A new method for the prediction of flash points for ternary miscible mixtures

Accepted Manuscript Title: A new method for the prediction of flash points for ternary miscible mixtures Author: Jie Cheng Yong Pan Xiaoya Song Juncheng Jiang Gaoyan Li Li Ding Hehe Chang PII: DOI: Reference:

S0957-5820(15)00047-6 http://dx.doi.org/doi:10.1016/j.psep.2015.02.019 PSEP 538

To appear in:

Process Safety and Environment Protection

Received date: Revised date: Accepted date:

8-5-2014 23-2-2015 26-2-2015

Please cite this article as: Cheng, J., Pan, Y., Song, X., Jiang, J., Li, G., Ding, L., Chang, H.,A new method for the prediction of flash points for ternary miscible mixtures, Process Safety and Environment Protection (2015), http://dx.doi.org/10.1016/j.psep.2015.02.019 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

A new method for the prediction of flash points for ternary miscible mixtures

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Jie Cheng, Yong Pan, Xiaoya Song, Juncheng Jiang, Gaoyan Li, Li Ding, Hehe Chang

Jiangsu Key Laboratory of Hazardous Chemicals Safety and Control, College of Urban Construction

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Highlights

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and Safety Engineering, Nanjing Tech University, Nanjing 210009, China

miscible mixtures.

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(1) We developed new empirical models to predict the flash points of ternary (2) Our models are based only on some common and accessible physicochemical

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parameters of pure components.

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(3) Our models are computationally inexpensive, easy to apply, and with good performance.

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(4) This study can provide a new way for predicting the flash points of miscible mixtures for engineering.

Abstract: The flash point is one of the most important physicochemical parameters used to 

Corresponding author. Tel.: +86-25-83587305; fax: +86-25-83587411. E-mail address: [email protected] (Y. Pan). Postal address: Mail Box 13, 200 North ZhongShan Rd, Nanjing 210009, China 1

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characterize the fire and explosion hazard for flammable liquids. The flash points of ternary miscible mixtures with different components and compositions were measured in this study. Four model input parameters, being normal boiling point, the standard enthalpy of vaporization, the average number of

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carbon atoms and the stoichiometric concentration of the gas phase for mixtures, were employed and calculated based on the theory of vapor-liquid equilibrium. Both multiple linear regression (MLR)

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and multiple nonlinear regression (MNR) methods were applied to develop prediction models for the flash points of ternary miscible mixtures. The developed predictive models were validated using data

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measured experimentally as well as taking data on flash points of ternairy mixtures from the

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literature. Results showed that the obtained average absolute error of both the MLR and the MNR model for all the datasets were within the range of experimental error of flash point measurements. It

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is shown that the presented models can be effectively used to predict the flash points of ternary mixtures with only some common physicochemical parameters.

1

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regressions; Prediction.

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Keywords: Ternary miscible mixtures; Flash point; Multiple linear regression; Multiple nonlinear

Introduction

The flash point (FP) of a pure liquid or mixtures of liquids is defined as the lowest temperature at which sufficient material is vaporized to form an ignitable mixture in air under the specified test conditions (AIChE / CCPS, 1993). It is one of the major properties used to assess the fire and explosion hazards of a flammable liquid. This parameter is of importance in the practice of safety considerations in storing, processing, and handling given compounds. The flash point of a substance can be obtained from the chemical manufacturer’s Material Safety Data Sheets (MSDSs) (MSDS, 2014), Lange’s Handbook of Chemistry (Dean, 1999) or from the Design Institute of Physical Properties (DIPPR) database of the American Institute of Chemical 2

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Engineers (DIPPR, 2006). The flash points of mixed liquids are less commonly reported. Experimental measurement is the most effective and reliable way to obtain flash point data of mixtures. However, it is also considered to be very dependent on the test apparatus and test methods

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(Gmehling and Rasmussen, 1982; Lance, 1979). Furthermore, the measuring process is resource- and time-consuming. For toxic and radioactive compounds, the process is more difficult or even

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impossible. Therefore, the development of theoretical prediction methods that are convenient and reliable in predicting the flash points of mixtures is desirably required and is an ongoing research

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field.

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Many researchers have developed prediction models to estimate the FP of binary liquid mixtures (Affens and McLaren, 1972; Gmehling and Rasmussen, 1982; Hanley, 1998; Liaw and Chiu, 2003,

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2006; Liaw et al., 2002, 2008a, 2008b; 1982; Vidal, 2006; White et al., 1997), but little for ternary mixtures. Recently, Liaw et al. (2004, 2009, 2010, 2011, Liaw and Chen, 2013; Liaw and Tsai, 2013)

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successfully proposed a series of mathematical models to predict the flash points of ternary mixtures

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for both non-ideal and ideal solutions. Furthermore, they found that the minimum flash point behavior (i.e., the flash points lower than those of the pure components) for a binary highly non-ideal

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solution may disappear on addition of a specific third component, such as the minimum flash point for methanol + methyl acrylate becomes less pronounced when a third component, methyl acetate, is added. If the mole fraction of methyl acetate is increased to 0.3, the minimum flash point increases, while when the mole fraction is increased to more than 0.4, the minimum flash point disappears (Liaw et al., 2004), e.g., the minimum flash point of the ternary mixture should be higher than the lower flash point of one of the three components to make it disappear, higher than two out of three flash points, etc. Meanwhile, empirical methods have also been developed for predicting the FP of mixtures. Garland and Malcolm (2002) developed a statistical model to predict the flash point of an organic acid-water solution. Kim and Lee (2010) proposed a predictive model based on the partial least 3

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squares (PLS) method, and the results were compared with those from the calculated methods using physicochemical laws such as Raoult's law and the Van Laar equation. Catoire et al. developed an equation based on pure compound predictions involving vaporization enthalpy, boiling point, and the

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number of carbon atoms (Catoire and Naudet, 2005) and later extended this equation to mixtures (Catoire et al., 2005, 2006).

