# Comparative Study on Theoretical and Machine Learning Methods for Acquiring Compressed Liquid Densities of 1,1,1,2,3,3,3-Heptafluoropropane (R227ea) via Song and Mason Equation, Support Vector Machine, and Artificial Neural Networks

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Section

#### 2.1. Theoretical Equation of State

^{−3}), k

_{B}is the Boltzmann constant, B

_{2}(T) is the second virial coefficient, α(T) is the contribution of the repulsive forces to the second virial coefficient, $G(\eta )$ is the average pair distribution function at contact for equivalent hard convex bodies [20], $\eta $ is the packing fraction. To the convex bodies, $G(\eta )$ can be adopted as follows [17,20]:

_{1}and γ

_{2}are values to reproduce the precise third and fourth virial coefficients, which can be estimated as [17,20]:

_{2}(T), α(T) and b(T) can be described in with the temperature of normal boiling point (T

_{nb}) and the density at normal boiling point (ρ

_{nb}) [17,20]:

_{1}= −0.086, α

_{2}= 2.3988, ${c}_{1}=0.5624$, and ${c}_{2}=1.4267$.

_{nb}, and ρ

_{nb}. γ can be obtained from fitting the experimental results, and T

_{nb}and ρ

_{nb}can be obtained from standard experimental data. According to previous studies, for R227ea, γ is 0.760 [20], T

_{nb}is 256.65 K [31] and ρ

_{nb}is 1535.0 kg·m

^{−3}[31]. Now we can only input the values of T (K) and P (bar) to Equation (1) and the calculated density of R227ea can be acquired.

#### 2.2. Support Vector Machine (SVM)

**Figure 1.**Main structure of a support vector machine (SVM) [35].

#### 2.3. Artificial Neural Networks (ANNs)

**Figure 2.**Schematic structure of an artificial neural network (ANN) for the prediction of compressed liquid densities of 1,1,1,2,3,3,3-heptafluoropropane (R227ea).

## 3. Results and Discussion

#### 3.1. Model Development

#### 3.1.1. Theoretical Model of the Song and Mason Equation

**Figure 3.**Theoretical calculated surface and experimental densities of R227ea. The surface represents the theoretical calculated results by Equations (1)–(9); black points represent the experimental results from Fedele et al. [1]; red crosses represent the experimental results from Ihmels et al. [24]; blue asterisks represent the experimental results from Klomfar et al. [25].

#### 3.1.2. Machine Learning Models

^{−3}) is set as the dependent variable. With the design that users can only input the values of the temperature and pressure to a developed model, we let the machine learning models in our study “learn” the existing data and make precise predictions. The experimental data of Fedele et al. [1], Ihmels et al. [24], and Klomfar et al. [25] were used for model developments respectively. In each model, 80% of the data were set as the training set, while 20% of the data were set as the testing set. The SVMs were developed by Matlab software (Libsvm package [42]) and the ANNs were developed by NeuralTools

^{®}software (trial version, Palisade Corporation, NY, USA). General regression neural network (GRNN) [43,44,45] and multilayer feed-forward neural networks (MLFNs) [46,47,48] were chosen as the learning algorithms of ANNs. Numbers of nodes in the hidden layer of MLFNs were set from 2 to 35. In this case study, the number of hidden layer was set as one. Trials of all ANNs were set as 10,000. All these settings of ANNs were set directly in the NeuralTools

^{®}software. Linear regression models were also developed for comparisons. To measure the performance of the model and make suitable comparisons, RMSE (for testing), training time, and prediction accuracy (under the tolerance of 30%) were used as indicators that evaluate the models. Model results using experimental data from Fedele et al. [1], Ihmels et al. [24], and Klomfar et al. [25] are shown in Table 1, Table 2, Table 3 and Table 4, respectively. Error analysis results are shown in Figure 4.

