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

Process Safety and Environmental Protection 9 5 ( 2 0 1 5 ) 102–113

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Process Safety and Environmental Protection journal homepage: www.elsevier.com/locate/psep

A new method for the prediction of flash points for ternary miscible mixtures 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 and Safety Engineering, Nanjing Tech University, Nanjing 210009, China

a r t i c l e

i n f o

a b s t r a c t

Article history:

The flash point is one of the most important physicochemical parameters used to character-

Received 8 May 2014

ize the fire and explosion hazard for flammable liquids. The flash points of ternary miscible

Received in revised form 23

mixtures with different components and compositions were measured in this study. Four

February 2015

model input parameters, being normal boiling point, the standard enthalpy of vaporiza-

Accepted 26 February 2015

tion, the average number of carbon atoms and the stoichiometric concentration of the gas

Available online 6 March 2015

phase for mixtures, were employed and calculated based on the theory of vapor–liquid equilibrium. Both multiple linear regression (MLR) and multiple nonlinear regression (MNR)

Keywords:

methods were applied to develop prediction models for the flash points of ternary miscible

Ternary miscible mixtures

mixtures. The developed predictive models were validated using data measured experimen-

Flash point

tally as well as taking data on flash points of ternairy mixtures from the literature. Results

Multiple linear regression

showed that the obtained average absolute error of both the MLR and the MNR model for

Multiple nonlinear regressions

all the datasets were within the range of experimental error of flash point measurements.

Prediction

It is shown that the presented models can be effectively used to predict the flash points of

Physicochemical parameters

ternary mixtures with only some common physicochemical parameters. © 2015 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

1.

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 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 (Gmehling and Rasmussen, 1982; Lance et al., 1979). Furthermore, the measuring process is resource- and time-consuming. For toxic and radioactive compounds, the process is more difficult or even 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 field. 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, 2006; Liaw et al., 2002, 2008a, 2008b; Vidal et al., 2006; White et al., 1997), but little for ternary mixtures.

Corresponding author at: Mail Box 13, 200 North ZhongShan Road, Nanjing 210009, China. Tel.: +86 25 83587305; fax: +86 25 83587411. E-mail address: [email protected] (Y. Pan). http://dx.doi.org/10.1016/j.psep.2015.02.019 0957-5820/© 2015 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

Process Safety and Environmental Protection 9 5 ( 2 0 1 5 ) 102–113

Recently, Liaw et al. (2004, 2009, 2010, 2011), Liaw and Chen (2013), Liaw and Tsai (2013) successfully proposed a series of mathematical models to predict the flash points of ternary mixtures 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 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 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 number of carbon atoms (Catoire and Naudet, 2005) and later extended this equation to mixtures (Catoire et al., 2005, 2006). The objective of this study is to provide a new empirical method to estimate the FP of ternary miscible mixtures as applied in hazard assessment and process design.

2.

Materials and methods

2.1.

Data sets

The data set for this study consists of 182 data points, all of which were determined 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 + n-heptane; n-octane + methanol + n-heptane; ethylene glycol monobutyl ether + methanol + n-heptane, and alcohol + toluene +butanone) with different compositions were measured using a Grabner FLPH Miniflash Tester (Grabner, Austria).

2.2. Determination and calculation of input parameters

number of carbon atoms in the vapor phase, and the 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) 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 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 available from the literature or databases. (Mashuga and Crowl, 2000). In this study, the normal boiling point (Tb ), the standard enthalpy of vaporization (vap Hm ), the average number of carbon atoms (n), and stoichiometric concentration (C) in the gas phase were selected as input parameters to characterize flash points of ternary mixed liquids. For liquid mixtures, the normal boiling point is the tem perature at which i Pi = 1 atm, where Pi denotes the partial pressure of the pure component i in the gas phase at the temperature Tb . Pi = xi i Psat , where xi denotes the mole fraci  tion of the pure component i in the liquid phase ( i xi = 1), ␥i denotes the activity coefficient of the pure component i in the liquid phase at temperature Tb . Psat denotes the satui rated 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 293.15–308.15 K, the evaporation enthalpy can be considered as a constant. The vapor pressures at three temperatures near 298.15 K (293.15 K, 303.15 K and 308.15 K) were calculated based on the Clausius-Clapeyron equation, the plot of ln P ∼ 1/T was drawn and hence, the standard evaporation enthalpy at 298.15 K was obtained from the slope of this straight line. 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 yi Ci , where ni is the number of carbon atoms for pure substance i, Ci is the stoichiometric concentration of the flammable component i and 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 phase at its boiling point (Catoire et al., 2006).