2

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miscible mixtures as applied in hazard assessment and process design. Materials and methods Data Sets

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2.1

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The objective of this study is to provide a new empirical method to estimate the FP of ternary

The data set for this study consists of 182 data points, all of which were determined

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experimentally. Based on the ASTM7094 standard, the flash points of five sets of common and frequently used mixtures (2-propanol + methanol + butanone; 1-amyl alcohol + methanol +

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n-heptane; n-octane + methanol + n-heptane; ethylene glycol monobutyl ether + methanol +

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n-heptane, and alcohol + toluene +butanone) with different compositions were measured using a Grabner FLPH Miniflash Tester (Grabner, Austria). Determination and calculation of input parameters

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2.2

There are many physicochemical properties associated with the flash point of mixed liquids. Most of the existing prediction models of flash points have employed some particular properties as input parameters, such as normal boiling point and lower flammable limit (LFL). Hshieh (1997) proposed a prediction model of flash points for silicone and general organic compounds using boiling points as input parameters. Jones (1998) developed a prediction model based on three correlated parameters, being enthalpy of vaporization, vapor pressure, and lower flammable limit to calculate the flash points of hydrocarbons, and this model worked well. Similarly, Catoire et al. (2005, 2006) developed equations to predict the flash points of both binary and ternary mixtures based on three correlated parameters, being the normal boiling point, the number of carbon atoms in the vapor phase, and the 4

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standard enthalpy of vaporization. The flash point of a liquid mixture is the temperature at which the vapor pressure curve crosses the lower flammable limit (LFL) (Mashuga and Crowl, 1998; Vidal et al., 2004). Kim and Lee (2010)

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employed the LFL, saturated vapor pressure and mixed mole ratio as input parameters to establish the flash point for mixed liquids. However, the lower flammable limit of a combustible vapor

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reduces slightly as the temperature increases, which will cause a small but certain error in the prediction of the flash point. In addition, not all lower flammable limit values of chemicals are

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available from the literature or databases. (Mashuga and Crowl, 2000).

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In this study, the normal boiling point ( Tb ), the standard enthalpy of vaporization (  vap H m ), the average number of carbon atoms ( n ), and stoichiometric concentration ( C ) in the gas phase were

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selected as input parameters to characterize flash points of ternary mixed liquids. For liquid mixtures, the normal boiling point is the temperature at which

 P =1atm , where Pi i

i

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denotes the partial pressure of the pure component i in the gas phase at the temperature Tb.

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Pi  xi i Pi sat ,where xi denotes the mole fraction of the pure component i in the liquid phase

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(  i xi  1 ),  i denotes the activity coefficient of the pure component i in the liquid phase at temperature Tb. Pi sat denotes the saturated vapor pressure of the pure component i at temperature Tb, which can be calculated according to the Antoine equation or obtained from literature. The UNIFAC method was employed to calculate the gas-liquid balance parameters of mixed liquids, and subsequently the normal boiling points of the mixed liquids were calculated by the VLECalc 1.3 program (http://vle-calc.com/index.html). Within the range of 293.15 K to 308.15 K, the evaporation enthalpy can be considered as a constant. The vapor pressures at three temperatures near 298.15K (293.15K, 303.15K and 308.15K) were calculated based on the Clausius-Clapeyron equation, the plot of ln P ~1/ T was drawn and hence, the standard evaporation enthalpy at 298.15K was obtained from the slope of this straight line. 5

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The average number of carbon atoms and the stoichiometric concentration in the gas phase for mixtures can be calculated by n   i yi ni and C   i yiCi , where ni is the number of carbon atoms for pure substance i, Ci is the stoichiometric concentration of the flammable component i and

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yi is the mole fraction of combustible component i in the vapor phase at the flash point . Because the flash point is unknown, yi is approximately defined as the mole fraction of component i in the vapor

Modeling methods

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2.3

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phase at its boiling point (Catoire et al., 2006).

In this paper, both the Multiple Linear Regression (MLR) and Multiple Nonlinear Regression

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(MNR) methods are applied for the prediction models of the flash point. In order to find the relationship between physicochemical properties and the flash points, four parameters of the

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mixtures were employed as input variables.

