# Accurate Predicting the Surface Tension of Vegetable-Oil-Based Fuels Using Gibbs’ Thermodynamic Additivity as an Alternative to QSPR

Authors: Nichaphat Sitthisuk; Kaokanya Sudaprasert; Suriya Phankosol; Vittaya Punsuvon
Journal: Journal of Chemistry
 (2023)

---

## Abstract

The surface tension of fuel oil is one important property of the engine combustion process which is correlative to other oil physical and chemical properties. Therefore, this research focuses on the predictive model of fuel oil surface tension through the outstanding alternative fuel for combustion engines such as vegetable oil which is biodegradable, less toxic, has potential in domestic production, and has energy content close to diesel fuel. The six economic saturated/unsaturated vegetable oil samples (rapeseed, sunflower, soybean, palm, corn, and grapeseed oil) are used as the comparison cases for the predictive model of surface tension. The proposed model is formulated by the concept of the Gibbs energy additivity method (GEAM) which apply two properties of oils composition such as the number of carbon atoms (z) and the number of double bonds (nd) of fatty acid at various temperatures to predict surface tension through the regression model. Three common regression models such as the Parachor model and two literature physical relation models are used as the comparative model at 293.15 K and 313.15 K, respectively. The comparison results of the predictive model show the acceptable predicting value in all predictive models, with an overall average absolute deviation (%AAD) of 0.38% for the proposed model, 2.28% for the Eliezer model, and 2.74% with the Esteban model at 293.15 K, while the overall %AAD at 313.15 K presents 0.21% for the proposed model and 9.08% with the Parachor model. However, the proposed model indicates the use of a combination between molecule structure and thermodynamic parameters by its model mismatch reduction and proves to be a handle for precisely predicting surface tension in the case of vegetable oil.


