussion

Kaminski et al. 33 showed that predicted PCP-SAFT parameters can be greatly improved in combination with a single experimental vapor pressure datum. To improve quality of predictions of equilibrium properties like vapor pressure, enthalpy of vaporization and liquid density, we combined predictive PCP-SAFT with the independent property prediction method COSMO-RS. We investigated the combination of predictive methods in property

19




vapor pressure 2-butyne 250% 200% 150% 100% 50% 0% 280 300 320 340 360 380 400 420 440 Temperature / K


enthalpy of vaporization 2-butyne 80% 60% 40% 20% 0% -20% -40% 280 300 320 340 360 380 400 420 440 Temperature / K liquid density 2-butyne deviation from experiment

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0% -5% -10% -15% -20% -25% -30% 280 300 320 340 360 380 400 420 440 Temperature / K

Figure 4: Deviations from experimental data for different properties over the considered temperature range for 2-butyne. The line types have the same meaning as in Fig. 1. The combination was done with vapor pressure at 244 K, which is not within the temperature range of our available experimental data.

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vapor pressure hexanal deviation from experiment

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300% 250% 200% 150% 100% 50% 0% -50% -100% 320 330 340 350 360 370 380 390 400 410 Temperature / K liquid density hexanal


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2% 1% 0% -1% -2% -3% 290

300 310 320 Temperature / K

330

340

Figure 5: Deviations from experimental data for different properties over the considered temperature range for hexanal. The line types have the same meaning as in Fig. 1. The combination was done with vapor pressure at 293 K, which is not within the temperature range of our available experimental data. The enthalpy of vaporization for the combination was calculated at 410 K.

21



vapor pressure 400% COSMO-RS

COSMO-RS

400%

200%

0%

200%

0% 0%

200%

400%

0%

predictive PCP-SAFT combination (parameter space)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 33

200%

400%

combination (parameter space)

400%

better than both in between worse than both

200%

0% 0%

200%

400%

predictive PCP-SAFT Figure 6: Scatterplot of relative RMSD of vapor pressure for each individual molecule in our test set. Errors for predictive PCP-SAFT, COSMO-RS and a combination of predictive PCP-SAFT with pvap (0.5 Tcrit ) and ∆H vap (0.7 Tcrit ) from COSMO-RS are shown in different projections. To guide the eye, coloring and different symbols are used to differentiate between combinations that are better, in between and worse than both original prediction methods.

22


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enthalpy of vaporization 120%

COSMO-RS

COSMO-RS

120%

80%

40%

0%

80%

40%

0% 0%

40%

80%

120%

0%

predictive PCP-SAFT combination (parameter space)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


40%

80%

120%

combination (parameter space)

120%

80% better than both in between worse than both

40%

0% 0%

40%

80%

120%

predictive PCP-SAFT Figure 7: Scatterplot of relative RMSD of enthalpy of vaporization for each individual molecule in our test set. Errors for predictive PCP-SAFT, COSMO-RS and a combination of predictive PCP-SAFT with pvap (0.5 Tcrit ) and ∆H vap (0.7 Tcrit ) from COSMO-RS are shown in different projections. To guide the eye, coloring and different symbols are used to differentiate between combinations that are better, in between and worse than both original prediction methods.

23



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

space and in parameter space. The combination in property space results in arithmetic property averaging if the variances of both prediction methods are equal. When 2 independent stochastic variables (e.g. predicted properties) with the same variance are arithmetically averaged, the variance of the average is half of the variance of the original variables. The √ standard deviation being the square root of the variance thus has a factor of 0.5 ≈ 0.7. This behaviour can be observed for the combination of vapor pressures in property space, cf. Tab. 1. For the combination in property space, a property predicted with COSMO-RS has only an influence on the same property (e.g. vapor pressure) at the same state point (e.g. temperature). In contrast, for the combination in parameter space, the information from COSMO-RS acts on the predicted PCP-SAFT parameters and thus has an effect on all further property calculations. When predicting properties that are known to have a high uncertainty or even systematic errors, e.g. ∆H vap from COSMO-RS at high temperatures, the combination in property space is likely to give poor results, while the combination in parameter space can be performed with data at other temperatures that are known to have a higher certainty. While for the combination in property space only one piece of information from COSMORS can be utilized, the combination in parameter space can be performed exploiting all the different information from COSMO-RS weighted by its uncertainty.

5

Conclusions

We show how to combine two independent predictive models in order to yield more accurate property predictions than those obtained by each individual model. The most intuitive approach of averaging the predicted properties in property space yields only marginal improvements. In contrast, the combination of PCP-SAFT parameters with vapor pressure, enthalpy of vaporization and density from COSMO-RS almost halves the error in vapor pressure prediction and also decreases errors in enthalpy of vaporization and liquid density

24


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prediction. There are 2 reasons for the large improvements which are both due to fact that the PCP-SAFT parameters themselves are adjusted and not just the predicted property: On the one hand, information on multiple different properties from COSMO-RS can be utilized simultaneously when combination in parameter space is performed. On the other hand, adjustment of the PCP-SAFT parameters affects all other calculated properties by PCP-SAFT, not just the property which is used in the adjustment. In particular, this allows to integrate information from each model in the range where it works best since the properties used for the adjustment of parameters can be chosen e.g. at temperatures known to have a lower uncertainty and lower systematic errors than at the temperature relevant for prediction.

Acknowledgement This work was performed as part of the Cluster of Excellence “Tailor-Made Fuels from Biomass” funded by the Excellence Initiative by the German federal and state governments to promote science and research at German universities.

Supporting Information Available The following files are available free of charge. • kaminski_kirgios_bardow_leonhard_supporting_information.xls: RMSD and parameters for each molecule and each combination of predictive methods This material is available free of charge via the Internet at http://pubs.acs.org/.

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