For the statistical analysis, the software package SPSS 19 (Pallant, 2010) was used and the linear

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fit was carried out with 95% confidence interval. Moreover, the data processing system (DPS)

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available in this software package was employed to perform nonlinear fitting of the relationship between flash points and input parameters, and the corresponding nonlinear prediction model was

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established. The nonlinear relationship between input parameters and flash points can be described as follows:

4

4

4

4

i 1

i 1

i 1

i 1

FP  a  ( ci X i )  b( ci X i )2  c( ci X i )3  d ( ci X i ) 4

(1)

where, a, b, c, and d are model structure parameters, X1 is the normal boiling point of the liquid mixture Tb (K), X2 is the standard enthalpy of vaporization of the liquid mixture  vap H m (kJ/mol) and X3 and X4 are the stoichiometric concentration and the average number of carbon atoms, respectively. 2.4

Model validation

Model validation is of crucial importance for developed empirical models. The widely used 6

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coefficient for determination ( R 2 ) can provide a reliable indication to test the robustness of the model. Thus, R 2 was employed in the present study to determine the calibration capability of the model. An F-test is a statistical test in which the test statistic has an F-distribution under the null

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hypothesis. It is most often used when comparing statistical models that have been fitted to a data set, in order to identify the model that best fits the population from which the data were sampled (Lomax

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and Hahs-Vaughn, 2013).Statistical significance plays a pivotal role in statistical hypothesis testing,

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where it is used to determine if a null hypothesis should be rejected or retained. To determine if a result is statistically significant, the p-value should be calculated, which is the probability of

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observing an effect given that the null hypothesis is true (Devore, 2011).

One of the most common methods to estimate how accurately a predictive model will perform is

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Cross-Validation (CV). A good CV result indicates a good robustness and high internal predictive ability of a model. In this work, the Leave-Many-Out (LMO, 20% out) Cross-Validation (Q2LMO) was

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employed.

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External validation is necessary to determine both the generalizability and the predictive ability of a developed empirical model. The available dataset was randomly divided into a training set and an

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external prediction set. The former was used for developing flash point prediction models, while the latter was used for checking the reliability of the developed models. In this work, the dataset was randomly divided into a training set (137 data points) and a test set (45 data points). The squared correlation coefficient for external validation (Q2ext) was employed as a traditional external validation criterion, which can be calculated as follows:

Q  1 2 ext

prediction



( yi  yˆi )2



( yi  ytr )

i 1 prediction i 1

(2)

2

where yi and yˆi indicate the experimental values and the predicted values of the test set, respectively. ytr indicates the average experimental value of the training set. 7

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The Y-Randomization test (Rücker et al., 2007) is also a widely used technique to ensure the robustness of prediction models, whereby the performances of the original model in data description are compared to that of models built for permuted (randomly shuffled) response, based on the

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original descriptor pool and the original model building procedure. The process is repeated 50~100 times. If all the randomized generated models have much lower R2 and Q2LMO values than that of the

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original model as expected, one may reasonably conclude that there is no chance correlation in the model development. Oppositely, if some resulting models have relatively high R2 and Q2LMO values,

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it implies that the original model may be obtained due to a chance correlation and thus can not be

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reasonably considered as an acceptable prediction model by the current modeling method. The Root Mean Square Error (RMSE) and Average Absolute Error (AAE) were determined to

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evaluate the predictive capability of the developed models. As the values of AAE and RMSE become smaller, the predictive capability of the model is stronger. Results and discussions Results of MLR models

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3.1

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The MLR procedure was performed on the training set. The dependent variable is the experimental

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flash points of the liquid mixtures, and independent variables are the normal boiling point (X1), the standard enthalpy of vaporization (X2), the average number of carbon atoms (X3) and the stoichiometric gas concentration (X4).

It should also be noted that Catoire and Naudet (2005) proposed an empirical equation for the estimation of flash points of pure compounds:

FP ( K )  1.477  Teb 0.79686  H vap 0.16845  n 0.05948

(3)

where Teb is the normal boiling point of the compound expressed in K, Hvap is the standard enthalpy of vaporization at 298.15 K of the compound expressed in kJ mol-1, and n is the number of carbon atoms. Besides, this model has been successfully applied to predict the flash points of some binary and ternary miscible mixtures (Catoire et al., 2005; 2006). When comparing to the model, a new 8

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variable which is the stoichiometric gas concentration (X4) was considered and included in this study. In order to verify the importance of the newly added parameter to the developed models and demonstrate how the addition of this parameter has improved the quality of the results, additional

‘stoichiometric gas concentration’ for comparison purpose.

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The corresponding MLR models for ternary mixtures are as follows:

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MLR and MNR models with only three input parameters were developed by omitting the parameter

Model with four parameters:

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 FP  18.993  0.554 X 1  1.566 X 2  1.868 X 3  1.852 X 4  2 n  137, R  0.902, f  304.364, SE  1.551, p  0.001

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Model with three parameters:

(4)

 FP  24.625  0.629 X 1  1.313 X 2  6.383 X 3  2 n  137, R  0.857, f  260.922, SE  1.838, p  0.001

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(5)

where, n is the number of data points used in training set, R2 is the multiple correlation coefficient,

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SE is the standard error for the model, f is the value of the F-test, and p is the significant probability

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of this equation. It can be seen from equations (4) and (5) that the significant probability of both

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models is less than 0.05, which indicates that the equations are of satisfactory statistical significance. Subsequently, the developed models were employed to predict the flash points of mixtures in the test set and the results are presented in Table 1. Plots of the predicted flash point values versus the experimental ones for both the training and test sets are shown in Figure 1. Table 1.