---

## Body

## 1. Introduction

Vegetable oil is an extracted plant oil which commonly in liquid form. The chemical composition of this oil consists of triacylglycerol (almost 95%) and diacylglycerol (approximately 5%) which is suitable for a cooking ingredient and widely used as an alternative fuel in the energy industry. Vegetable oil feedstocks for the industrial plant are generally applied following the agricultural potential in each topography. For example, a large amount of palm productivity in Asia leads to the use of palm oil as industrial raw material. Moreover, the by-product and waste residual from any industry can add value by using it as the main raw material for the alternative energy industry. Moreover, the energy content of vegetable oils is also close to diesel fuel which leads to using vegetable oils in the engine. Many research attempt to apply vegetable oils in the form of blending fuel such as the use of soybean oil, rapeseed oil, palm oil, and palm kernel oil with a combining of alcohols and several cosolvents, as reported by Jin et al. [1–3]. According to Jin et al. [1–3], solubilization and fuel properties in the blends of alcohols and vegetable oils were studied in order to improve engine performance. This reason demonstrates the significant effect of raw material vegetable oils and other chemical compounds on fuel properties. Thus, the trend of fuel properties improvement was focused on in many research.In the past, vegetable oils have been used in several applications (i.e., direct-used vegetable oil in low-speed diesel engines, biodiesel, and biodiesel from the hydrogenation process (H-FAMES)) [4, 5]. However, the use of this fuel is needed to consider its fuel properties as per the previous fuel properties improvement for example iodine value, saponification, viscosity, density, and surface tension. Surface tension is the cohesive interaction between the molecules in the liquid phase. The liquid molecules are attracted to each other and exert a net force pulling themselves together. Surface tension is one of the important physical properties of biodiesel that directly affects the combustion of the injector in the diesel engine. The appropriate surface tension in the oil is well-known as necessary for the design of experimental tools, equipment, and transport processes, respectively.Surface tension is well-known as a direct variation in temperature. Oh and Let [6] studied the surface tension of palm oil. It was found that the surface tension decreased in a linear trend while the temperature increased, as shown in relation to the following equation:(1)γ=a1+b1T,where a1 and b1 are regression coefficients for the linear relation of chemical and physical properties. Moreover, surface tension is also correlated with other physical properties, especially the viscosity and density of substances. Pelofsky [7] proposed the following relationship between surface tension and absolute viscosity, which is, apparently, valid for a large variety of liquids in the following equation:(2)γ=A.e−B/ηi.Esteban et al. [8] tested the surface tension of eight vegetable oils at a temperature of 10–140°C and proposed the equation to predict the surface tension of vegetable oils from the density of vegetable oils as shown in the following equation:(3)γ=a2+b2ρ,where a2 and b2 are regression coefficients for the linear relation of chemical and physical properties.The Parachor model was developed by Sugden [9] which described the relation of surface tension with temperature in some functional groups. In this work, Sugden’s Parachor value for some pure methyl ester is introduced inTable 1 to predict the pure component surface tension. The Sugden expression is given as follows:(4)σ1/4=PchρM,where σ is the surface tension of the pure component (mN/m), (Pch) is the Parachor of the pure component, ρ is the density of the liquid phase in g/mL, and M is the molecular weight in g/mol. However, the simple function of the surface tension for the pure component cannot be described as the surface tension of mixtures which is caused by the nonunity of the bulk. In fact, the surface tension of a mixture is normally in the form of a complex molecular structure. Thus, the average method such as a Dalton-typemass-average equation is necessary to predict the surface tension of a mixture. The Parachor model for mixture liquid is shown as follows:(5)σm=∑i=1nσiyi,where σm is the mean surface tension of the mixture (N/m), σi is the surface tension of component i (N/m), and yi is the mass fraction of component i.Table 1
Sugden’s Parachor and physicochemical property in each pure methyl ester of triglyceride branch [9].
Pure methyl ester of triglyceride branchParachorMolecular weight (g/mol)Density (g/mL)C16:0762284.40.86C18:0840312.50.86C16:1751282.40.87C18:1829310.50.87C18:2818308.50.88C18:3807306.50.89Phankosol et al. [10] developed a linear regression model on the concept of the Gibbs free energy following the Martin theory [11]. The developed model extends the relation of Gibbs free energy, enthalpy, and entropy in order to estimate the rate of the surface tension change with temperature in equation (6) as a simplified model followed by Halvorsen et al. [12](6)γ=60.211−0.4307z−0.1125TK−0.00207zTK+3.676nd−0.00893ndTK,where a=HfS/∆A,b=∆Sfs/∆A,c=δHs/∆A,d=δTSs/∆A,e=∆Hdbs/∆A,f=∆Sdbs/∆A,f,nd, and A are the function group, the number of double bonds, and the surface area.In the Melo-Espinosa et al. [13] research, the estimated oil surface tension model from the fatty acid composition is developed by using a multiple linear regression model and the experimental data of oil surface tension at 20°C [12]. The model is formulated based on the relation of surface tension and fatty acid composition in 15 samples of vegetable oils and is shown in the following equation:(7)γ=−0.201777La+0.276565P+0.300558Ot+0.321542O+0.333744Li+0.363275Ln+0.394795Ri+0.6142218S+2.002772M,where La, P, Ot, O, Li, Ln, Ri, S, and M are the independent variables of fatty acid composition (La = lauric acid (C12:0), P = palmitic acid (C16:0), Ot = oleic acid (C18:1) trans, O = oleic acid (C18:1) cis, Li = linoleic acid (C18:2), Ln = linolenic acid (C18:3), Ri = ricinoleic acid (C18:1), S = stearic acid (C18:0), and M = myristic acid (C14:0)).According to all the above equations, surface tension is an important physical property that is associated with several variables such as viscosity, density, temperature, and chemical composition in each vegetable oil (for example, the number of carbon atoms;z and number of double bonds; nd). However, equation (6) was used to predict only the surface tension of biodiesels and FAMEs.Although the equation for predicting surface tension is commonly studied but in vegetable oils is rarely presented. As a result, many studies have studied the effects of the chemical composition of various vegetable oils on surface tension. Thus, Gibbs free energy is used to correlate the various molecular forces to estimate the physical and chemical properties of fatty acids, especially in vegetable oil. This correlation led to the development of equations that are linked to the thermodynamic theory of all variables and also in an easy-to-use form. Thus, the developed correlation equations following Halvorsen et al. [12] is highly accurate estimation for the complex chemical composition of various vegetable oils.


### 1.1. Theory

Based on the surface tension theorem, surface tension is closely related to the intermolecular force of its chemical structure. According to the number of the carbon atomic chain, the surface tension model expands the development of the relationship between surface tension and Gibbs free energy (free energy of interfacial interaction: G) can be illustrated in the following equation:(8)γ=∂Gs∂AT,P,ni,when γ is the surface tension, G is the Gibbs free energy, A is the surface area, T is the temperature, P is the pressure, and ni is the molar mass of each fatty acid composition.In the case of the triglyceride structure (triglyceride; TG) in vegetable oil, it was the carbon-based chemical structure slightly the same as biodiesel oils but it has a difference in its structural groups that consists of 1 glycerol group and 3 fatty acids molecules in Figure1. The total Gibbs free energy can be analyzed in the summation of sub-Gibbs free energy of glycerol (ΔG) and sub-Gibbs free energy of fatty acids molecules (δG).Figure 1
Molecular structure of triglycerides.From Martin’s equation Martin [11] modified by Phankosol et al. [10] shown in equation (9), TG divided the structural groups in each molecule functional group as glycerol(f), CH, CH2, and CH3 in order to describe the total Gibb free energy transformation of the liquid-to-gas transition.