2.3. 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

103

Modeling methods

In this paper, both the multiple linear regression (MLR) and multiple nonlinear regression (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 mixtures were employed as input variables. For the statistical analysis, the software package SPSS 19 (Pallant, 2010) was used and the linear fit was carried out with 95% confidence interval. Moreover, the data processing system (DPS) 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 established. The nonlinear relationship

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between input parameters and flash points can be described as follows:

FP = a +

 4 

 ci Xi

i=1

+c

 4 

 4 

+b

ci Xi

i=1

 4 

3

ci Xi

2

+d

i=1

4 ci Xi

(1)

i=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 Hm (kJ mol−1 ) 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 coefficient for determination (R2 ) can provide a reliable indication to test the robustness of the model. Thus, R2 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 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 and Hahs-Vaughn, 2013). Statistical significance plays a pivotal role in statistical hypothesis testing, 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 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 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 2 (LMO, 20% out) Cross-Validation (QLMO ) was employed. 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 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 2 ) squared correlation coefficient for external validation (Qext was employed as a traditional external validation criterion, which can be calculated as follows:

prediction 2 Qext =1−

3.

Results and discussions

3.1.

Results of MLR models

The MLR procedure was performed on the training set. The dependent variable is the experimental 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: +0.79686 +0.16845 FP(K) = 1.477 × Teb × Hvap × n−0.05948

(yi − yˆ i )

i=1

(yi − y¯ tr )

2

(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 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 MLR and MNR models with only three input parameters were developed by omitting the parameter ‘stoichiometric gas concentration’ for comparison purpose. The corresponding MLR models for ternary mixtures are as follows: Model with four parameters:



FP = 18.993 + 0.554X1 + 1.566X2 − 1.868X3 + 1.852X4 n = 137, R2 = 0.902, f = 304.364, SE = 1.551, p < 0.001 (4)

2

i=1

prediction

2 generated models have much lower R2 and QLMO values than that of the original model as expected, one may reasonably conclude that there is no chance correlation in the model development. Oppositely, if some resulting models have rel2 atively high R2 and QLMO values, it implies that the original model may be obtained due to a chance correlation and thus cannot be 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 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.

(2)

where yi and yˆ i indicate the experimental values and the predicted values of the test set, respectively. y¯ tr indicates the average experimental value of the training set. 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 original descriptor pool and the original model building procedure. The process is repeated 50–100 times. If all the randomized

Model with three parameters:



FP = 24.625 + 0.629X1 + 1.313X2 − 6.383X3 n = 137, R2 = 0.857, f = 260.922, SE = 1.838, p < 0.001 (5)

where n is the number of data points used in training set, R2 is the multiple correlation coefficient, SE is the standard error for the model, f is the value of the F-test, and p is the significant probability of this equation. It can be seen from Eqs. (4) and

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(5) that the significant probability of both 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 Fig. 1.

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. Interestingly, it can be seen from Table 2 that the R2 values of the training set for both 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, 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 parameters and three parameters. Consequently, the second-order nonlinear models with 95% confidence

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

Volume percentage v1

v2

Methanol (1) + n-heptane (2) + n-octane (3)

0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.5 0.6

0.2 0.6 0.2 0.5 0.1 0.5 0.3 0.1 0.3

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

0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.5 0.6

Experimental FP/K

Predicted FP/K (models with four parameters)

Predicted FP/K (models with three parameters)

Predicted FP/K Catoire and Naudet (2005)