Predicted flash point values for the test set by both the MLR and MNR models as well as Catoire and Naudet’s (2005) model

Ternary mixtures methanol (1) + n-heptane (2) + n-octane (3)

Volume percentage

Experimental FP /K

Predicted FP / K (models with four parameters)

Predicted FP / K (models with three parameters)

Predicted FP / K

MLR

MNR

MLR

MNR

272.35

270.98

271.17

270.88

270.52

Catoire and Naudet (2005) 269.63

0.6

269.15

267.42

267.65

267.68

267.97

265.91

0.2

0.2

273.45

272.04

272.24

271.94

272.22

270.46

0.2

0.5

268.45

269.6

269.81

269.69

270.17

267.95

v1

v2

0.1

0.2

0.1

9

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0.3

methyl alcohol (1) + n-heptane (2) + butyl cellosolve (3)

273.53

273.24

273.62

271.75

0.3

0.5

268.45

268.02

268.25

268.22

269.08

266.23

0.4

0.3

270.45

269.48

269.7

269.72

270.39

267.75

0.5

0.1

274.55

272.29

272.49

272.44

272.94

270.66

0.6

0.3

269.55

267.16

267.4

267.32

268.31

265.31

0.1

0.2

277.25

278.42

278.65

280.17

277.4

278.31

0.6

269.35

269.2

269.42

269.48

268.76

267.95

0.2

275.35

275.69

275.88

277.2

275.43

275.24

0.2

0.5

268.35

268.64

268.86

269.1

268.96

267.24

0.3

0.1

279.35

279.4

279.64

280.43

278.94

279.18

0.3

0.5

267.35

267.38

267.63

267.69

268.2

265.75

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0.1 0.2

0.3

268.25

269.45

269.67

270.05

270

268.07

0.1

274.25

276.85

277.05

277.17

276.29

276.06

0.6

0.3

266.25

267.14

267.38

267.41

268.16

265.42

0.1

0.2

270.95

269.81

270.03

269.08

267.11

268.45

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0.4 0.5

0.6

274.35

273.66

273.88

272.1

270.72

271.67

0.2

272.15

271.19

271.4

270.78

269.3

269.62

0.2

0.5

274.55

274.14

274.36

273.05

272.41

271.95

0.3

0.1

271.15

271.66

271.86

271.63

270.29

270.13

0.3

0.5

275.55

275.71

275.93

274.63

274.79

273.18

274.86

275.07

274.35

274.07

272.69

274.3

274.5

274.45

273.67

272.59

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0.1 0.2

0.4

0.3

275.15

0.5

0.1

275.15

0.6

0.3

278.15

278.31

278.56

277.67

278.85

275.63

0.1

0.2

272.95

270.82

271.02

270.83

269.28

269.07

0.1

0.6

0.2

0.2

277.15

275.56

275.77

275.7

275.23

273.66

273.15

272.3

272.49

272.51

271.11

270.6

0.2

0.5

276.15

275.89

276.1

276.16

275.56

274.18

0.3

0.1

272.15

272.91

273.09

272.95

271.51

271.22

0.3

0.5

280.15

278.12

278.34

278.11

277.88

276.51

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methyl alcohol (1) + isopropanol (2) + 2 - butanone (3)

273.34

d

ethyl alcohol (1) + methylbenzene (2) +2 -butanone (3)

275.45

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methanol (1) + n-heptane (2) + n-amyl alcohol (3)

0.1

0.4

0.3

276.05

276.87

277.08

276.52

275.74

275.23

0.5

0.1

275.15

276.11

276.31

275.37

274.25

274.38

0.6

0.3

281.15

281.76

282.04

279.82

279.71

280.03

0.1

0.2

275.35

279.05

279.29

279.8

278.04

278.13

0.1

0.6

269.35

270.61

270.84

270.04

270.48

268.56

0.2

0.2

273.35

276.02

276.23

276.82

276.32

274.76

0.2

0.5

268.25

269.68

269.91

269.46

270.51

267.57

0.3

0.1

277.35

280.17

280.42

280.78

280.72

279.13

0.3

0.5

267.35

266.52

266.78

266.65

267.37

264.78

0.4

0.3

268.35

267.96

268.19

268.48

268.63

266.53

0.5

0.1

274.35

275.37

275.56

275.88

274.93

274.51

0.6

0.3

266.35

266.51

266.76

266.8

267.65

264.82

10

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290

( a)

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280

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270

260 260

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Predicted flash point values (K)

training set n=137 test set n=45

270

280

290

Experimental flash point values (K)

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290

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260 260

d

270

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280

(b)

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Predicted flash point values (K)

training set n=137 test set n=45

270

280

290

Experimental flash point values (K)

Figure 1. Comparison between the predicted and experimental FP values of the MLR models. a: model with four parameters; b: model with three parameters

3.2

Results of MNR models

The same training set, test set and physicochemical parameters used as the input parameters in the MLR modeling were also employed for the MNR modeling for comparison purpose. To study the influence of the nonlinear order on the model prediction effects, both second-order and third-order models were built. The main performance parameters of the different nonlinear models are presented in Table 2. Table 2.