## 1.1. Theory

Based on the surface tension theorem, surface tension is closely related to the intermolecular force of its chemical structure. According to the number of the carbon atomic chain, the surface tension model expands the development of the relationship between surface tension and Gibbs free energy (free energy of interfacial interaction: G) can be illustrated in the following equation:(8)γ=∂Gs∂AT,P,ni,when γ is the surface tension, G is the Gibbs free energy, A is the surface area, T is the temperature, P is the pressure, and ni is the molar mass of each fatty acid composition.In the case of the triglyceride structure (triglyceride; TG) in vegetable oil, it was the carbon-based chemical structure slightly the same as biodiesel oils but it has a difference in its structural groups that consists of 1 glycerol group and 3 fatty acids molecules in Figure1. The total Gibbs free energy can be analyzed in the summation of sub-Gibbs free energy of glycerol (ΔG) and sub-Gibbs free energy of fatty acids molecules (δG).Figure 1
Molecular structure of triglycerides.From Martin’s equation Martin [11] modified by Phankosol et al. [10] shown in equation (9), TG divided the structural groups in each molecule functional group as glycerol(f), CH, CH2, and CH3 in order to describe the total Gibb free energy transformation of the liquid-to-gas transition.


## 2. Materials and Methods

The materials and methods section should contain sufficient detail so that all procedures can be repeated. It may be divided into headed subsections if several methods are described.(9)∆Gi,N=∆Gf,N+zNδG+nd,NδGdb,where zN and nd,N are the number of carbon atoms and number of double bonds in the chain hydrocarbon and ∆Gf,N, δG, and δGdb are the Gibbs free energy of a glycerol group, the change in free energy/carbon atom, and double bond, respectively.For triglycerides, the free energy of interfacial interaction∆GTG is the sum of free energies of interfacial interaction of individual glycerol, and three fatty acids contributed to the mixture.(10)∆GTG=1N∑i=1N∆Gi,NwhereforTG;N=3.Equation (11) is obtained by combining equations (9) and (10)(11)∆GTG=1N∑i=1N∆Gf,N+1N∑i=1NzNδG+1N∑i=1Nnd,NδGdb.From the Gibbs free energy of a functional group, carbon atom, and double bond molecule average form, the equations are shown in the following equations:(12)∆Gf,ave=∆Gf,1=∆Gf,2=∆Gf,3=∆Gf,N,(13)zave∆Gave=1N∑i=1NzNδG,(14)nd,ave∆Gdb,ave=1N∑i=1Nnd,NδGdb.The (11) is rearranged in form of (15) by replacing the average form of equation (12) with (14).(15)∆GTG=∆GFAave+zFAave∆GFAave+nFAave∆GdbFAave.Applying the Matin equation for vegetable oils, the Gibbs free energy of vegetable oils is shown in the average form of the Gibbs free energy of triglyceride in the following equation:(16)∆Goil=1M∑j=1M∆GTGwhereforvegetableoil;M=amountofTGquantity.Equation (16) is substituted by (15) and shown in each average term of the following equation:(17)∆Goil=1M∑j=1M∆Gfave,j+1M∑j=1MzTGave,j∆GTGave,j+1M∑j=1MndTGave,j∆GdbTGave,j.Equation (17) is reduced to the following equation:(18)∆Goil=∆Gfave+zave∆Gave+ndave∆Gdbave,where(19)∆Gfave=1M∑j=1M∆Gfave,j,∆Gave=1M∑j=1M∆GTGave,j,∆Gdbave=1M∑j=1M∆GdbTGave,j,zave=1M∑j=1MzTGave,j,ndave=1M∑j=1MndTGave,j.The surface tension of vegetable oil and Gibbs free energy relation to the following equation:(20)γoil=∆Goil∆A.Expanding equation (19) to the enthalpic and entropic form ∆G=∆H−T∆S, one gets,(21)γoil=a+bzave+cT+dzaveT+end,ave+fnd,aveT,where a=∆Hf/∆A,b=∆H/∆A,c=−∆Sf/∆A,d=−∆S/∆A,e=∆Hdb/∆A,f=∆Hdb/∆A and γoil is surface tension of each triglycerides(i) (mN/m) which can estimate from average surface tension of all triglyceride(i). This average method is based on Dalton’s atomic theory which concludes the mixture components are the rearrangement of the constant pure component in the mixture. Alternatively, the surface tension of vegetable oil can be estimated by (21) if the carbon numbers and number of double bonds of fatty acids in vegetable oils are average and the mixture is arbitrarily treated as a single triglyceride.