MLR

MNR

MLR

MNR

272.35 269.15 273.45 268.45 275.45 268.45 270.45 274.55 269.55

270.98 267.42 272.04 269.6 273.34 268.02 269.48 272.29 267.16

271.17 267.65 272.24 269.81 273.53 268.25 269.7 272.49 267.4

270.88 267.68 271.94 269.69 273.24 268.22 269.72 272.44 267.32

270.52 267.97 272.22 270.17 273.62 269.08 270.39 272.94 268.31

269.63 265.91 270.46 267.95 271.75 266.23 267.75 270.66 265.31

0.2 0.6 0.2 0.5 0.1 0.5 0.3 0.1 0.3

277.25 269.35 275.35 268.35 279.35 267.35 268.25 274.25 266.25

278.42 269.2 275.69 268.64 279.4 267.38 269.45 276.85 267.14

278.65 269.42 275.88 268.86 279.64 267.63 269.67 277.05 267.38

280.17 269.48 277.2 269.1 280.43 267.69 270.05 277.17 267.41

277.4 268.76 275.43 268.96 278.94 268.2 270 276.29 268.16

278.31 267.95 275.24 267.24 279.18 265.75 268.07 276.06 265.42

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

0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.5 0.6

0.2 0.6 0.2 0.5 0.1 0.5 0.3 0.1 0.3

270.95 274.35 272.15 274.55 271.15 275.55 275.15 275.15 278.15

269.81 273.66 271.19 274.14 271.66 275.71 274.86 274.3 278.31

270.03 273.88 271.4 274.36 271.86 275.93 275.07 274.5 278.56

269.08 272.1 270.78 273.05 271.63 274.63 274.35 274.45 277.67

267.11 270.72 269.3 272.41 270.29 274.79 274.07 273.67 278.85

268.45 271.67 269.62 271.95 270.13 273.18 272.69 272.59 275.63

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

0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.5 0.6

0.2 0.6 0.2 0.5 0.1 0.5 0.3 0.1 0.3

272.95 277.15 273.15 276.15 272.15 280.15 276.05 275.15 281.15

270.82 275.56 272.3 275.89 272.91 278.12 276.87 276.11 281.76

271.02 275.77 272.49 276.1 273.09 278.34 277.08 276.31 282.04

270.83 275.7 272.51 276.16 272.95 278.11 276.52 275.37 279.82

269.28 275.23 271.11 275.56 271.51 277.88 275.74 274.25 279.71

269.07 273.66 270.6 274.18 271.22 276.51 275.23 274.38 280.03

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

0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.5 0.6

0.2 0.6 0.2 0.5 0.1 0.5 0.3 0.1 0.3

275.35 269.35 273.35 268.25 277.35 267.35 268.35 274.35 266.35

279.05 270.61 276.02 269.68 280.17 266.52 267.96 275.37 266.51

279.29 270.84 276.23 269.91 280.42 266.78 268.19 275.56 266.76

279.8 270.04 276.82 269.46 280.78 266.65 268.48 275.88 266.8

278.04 270.48 276.32 270.51 280.72 267.37 268.63 274.93 267.65

278.13 268.56 274.76 267.57 279.13 264.78 266.53 274.51 264.82

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Fig. 1 – Comparison between the predicted and experimental FP values of the MLR models. (a) Model with four parameters; (b) model with three parameters. levels were regarded as adequate models in this study, and the corresponding nonlinear empirical equations are as follows: Model with four parameters:

⎧   2 ⎪ FP = 99.3379 + ci Xi + 0.0091 ci Xi ⎪ ⎨  ci Xi = 0.1042X1 − 0.5806X2 − 0.6783X3 + 0.6860X4 ⎪ ⎪ ⎩ 2

n = 137, R = 0.9022, f = 241.7644, SE = 1.503, p < 0.001 (6) Model with three parameters:

⎧   2 ⎪ ci Xi + 0.0011 ci Xi FP = 682.121 − 4.347 ⎪ ⎨  ci Xi = 0.481X1 + 2.376X2 − 5.575X3 ⎪ ⎪ ⎩ 2

n = 137, R = 0.823, f = 306.436, SE = 2.035, p < 0.001 (7)

As can be seen from Eqs. (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 Fig. 2.

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

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Table 2 – The performance parameters of nonlinear models with different orders. Model

Training set

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

Test set 2 Qext

R2 0.902 0.902 0.823 0.808

Maximum absolute error (K)

0.863 0.745 0.774 0.797

Average absolute error (K)

3.94 10.10 4.03 8.41

1.12 6.00 1.64 3.28

Fig. 2 – Comparison between the predicted and experimental FP values of the MNR models. (a) Model with four parameters; (b) model with three parameters. Table 3 – Main prediction performance parameters of the MLR and MNR models applied on the training and test sets. Model

Training set

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

Test set

R

AAE

RMSE

2 Qext

0.902 0.902 0.857 0.823

1.200 1.201 1.482 1.637

1.524 1.492 1.811 2.019

0.866 0.863 0.796 0.774

2

AAE

RMSE

1.111 1.124 1.405 1.489

1.400 1.418 1.720 1.808

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Fig. 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.