The performance parameters of nonlinear models with different orders 11

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Training set

Model Second-order (Model with four parameters) Third-order (Model with four parameters) Second-order (Model with three parameters) Third-order (Model with three parameters)

R2

Q2ext

0.902 0.902 0.823 0.808

0.863 0.745 0.774 0.797

Test set Maximum absolute error (K) 3.94 10.10 4.03 8.41

Average absolute error (K) 1.12 6.00 1.64 3.28

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Interestingly, it can be seen from Table 2 that the R2 values of the training set for both

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second-order and third-order four parameters models are the same. The phenomenon may be just a coincidence, since the other performance parameters of the two models are quite different. Moreover,

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from Table 2, it can be seen that the maximum absolute error and average absolute error of the third-order models are larger than those of the second-order models, for both the models with four

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parameters and three parameters. Consequently, the second-order nonlinear models with 95% confidence levels were regarded as adequate models in this study, and the corresponding nonlinear

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empirical equations are as follows: Model with four parameters:

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d

 FP  99.3379  ( ci X i )  0.0091( ci X i )2   c i X i  0.1042 X 1  0.5806 X 2  0.6783 X 3  0.6860 X 4  2 n  137, R  0.9022, f  241.7644, SE  1.503, p  0.001

(6)

Model with three parameters:

 FP  682.121-4.347( ci X i )  0.0011( ci X i )2   c i X i  0.481X 1 +2.376X 2  5.575 X 3  2 n  137, R  0.823, f  306.436, SE  2.035, p  0.001

(7)

As can be seen from equations (6) and (7), both models are statistically significant and reliable with satisfactory correlation coefficients. Subsequently, the developed models were employed to predict the flash points of mixtures in the test set and the results are presented in Table 1. Plots of the predicted flash point values versus the experimental ones for both the training and test sets are shown in Figure 2.

12

Page 12 of 25

290

(a)

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280

270

260 260

270

280

290

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Experimental flash point values (K) 290

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(b)

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280

270

260 260

te

d

Predicted flash point values (K)

training set n=137 test set n=45

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Predicted flash point values (K)

training set n=137 test set n=45

270

280

290

Experimental flash point values (K)

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Figure 2. Comparison between the predicted and experimental FP values of the MNR models. a: model with four parameters; b: model with three parameters

3.3

Analysis of results

Table 3 presents the main prediction performance parameters of both MLR and MNR models with four and three input parameters on the training and test sets. For all these four models, the AAE values for both the training and test sets are within the experimental error of FP determination, which is around ±10 K (Botros and Atkenson, 1991; Moody and Darken, 1989). Furthermore, the RMSE and AAE values of the models for both training and test sets were not only low but also similar, which suggests that the proposed models have both predictive ability and generalization capability. Table 3.

Main prediction performance parameters of the MLR and MNR models applied on the training and test sets 13

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Training set

Model R2 0.902 0.902 0.857 0.823

MLR (Model with four parameters) MNR (Model with four parameters) MLR (Model with three parameters ) MNR (Model with three parameters)

AAE 1.200 1.201 1.482 1.637

Test set Q2ext 0.866 0.863 0.796 0.774

RMSE 1.524 1.492 1.811 2.019

AAE 1.111 1.124 1.405 1.489

RMSE 1.400 1.418 1.720 1.808

ip t

Moreover, it can be seen from Table 3 that, for the MLR models with four or three parameters, the resulting AAE values of test set were 1.111K and 1.405K, respectively. For the MNR models, the

cr

AAE values were 1.124K and 1.489K, respectively. It can thus be concluded that, for both the MLR

us

model and MNR model, the prediction errors of the models with four input parameters are obviously lower than those of models with three input parameters. Moreover, the differences between other

an

performance parameters of these models with different input parameters also indicated the superiority of the models with four input parameters. By comparing these performance parameters

M

between the models, the obvious differences indicated that the addition of the new parameter of stoichiometric gas concentration (X4) has obviously improved the quality of the prediction models,

d

which also demonstrates the important contribution of the new parameter towards the predictivity of

te

the developed models for predicting the FP of ternary mixtures. Consequently, both the developed

Ac ce p

MLR and MNR models with four input parameters were recommended here and further validated for their stability and applicability, as well as compared to previous works. 3.4

Stability validation of the models

Subsequently, all the developed models were tested for chance correlation to further analyze the model stability. Firstly, a Y-randomization test was performed on the training set for 100 times for both the MLR and MNR models. As expected, all generated models produced low R2 and low Q2LMO values. The obtained maximum R2 and Q2LMO values of the generated models were 0.097 and 0.089 , 0.117 and 0.105 for the MLR and MNR models with four parameters, 0.231 and 0137, 0.153 and 0.223 for the MLR and MNR models with three parameters, respectively, all of which were much lower than the corresponding ones calculated when the dependent variables were not scrambled. It 14

Page 14 of 25

can thus be concluded that the correct dependent variables were used to generate reasonable models. The predicted residual distributions of the MLR and MNR models are shown in Figure 3 and Figure 4. It can be seen from these figures that the prediction residuals are randomly distributed on

ip t

both sides of the baseline, and no obvious regularity exists. This indicates that there are no systematic errors in the development of the two models. training set n=137 test set n=45

cr

10 (a)

us

0

an

Residuals

5

M

-5

-10

260

270 280 Experimental flash point values (K)

10

te

0

Ac ce p

Residuals

5

(b)

d

training set n=137 test set n=45

290

-5

-10 260

270

280

290

Experimental flash points (K)

Figure 3. Plot of the residuals versus the experimental flash point values for the MLR models. a: model with four parameters; b: model with three parameters

15

Page 15 of 25

10 training set n=137 test set n=45

(a)

ip t

0

-5

-10

270 280 Experimental flash point values (K)

10

training set n=137 test set n=45

an

(b)

M

Ac ce p

-10 260

d

0

te

Residuals

5

-5

290

us

260

cr

Residuals

5

270

280

290

Experimental flash point values (K)