**Figure 4.**Root mean square error (RMSE) versus number of nodes of multilayer feed-forward neural networks (MLFNs). Bars represent the RMSEs; black dashed lines represent the RMSEs of general regression neural network (GRNN) and support vector machine (SVM). (

**a**) Machine learning models for data provided by Fedele et al. [1]; (

**b**) machine learning models for data provided by Ihmels et al. [24]; (

**c**) machine learning models for data provided by Klomfar et al. [25]; and (

**d**) machine learning models for data provided by all the three experimental reports [1,24,25].

Model Type | RMSE (for Testing) | Training Time | Prediction Accuracy |
---|---|---|---|

Linear Regression | 10.90 | 0:00:01 | 85.0% |

SVM | 0.11 | 0:00:01 | 100% |

GRNN | 1.62 | 0:00:01 | 100% |

MLFN 2 Nodes | 1.13 | 0:03:46 | 100% |

MLFN 3 Nodes | 0.40 | 0:04:52 | 100% |

MLFN 4 Nodes | 0.25 | 0:06:33 | 100% |

MLFN 5 Nodes | 0.37 | 0:07:25 | 100% |

MLFN 6 Nodes | 0.59 | 0:10:38 | 100% |

MLFN 7 Nodes | 0.47 | 0:13:14 | 100% |

MLFN 8 Nodes | 0.32 | 0:14:10 | 100% |

… | … | … | … |

MLFN 29 Nodes | 0.13 | 2:00:00 | 100% |

MLFN 30 Nodes | 0.16 | 2:00:00 | 100% |

MLFN 31 Nodes | 0.10 | 2:00:00 | 100% |

MLFN 32 Nodes | 0.15 | 2:00:00 | 100% |

MLFN 33 Nodes | 0.13 | 2:00:00 | 100% |

MLFN 34 Nodes | 0.12 | 2:00:00 | 100% |

MLFN 35 Nodes | 0.13 | 2:00:00 | 100% |

Model Type | RMSE (for Testing) | Training Time | Prediction Accuracy |
---|---|---|---|

Linear Regression | 86.33 | 0:00:01 | 63.4% |

SVM | 6.09 | 0:00:01 | 100% |

GRNN | 14.77 | 0:00:02 | 96.2% |

MLFN 2 Nodes | 35.41 | 0:02:18 | 82.7% |

MLFN 3 Nodes | 16.84 | 0:02:55 | 96.2% |

MLFN 4 Nodes | 12.14 | 0:03:38 | 96.2% |

MLFN 5 Nodes | 10.67 | 0:04:33 | 96.2% |

MLFN 6 Nodes | 8.35 | 0:04:54 | 98.1% |

MLFN 7 Nodes | 14.77 | 0:06:06 | 96.2% |

MLFN 8 Nodes | 13.06 | 3:19:52 | 96.2% |

… | … | … | … |

MLFN 29 Nodes | 25.46 | 0:31:00 | 90.4% |

MLFN 30 Nodes | 24.25 | 0:34:31 | 90.4% |

MLFN 31 Nodes | 21.23 | 0:42:16 | 90.4% |

MLFN 32 Nodes | 13.40 | 3:38:17 | 96.2% |

MLFN 33 Nodes | 24.84 | 0:47:06 | 90.4% |

MLFN 34 Nodes | 20.65 | 0:53:14 | 90.4% |

MLFN 35 Nodes | 22.46 | 0:58:16 | 90.4% |

Model Type | RMSE (for Testing) | Training Time | Prediction Accuracy |
---|---|---|---|

Linear Regression | 15.87 | 0:00:01 | 94.1% |

SVM | 13.93 | 0:00:01 | 94.1% |

GRNN | 9.53 | 0:00:01 | 100% |

MLFN 2 Nodes | 2.72 | 0:01:13 | 100% |

MLFN 3 Nodes | 5.10 | 0:01:19 | 100% |

MLFN 4 Nodes | 14.05 | 0:01:36 | 94.1% |

MLFN 5 Nodes | 2.77 | 0:02:25 | 100% |

MLFN 6 Nodes | 2.85 | 0:02:31 | 100% |

MLFN 7 Nodes | 15.72 | 0:03:15 | 94.1% |

MLFN 8 Nodes | 3.46 | 0:03:40 | 100% |

… | … | … | … |

MLFN 29 Nodes | 68.34 | 0:15:03 | 82.4% |

MLFN 30 Nodes | 47.09 | 0:17:58 | 82.4% |

MLFN 31 Nodes | 52.60 | 0:22:01 | 82.4% |

MLFN 32 Nodes | 40.03 | 0:27:46 | 82.4% |

MLFN 33 Nodes | 20.69 | 0:39:27 | 94.1% |

MLFN 34 Nodes | 352.01 | 0:56:26 | 11.8% |

MLFN 35 Nodes | 145.61 | 5:01:57 | 11.8% |

Model Type | RMSE (for Testing) | Training Time | Prediction Accuracy |
---|---|---|---|