±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. 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 1.111 K and 1.405 K, respectively. For the MNR models, the AAE values were 1.124 K and 1.489 K, respectively. It can thus be concluded that, for both the MLR 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 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 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, which also demonstrates the important contribution of the new parameter toward the predictivity of the developed models for predicting the FP of ternary mixtures. Consequently, both the developed 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, 2 values. all generated models produced low R2 and low QLMO 2 2 The obtained maximum R and QLMO values of the generated models were 0.097 and 0.089, 0.117 and 0.105 for the MLR

Process Safety and Environmental Protection 9 5 ( 2 0 1 5 ) 102–113

109

Fig. 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. and MNR models with four parameters, 0.231 and 0.137, 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 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 Figs. 3 and 4. It can be seen from these figures that the prediction residuals are randomly distributed on 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. 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 developed models. These 53 different data points cover five sets of common mixtures: methanol + toluene + 2,2,4trimethylpentane; methanol + decane + acetone; cyclohexanone + p-xylene + methanol; ethanol + toluene + ethylacetate; methanol + ethanol + acetone. After calculating the four 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 4. It can be seen from Table 4 that most of the predicted values agreed well with the experimental ones, with the maximum absolute error being 8.62 K for the MLR model and 9.31 K for the MNR model, respectively. Besides, the predicted AAE and RMSE values were calculated for this new external test set.

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Table 4 – Predicted flash point values for the additional external test set by the MLR and MNR models (with four parameters). Ternary mixtures

Mole fraction

Experimental FP/K

x1

x2

Methanol (1) + toluene (2) + 2,2,4trimethylpentane (3)

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 0.5 0.5 0.6 0.6 0.6 0.7 0.7 0.8

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 0.3 0.4 0.1 0.2 0.3 0.1 0.2 0.1

262.95 263.85 264.25 264.75 265.65 266.85 268.25 270.25 261.95 262.45 263.85 264.45 265.35 267.75 269.75 261.85 263.25 264.65 265.85 267.05 268.85 262.15 263.25 264.25 265.75 268.55 262.65 263.35 264.85 268.05 262.45 263.95 267.05 263.35 265.45 262.85

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

0.1 0.2 0.3 0.3 0.4 0.4 0.8

0.7 0.1 0.5 0.6 0.3 0.5 0.1

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

0.25 0.65

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

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

MLR Predicted FP/K

MNR

Absolute error/K

Predicted FP/K

Absolute error/K

263.86 265.07 266.25 267.40 268.48 269.49 270.39 271.13 261.84 262.77 263.69 264.55 265.35 266.03 266.54 261.94 262.76 263.54 264.24 264.81 265.19 262.42 263.16 263.82 264.35 264.64 262.96 263.62 264.13 264.35 263.48 264.00 264.16 263.89 264.00 263.85

−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 0.72 3.70 −1.03 −0.05 2.89 −0.54 1.45 −1.00

267.51 266.86 266.27 265.76 265.33 265.02 264.88 264.97 266.62 266.14 265.72 265.40 265.21 265.19 265.44 265.60 265.25 264.98 264.86 264.93 265.34 264.73 264.49 264.40 264.54 265.08 263.99 263.90 264.08 264.76 263.38 263.56 264.38 262.96 263.92 263.29

−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 0.77 3.29 −0.93 0.39 2.67 0.39 1.53 −0.44

260.65 257.75 263.95 269.65 262.65 269.65 275.15

267.76 256.82 261.00 261.62 259.33 261.03 271.74

−7.11 0.93 2.95 8.03 3.32 8.62 3.41

260.55 268.02 266.21 267.60 267.39 270.30 273.47

0.10 −10.27 −2.26 2.05 −4.74 −0.65 1.68

0.2 0.15

281.15 290.15

282.68 291.17

−1.53 −1.02

280.07 289.02

1.08 1.13

0.55 0.5

0.15 0.25

269.82 270.37

269.22 272.04

0.60 −1.67

268.52 271.54

1.30 −1.17

0.1 0.8 0.7 0.1 0.6 0.3

0.8 0.1 0.1 0.7 0.1 0.1

274.45 273.15 267.55 268.85 264.45 258.35

273.59 271.30 265.02 270.77 264.32 257.32

0.86 1.85 2.53 −1.92 0.13 1.03

273.07 273.70 269.67 269.24 267.67 256.03

1.38 −0.55 −2.12 −0.39 −3.22 2.32

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, instead of a complex nonlinear one, may exist