Figure 4. Plot of the residuals versus the experimental flash point values for the MNR models. a: model with four parameters; b: model with three parameters

It can be reasonably concluded that both presented models are valid models and can be effectively used to predict the flash points of ternary mixtures. 3.5

Applicability validation of the models

As mentioned previously, there are more experimental flash point data available in literature on ternary mixtures. In this section, in order to further verify the applicability of the developed models, a new dataset containing 53 available flash point data points was introduced, all of which were taken from published literature (Catoire et al., 2005, 2006; Liaw et al., 2009). The new dataset was then employed as an additional external test set to verify the predictivity and applicability of the 16

Page 16 of 25

developed models. These 53 different data points cover five sets of common mixtures: methanol + toluene + 2,2, 4-trimethylpentane; methanol + decane + acetone; cyclohexanone + p-xylene + methanol; ethanol + toluene + ethylacetate; methanol + ethanol + acetone. After calculating the four

ip t

input parameters for these 53 samples, both the MLR and MNR models (with four parameters) were employed to predict the flash point values of these samples. The predicted results are shown in Table

Table 4.

cr

4.

Predicted flash point values for the additional external test set by the MLR and MNR

Predicted FP /K 263.86 265.07 266.25 267.40 268.48 269.49 270.39 271.13 261.84 262.77 263.69 264.55 265.35 266.03 266.54 261.94 262.76 263.54 264.24 264.81 265.19 262.42 263.16 263.82 264.35 264.64 262.96 263.62

Absolute error /K -0.91 -1.22 -2.00 -2.65 -2.83 -2.64 -2.14 -0.88 0.11 -0.32 0.16 -0.10 0.00 1.72 3.21 -0.09 0.49 1.11 1.61 2.24 3.66 -0.27 0.09 0.43 1.40 3.91 -0.31 -0.27

an

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.1 0.2

MLR

262.95 263.85 264.25 264.75 265.65 266.85 268.25 270.25 261.95 262.45 263.85 264.45 265.35 267.75 269.75 261.85 263.25 264.65 265.85 267.05 268.85 262.15 263.25 264.25 265.75 268.55 262.65 263.35

M

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3 0.3 0.3 0.4 0.4 0.4 0.4 0.4 0.5 0.5

d

x2

Ac ce p

methanol(1)+toluene (2)+2,2,4-trimethylpentane (3)

x1

Experimental FP/K

te

Ternary mixtures

Mole fraction

us

models (with four parameters)

MNR Predicted FP /K 267.51 266.86 266.27 265.76 265.33 265.02 264.88 264.97 266.62 266.14 265.72 265.40 265.21 265.19 265.44 265.60 265.25 264.98 264.86 264.93 265.34 264.73 264.49 264.40 264.54 265.08 263.99 263.90

Absolute error /K -4.56 -3.01 -2.02 -1.01 0.32 1.83 3.37 5.28 -4.67 -3.69 -1.87 -0.95 0.14 2.56 4.31 -3.75 -2.00 -0.33 0.99 2.12 3.51 -2.58 -1.24 -0.15 1.21 3.47 -1.34 -0.55

17

Page 17 of 25

te

methanol (1)+ ethanol (2)+ acetone (3)

269.22 272.04 273.59 271.30 265.02 270.77 264.32 257.32

0.77 3.29 -0.93 0.39 2.67 0.39 1.53 -0.44 0.10 -10.27 -2.26 2.05 -4.74 -0.65 1.68 1.08 1.13 1.30 -1.17 1.38 -0.55 -2.12 -0.39 -3.22 2.32

ip t

264.08 264.76 263.38 263.56 264.38 262.96 263.92 263.29 260.55 268.02 266.21 267.60 267.39 270.30 273.47 280.07 289.02 268.52 271.54 273.07 273.70 269.67 269.24 267.67 256.03

us

cr

0.72 3.70 -1.03 -0.05 2.89 -0.54 1.45 -1.00 -7.11 0.93 2.95 8.03 3.32 8.62 3.41 -1.53 -1.02 0.60 -1.67 0.86 1.85 2.53 -1.92 0.13 1.03

an

ethanol(1) + toluene (2) + ethylacetate (3)

271.74 282.68 291.17

M

cyclohexanone (1)+ p-xylene (2)+ methanol(3)

264.13 264.35 263.48 264.00 264.16 263.89 264.00 263.85 267.76 256.82 261.00 261.62 259.33 261.03

264.85 268.05 262.45 263.95 267.05 263.35 265.45 262.85 260.65 257.75 263.95 269.65 262.65 269.65 275.15 281.15 290.15 269.82 270.37 274.45 273.15 267.55 268.85 264.45 258.35

d

methanol (1)+decane (2)+acetone (3)

0.5 0.3 0.5 0.4 0.6 0.1 0.6 0.2 0.6 0.3 0.7 0.1 0.7 0.2 0.8 0.1 0.1 0.7 0.2 0.1 0.3 0.5 0.3 0.6 0.4 0.3 0.4 0.5 0.8 0.1 0.25 0.2 0.65 0.15 0.55 0.15 0.5 0.25 0.1 0.8 0.8 0.1 0.7 0.1 0.1 0.7 0.6 0.1 0.3 0.1

It can be seen from Table 4 that most of the predicted values agreed well with the experimental