### 2.1. Methodology

#### 2.1.1. The Average Number of Carbon Atom and Double Bond

The average number of carbon atom (zave) and the number of double bond (nd(ave)) of fatty acid for biodiesel was presented in term of z and nd and they were calculated according to the following equations:(22)zave=∑i=1zxizi∑i=1zxi,(23)nd,ave=∑i=1zxind,ave∑i=1zxi,where xi is the mass or mole fraction of fatty acid ester; zi and nd(i) are the number of carbon atoms and double bond of the i fatty acids.


#### 2.1.2. The Numeric Constant Coefficients Estimation

Numeric constant coefficients of (24) were determined by multiple linear regression of Phankosol et al. [10]. The surface tension data of vegetable oils at different temperatures were obtained from Esteban et al. [8]. However, all the six numeric constants a, b, c, d, e, and f are 60.211, −0.4307, −0.1125, 0.00207, 3.676, and −0.00893, respectively. Substitution these numeric values into equation (23), we get(24)γ=60.211−0.4307z−0.1125T+0.00207zT+3.676nd−0.00893ndT.


#### 2.1.3. Statistical Analysis

The percent of average absolute deviation (%AAD) and %Bias were used for statistical analysis as shown in the following equations:(25)AAD=γexp−γest/γexp×100n,(26)Bias=γexp−γest/γexp×100n,where γexp and γest are the experimental and predicted surface tensions, respectively. n is the number of data points.


## 2.1. Methodology

### 2.1.1. The Average Number of Carbon Atom and Double Bond

The average number of carbon atom (zave) and the number of double bond (nd(ave)) of fatty acid for biodiesel was presented in term of z and nd and they were calculated according to the following equations:(22)zave=∑i=1zxizi∑i=1zxi,(23)nd,ave=∑i=1zxind,ave∑i=1zxi,where xi is the mass or mole fraction of fatty acid ester; zi and nd(i) are the number of carbon atoms and double bond of the i fatty acids.


### 2.1.2. The Numeric Constant Coefficients Estimation

Numeric constant coefficients of (24) were determined by multiple linear regression of Phankosol et al. [10]. The surface tension data of vegetable oils at different temperatures were obtained from Esteban et al. [8]. However, all the six numeric constants a, b, c, d, e, and f are 60.211, −0.4307, −0.1125, 0.00207, 3.676, and −0.00893, respectively. Substitution these numeric values into equation (23), we get(24)γ=60.211−0.4307z−0.1125T+0.00207zT+3.676nd−0.00893ndT.


### 2.1.3. Statistical Analysis

The percent of average absolute deviation (%AAD) and %Bias were used for statistical analysis as shown in the following equations:(25)AAD=γexp−γest/γexp×100n,(26)Bias=γexp−γest/γexp×100n,where γexp and γest are the experimental and predicted surface tensions, respectively. n is the number of data points.


## 2.1.1. The Average Number of Carbon Atom and Double Bond

The average number of carbon atom (zave) and the number of double bond (nd(ave)) of fatty acid for biodiesel was presented in term of z and nd and they were calculated according to the following equations:(22)zave=∑i=1zxizi∑i=1zxi,(23)nd,ave=∑i=1zxind,ave∑i=1zxi,where xi is the mass or mole fraction of fatty acid ester; zi and nd(i) are the number of carbon atoms and double bond of the i fatty acids.


## 2.1.2. The Numeric Constant Coefficients Estimation

Numeric constant coefficients of (24) were determined by multiple linear regression of Phankosol et al. [10]. The surface tension data of vegetable oils at different temperatures were obtained from Esteban et al. [8]. However, all the six numeric constants a, b, c, d, e, and f are 60.211, −0.4307, −0.1125, 0.00207, 3.676, and −0.00893, respectively. Substitution these numeric values into equation (23), we get(24)γ=60.211−0.4307z−0.1125T+0.00207zT+3.676nd−0.00893ndT.