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Table 5 – Predicted flash point values for the literature mixtures by the MLR and MNR models (with four parameters) as well as ). Ternary mixtures

Cyclohexanone (1) + p-xylene (2) + methanol (3) (Catoire et al., 2005) Ethanol (1) + toluene (2) + ethylacetate (3) (Catoire et al., 2005) Methanol (1) + ethanol (2) + acetone (3) (Catoire et al., 2006)

Mole fraction x1

x2

0.25 0.65

0.2 0.15

0.55 0.5 0.1 0.8 0.7 0.1 0.6 0.3

Experimental FP/K

MLR

Catoire et al.’s models (2005, 2006)

Predicted FP/K

Absolute error/K

Predicted FP/K

281.15 290.15

282.68 291.17

−1.53 −1.02

280.07 289.02

1.08 1.13

280.75 287.75

0.4 2.4

0.15 0.25

269.82 270.37

269.22 272.04

0.60 −1.67

268.52 271.54

1.30 −1.17

269.35 269.75

0.47 0.62

0.8 0.1 0.1 0.7 0.1 0.1

274.45 273.15 267.55 268.85 264.45 258.35

273.59 271.30 265.02 270.77 264.32 257.32

0.86 1.85 2.53 −1.92 0.13 1.03

273.07 273.70 269.67 269.24 267.67 256.03

1.38 −0.55 −2.12 −0.39 −3.22 2.32

271.65 272.45 267.05 265.85 263.45 256.95

2.8 0.7 0.5 3.0 1.0 1.4

between the selected physicochemical parameters and flash points of ternary mixtures.

3.6.

Model comparison

3.6.1.

Comparison between MLR and MNR models

The performance comparison of the MLR and MNR models is shown in Table 3, where the results obtained from the MLR model are very close to or even a little better than those obtained from the 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 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 relationships. Moreover, the MLR model provides a simple mathematical equation, which is convenient to apply and has a better model interpretability. As such the MLR model is recommended for practical engineering applications.

3.6.2.

MNR

Comparison with previous works

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. For Catoire et al.’s dataset (2005, containing two different mixtures of cyclohexanone + p-xylene + 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 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 + acetone), the resulting AAE and RMSE values calculated by the models presented in this work were 1.4 and 1.6 for the MLR model, and 1.7 and 1.9 for the MNR model, respectively, while the values calculated by Catoire et al.’s model (2006) were 1.6 and 1.8, respectively. It can

Absolute error/K

Predicted FP/K

Absolute error/K

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 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 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 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 literature were performed. As is well known, most predictive models were developed based on Le 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 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 specific ternary mixtures. It should be noted that the non-ideality of the liquid phase is accounted for 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 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

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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 + 2,2,4trimethylpentane (2) Methanol + acetone + decane

Activity coefficient model (NRTL,UNIQUAC)

Saldana et al. (2013)

Not specified

Catoire et al. (2005, 2006)

(1) Ethanol + toluene + ethylacetate (2) Methanol + ethanol + acetone

QSPR model based on additive molecular descriptors Mathematical regression models based on three common physicochemical parameters

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

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. 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, 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 ability leading to satisfactory prediction performance. However, these methods also suffer some 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, 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 physicochemical parameters of these mixtures were calculated as input parameters to develop

MLR and MNR models based on four common physicochemical parameters

Comments The model provides satisfactory predictions, which can accurately predict the flash points of the studied mixtures The model presents a lower prediction accuracy 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 The models have satisfactory predictivity and applicability, which can give reasonable predictions for new ternary mixtures

empirical models for predicting the FP of ternary mixtures. Particularly, the contribution of the parameter ‘stoichiometric gas concentration’ toward the predictivity of the developed models was 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 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 ternary mixtures. Moreover, the presented models were further compared with previous ones 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 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), Natural Science Fund of the Jiangsu Higher Education Institutions of China (no. 12KJA620001), and Priority Academic Program Development of Jangsu Higher Education Institutions. Yong Pan acknowledges the sponsorship of Qing Lan Project.

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