Ac ce p

ones, with the maximum absolute error being 8.62K for the MLR model and 9.31K for the MNR model, respectively. Besides, the predicted AAE and RMSE values were calculated for this new external test set. The results showed that the obtained AAE and RMSE values for both models were as low as possible, which were 2.164 K and 3.063 K for the MLR model, and 2.601 K and 3.411 K for the MNR model, respectively. For both models, the AAE values for the new test set are within the experimental error of FP determination (around ±10 K) (Botros and Atkenson, 1991; Moody and Darken, 1989), which again demonstrated the satisfactory predictivity and applicability of the developed models. Furthermore, both the AAE and RMSE values for the MLR model were lower than those for the MNR model for the new test set. This phenomenon strongly suggests that a simple linear relationship, 18

Page 18 of 25

instead of a complex nonlinear one, may exist between the selected physicochemical parameters and flash points of ternary mixtures.

3.6.1

Model comparison Comparison between MLR and MNR models

ip t

3.6

The performance comparison of the MLR and MNR models is shown in Table 3, where the results

cr

obtained from the MLR model are very close to or even a little better than those obtained from the

us

MNR model for both the training and test set. This phenomenon strongly verified the aforementioned assumption that a simple linear relationship, instead of a complex nonlinear one, may exist between

an

the selected physicochemical parameters and flash points of the ternary mixtures. Besides, the MLR method is good enough in solving this kind of problems and can effectively describe the potential

M

relationships. Moreover, the MLR model provides a simple mathematical equation, which is

for practical engineering applications.

Comparison with previous works

te

3.6.2

d

convenient to apply and has a better model interpretability. As such the MLR model is recommended

Ac ce p

Comparisons have also been made between the obtained models and previous ones. Firstly, the prediction ability of the developed models was compared with the ones provided in the aforementioned three literature sources (Catoire et al., 2005, 2006; Liaw et al., 2009). The models were employed to predict the FP of mixtures mentioned in literature. The predicted values of the FP for both MLR and MNR models (with four parameters) of the mixtures can be seen in detail in Table 5. Table 5.

Predicted flash point values for the literature mixtures by the MLR and MNR models (with four parameters) as well as Catoire et al.’s models (2005, 2006)

Ternary mixtures cyclohexanone (1)+ p-xylene (2)+ methanol(3)

Mole fraction x1

x2

0.25

0.2

Catoire et al.’s models (2005, 2006) Experimental FP/K Predicted Absolute Predicted Absolute Predicted Absolute FP /K error /K FP /K error /K FP /K error /K MLR

281.15

282.68

MNR

-1.53

280.07

1.08

280.75

0.4

19

Page 19 of 25

0.15

290.15

291.17

-1.02

289.02

1.13

287.75

2.4

0.55

0.15

269.82

269.22

0.60

268.52

1.30

269.35

0.47

0.5

0.25

270.37

272.04

-1.67

271.54

-1.17

269.75

0.62

0.1

0.8

274.45

273.59

0.86

273.07

0.8 0.7 0.1 0.6

0.1 0.1 0.7 0.1

273.15 267.55 268.85 264.45

271.30 265.02 270.77 264.32

1.85 2.53 -1.92 0.13

273.70 269.67 269.24 267.67

ip t

0.3

0.1

258.35

257.32

1.03

256.03

1.38

271.65

2.8

-0.55 -2.12 -0.39 -3.22

272.45 267.05 265.85 263.45

0.7 0.5 3.0 1.0

2.32

256.95

1.4

cr

methanol (1)+ ethanol (2)+ acetone (3) （Catoire et al., 2006)

0.65

us

(2)+ methanol(3) （Catoire et al., 2005) ethanol(1) + toluene (2) + ethylacetate (3) （Catoire et al., 2005)

For Catoire et al.’s dataset (2005, containing two different mixtures of cyclohexanone + p-xylene

an

+ methanol and ethanol + toluene + ethylacetate), the resulting AAE and RMSE values calculated by the models presented in this work were 1.2 and 1.3 for the MLR model, and 1.2 and 1.2 for the MNR

M

model, respectively, while the values calculated by Catoire et al.’s model (2005) were 1.0 and 1.3, respectively. As for Catoire et al.’s dataset (2006, containing mixtures of methanol + ethanol +

d

acetone), the resulting AAE and RMSE values calculated by the models presented in this work were

te

1.4 and 1.6 for the MLR model, and 1.7 and 1.9 for the MNR model, respectively, while the values

Ac ce p

calculated by Catoire et al.’s model (2006) were 1.6 and 1.8, respectively. It can thus be concluded that the results obtained with the models developed in this work are very close and comparable to those obtained by the models mentioned in literature on the same dataset. As for Liaw et al.’s work (2009), it should be noted that some difficulties arose when performing the comparison, since the predicted FP values of the samples (methanol + toluene +2, 2, 4-trimethylpentane and methanol + decane + acetone) using Liaw et al’s model are not mentioned, making a comparison of the predicted results between the two methods difficult. Furthermore, the models mentioned in literature were used to predict the FP of the mixtures used in this work for comparison. As has been mentioned above, the empirical equation proposed by Catoire and Naudet (2005) has been successfully applied to predict the flash points of some binary and ternary miscible mixtures (Catoire et al., 2005; 2006). Thus in this study, the equation was also 20

Page 20 of 25

applied for calculating the FP of mixtures used in the test set mentioned in this work. The predicted values can be seen in detail in Table 1. The resulting R2, AAE and RMSE values are 0.717, 1.977 and 2.285, respectively. All these performance parameters indicate that the models reported in this work

ip t

perform better than the models mentioned in literature on the same test set. Based on these findings it can be concluded that the models reported in this work are similar or

cr

better in predicting the FP of ternary mixtures, compared to the selected models.