Linear Regression | 96.42 | 0:00:01 | 93.0% |

SVM | 15.79 | 0:00:02 | 99.2% |

GRNN | 92.33 | 0:00:02 | 93.0% |

MLFN 2 Nodes | 39.70 | 0:06:50 | 96.1% |

MLFN 3 Nodes | 25.03 | 0:08:36 | 97.7% |

MLFN 4 Nodes | 22.65 | 0:10:06 | 99.2% |

MLFN 5 Nodes | 73.84 | 0:13:49 | 93.0% |

MLFN 6 Nodes | 23.64 | 0:17:26 | 99.2% |

MLFN 7 Nodes | 65.74 | 0:14:39 | 93.8% |

MLFN 8 Nodes | 55.32 | 0:16:18 | 93.8% |

… | … | … | … |

MLFN 29 Nodes | 164.54 | 0:52:29 | 89.1% |

MLFN 30 Nodes | 136.96 | 0:37:38 | 89.8% |

MLFN 31 Nodes | 168.13 | 0:41:35 | 89.1% |

MLFN 32 Nodes | 88.25 | 0:50:43 | 93.0% |

MLFN 33 Nodes | 143.65 | 2:30:12 | 89.8% |

MLFN 34 Nodes | 163.78 | 1:00:17 | 89.1% |

MLFN 35 Nodes | 166.92 | 0:44:16 | 89.1% |

#### 3.2. Evaluation of Models

#### 3.2.1. Comparison between Machine Learning Models and the Equation of State

**Figure 5.**Predicted values versus actual values in testing processes using machine learning models. (

**a**) The SVM for data provided by Fedele et al. [1]; (

**b**) the SVM for data provided by Ihmels et al. [24]; (

**c**) the MLFN-2 for data provided by Klomfar et al. [25]; and (

**d**) the SVM for data provided by all the three experimental reports [1,24,25].

Item | RMSE in Training | RMSE in Testing |
---|---|---|

SVM for data provided by Fedele et al. [1] | N/A | 0.11 |

SVM for data provided by Ihmels et al. [24] | N/A | 6.09 |

MLFN-2 for data provided by Klomfar et al. [25] | 11.81 | 2.72 |

SVM for all data [1,24,25] | N/A | 15.79 |

Theoretical calculation for data provided by Fedele et al. [1] | N/A | 196.26 |

Theoretical calculation for data provided by Ihmels et al. [24] | N/A | 372.54 |

Theoretical calculation for data provided by Klomfar et al. [25] | N/A | 158.54 |

#### 3.2.2. Comparison between Conventional Measurement Methods and Machine Learning

**Figure 6.**Apparatus scheme of density measuring for R227ea [1]. VTD represents the vibrating tube densimeter; PM represents the frequency meter; DAC represents the data acquisition and control; MT represents the temperature measurement sensor; M represents the multi-meter; LTB represents the liquid thermostatic bath; HR represents the heating resistance; SB represents the sample bottle; PG represents the pressure gauge; VP represents the vacuum pump; SP represents the syringe pump; NC represents the cylinder.

## 4. Conclusions

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Li, H.; Tang, X.; Wang, R.; Lin, F.; Liu, Z.; Cheng, K. Comparative Study on Theoretical and Machine Learning Methods for Acquiring Compressed Liquid Densities of 1,1,1,2,3,3,3-Heptafluoropropane (R227ea) via Song and Mason Equation, Support Vector Machine, and Artificial Neural Networks. *Appl. Sci.* **2016**, *6*, 25.
https://doi.org/10.3390/app6010025

**AMA Style**

Li H, Tang X, Wang R, Lin F, Liu Z, Cheng K. Comparative Study on Theoretical and Machine Learning Methods for Acquiring Compressed Liquid Densities of 1,1,1,2,3,3,3-Heptafluoropropane (R227ea) via Song and Mason Equation, Support Vector Machine, and Artificial Neural Networks. *Applied Sciences*. 2016; 6(1):25.
https://doi.org/10.3390/app6010025

**Chicago/Turabian Style**

Li, Hao, Xindong Tang, Run Wang, Fan Lin, Zhijian Liu, and Kewei Cheng. 2016. "Comparative Study on Theoretical and Machine Learning Methods for Acquiring Compressed Liquid Densities of 1,1,1,2,3,3,3-Heptafluoropropane (R227ea) via Song and Mason Equation, Support Vector Machine, and Artificial Neural Networks" *Applied Sciences* 6, no. 1: 25.
https://doi.org/10.3390/app6010025