Subsequently more detailed comparisons between the presented models and models reported in

us

literature were performed. As is well known, most predictive models were developed based on Le

an

Chatelier’s rule and the theory of vapor-liquid equilibrium. Certainly, these models provide high predictive accuracy and are also theoretically reliable since every parameter in the model has a

M

physical meaning. However, regarding the applicability efficiency and range, previous models for predicting the FP of ternary mixtures are only able to estimate the FP of aqueous-organic mixtures or

d

specific ternary mixtures. It should be noted that the non-ideality of the liquid phase is accounted for

te

by liquid-phase coefficients by means of thermodynamic models, such as Wilson (Wilson, 1964), NRTL (Renon and Prausnitz, 1968), van Laar equations (Poling et al, 2001), or UNIFAC (Gmehling

Ac ce p

and Rasmussen, 1982). Moreover, the calculation of these parameters sometimes may be quite complex and difficult. One of the most obvious advantages of the models presented in this work is that these models are conceptually simple and convenient to apply. In addition, these models can be tailored in order to fit any available experimental data on the condition that proper validation data sets are used. The comparison details are presented in Table 6. Table 6.

Comparisons of different models for predicting the flash points of ternary mixtures

Models

Ternary mixtures

Methods

Liaw et al. (2009)

(1) methanol + toluene +2,2,4-trimethylpentane (2) methanol + acetone + decane

activity coefficient model (NRTL,UNIQUAC)

Saldana et al. (2013)

not specified

QSPR model based on additive molecular descriptors

Comments The model provides satisfactory predictions, which can accurately predict the flash points of the studied mixtures The model presents a lower prediction accuracy

21

Page 21 of 25

MLR and MNR models based on four common physicochemical parameters

The models have satisfactory predictivity and applicability, which can give reasonable predictions for new ternary mixtures

ip t

This work

(1) methanol + n-heptane + n-octane (2) methanol + n-heptane + n-amyl alcohol (3) ethyl alcohol + methylbenzene +2 -butanone (4) methyl alcohol + isopropanol + 2 - butanone (5) methyl alcohol + n-heptane + butyl cellosolve

cr

(1) ethanol + toluene + ethylacetate (2) methanol + ethanol + acetone

The models present good compatibility for predicting the flash points of mixtures belonging to the same organic group but are not suitable for mixtures with different organic groups

us

Catorie et al. (2005,2006)

mathematical regression models based on three common physicochemical parameters

an

Finally, it should also be noted that, besides MLR and MNR, the literature contains many other machine learning methods, such as neural networks, support vector machine, and genetic algorithms,

M

which have been successfully applied in various studying fields. These methods possess some obvious superiority in handling complex problems, especially for their strong nonlinear modeling

d

ability leading to satisfactory prediction performance. However, these methods also suffer some

te

disadvantages, such as overtraining, resulting untransparent models and being inconvenient to apply. Thus in this study, in order to obtain intuitive, transparent and easy applicable prediction models,

Ac ce p

only the more intuitive MLR and MNR methods were chosen to build the linear and nonlinear models, respectively. The prediction results are satisfactory. Certainly, if we need to further improve the accuracy of prediction models without considering the simplicity and interpretability, these more powerful modeling methods could be introduced to develop more predictable models as future work. Meanwhile, the main purpose of this work is to provide a new empirical method to estimate the FP of ternary miscible mixtures by employing particular characteristic physicochemical parameters. The proposed method has also been successfully employed to predict other properties for mixtures, such as auto-ignition temperature (Ye et al, 2014) and flammability limits (Li et al, 2013). 4

Conclusions The flash points of ternary mixtures were measured experimentally. Four characteristic 22

Page 22 of 25

physicochemical parameters of these mixtures were calculated as input parameters to develop empirical models for predicting the FP of ternary mixtures. Particularly, the contribution of the parameter ‘stoichiometric gas concentration’ towards the predictivity of the developed models was

ip t

verified and emphasized. Model validation was performed to check the stability and predictive capability of the presented models. New dataset from literature was employed as an additional

cr

external test set to further verify the applicability of the models. Results showed that the presented models with four parameters are valid, predictive and can give reasonable predictions for new

us

ternary mixtures. Moreover, the presented models were further compared with previous ones

an

reported in literature, and the results showed that the models developed in this work are at least comparable to those mentioned in literature. Particularly, the presented MLR model is more

M

convenient to apply, since it’s more simple and interpretable. Acknowledgements

This research was supported by National Natural Science Fund of China (No. 21436006, 21006045),

d

Natural Science Fund of the Jiangsu Higher Education Institutions of China (No. 12KJA620001),

te

and Priority Academic Program Development of Jangsu Higher Education Institutions. Yong Pan acknowledges the sponsorship of Qing Lan Project.

Ac ce p

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AIChE.14(1): 135-144. DOI: 10.1002/aic.690140124. Rücker, C., Rücker G., & Meringer M., 2007. y-Randomization and its variants in QSPR/QSAR. J. Chem. Inf.

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