## 2.1.3. Statistical Analysis

The percent of average absolute deviation (%AAD) and %Bias were used for statistical analysis as shown in the following equations:(25)AAD=γexp−γest/γexp×100n,(26)Bias=γexp−γest/γexp×100n,where γexp and γest are the experimental and predicted surface tensions, respectively. n is the number of data points.


## 3. Results and Discussion

The predicted surface tensions of six vegetable oils including rapeseed, sunflower, soybean, palm, corn, and grapeseed at the temperature interval from 293.15 K up to 493.15 K agree well with the reference values, thez(ave) and nd(ave) calculated according to (24) for each vegetable oils are included in Table 2, and the average of z(ave) and nd(ave) are 17.06–17.90 and 0.60–1.58, respectively. The predictive model is estimated through chemical properties and temperature in order to describe the rate of change in surface tension. The six proposed vegetable oils are used as the sample cases in model prediction and describe their composition and the average number of the double bond and atom carbon in Table 2.Table 2
The proposed vegetable oil composition and its average number of carbon atoms, number of double bonds, saponification number, and iodine value.
Vegetable oilsMass fraction of methyl ester compositionz(ave)nd(ave)SNIVC16:0C16:1C18:0C18:1C18:2C18:3Rapeseed4.810.291.4265.4819.508.5017.901.30190.15111.65Sunflower6.700.322.9038.6751.290.1217.861.42191.19122.18Soybean11.180.352.9125.1955.375.0017.771.51191.98130.35Palm46.560.433.8438.4910.500.1817.060.60197.6051.69Corn11.400.341.7432.7553.280.5017.771.41192.09121.62Grapeseed7.200.103.9020.2068.410.1817.851.58191.45135.89The proposed model is based on the total Gibbs energy (G) theorem that is associated with a chemical functional group. In the proposed model, the average Gibbs energy is used as the representative of the Gibbs energy in each functional group in order to evaluate the total Gibbs energy. This total Gibbs energy model can rearrange in the simple form of linear expression that the linear relationship was supported by the Gibbs energy model of Krisanangkura et al. [14] and the biodiesel viscosity model of Chum-in et al. [15]. The coefficient of the linear model for this model at different temperatures is evaluated by the multiple regression of surface tension and chemical property terms of each vegetable oil.The predicted surface tension of the proposed model demonstrates the model mismatch in terms of %AAD in Table3. The palm oil which had the lowest z(ave) (17) and nd(ave) (0.6) is present the lowest %AAD average (0.21%) while other vegetable oils had the higher of z(ave) (17.8–17.9) and nd(ave) (1.3–1.6) give the high %AAD (0.26–0.5%). Generally, the high z(ave) and nd(ave) give the high %AAD due to the unpredictable structure of long chain molecule and double bond located in the model. However, the soybean and corn oil %AAD in Table 3 are shown to be high (0.4–0.5%) at z(ave) (17.8). The reason for this abnormally high %AAD is the high ratio between high unsaturation degree (C18:2 and C18:3) and low unsaturation degree (C18:1) in both oils compared with other sample oils. Normally, double and triple bonds are characterized as a high nonlinear relation in surface tension terms. Thus, it led to a mismatch in the linear relation model.Table 3
The predicted surface tension of the proposed model compared in the case of 6 vegetable oils at 293.15–413.15 K.
Vegetable oilsz(ave)nd(ave)Temperature (K)293.15313.15333.15353.15373.15393.15413.15Rapeseed17.901.3033.7032.1730.6529.1227.6026.0724.55(0.30)(−0.22)(−0.49)(−0.07)(0.00)(0.11)(0.61)Sunflower17.861.4233.7532.2430.7429.2327.7326.2224.71(0.74)(0.19)(−0.13)(−0.10)(−0.11)(0.30)(−0.36)Soybean17.771.5133.7632.2830.8029.3227.8326.3524.87(0.41)(0.37)(−0.33)(−0.41)(−0.47)(−0.57)(−0.92)Palm17.060.60—31.4029.9828.5627.1525.7324.31—(0.32)(0.07)(−0.21)(−0.18)(−0.12)(0.37)Corn17.771.4133.7032.2130.7229.2327.7426.2524.76(0.30)(−0.03)(−0.39)(−0.79)(−0.51)(−0.57)(−0.24)Grapeseed17.851.5833.8532.3530.8629.3727.8726.3824.89(0.15)(−0.15)(−0.19)(−0.58)(−0.25)(0.08)(0.84)In order to verify the model, the proposed model is also compared in two steps. Step one is the comparison value of linear coefficientsa and b in the form of the relationship between surface tension and temperature as per equation (1) and the model mismatch by using the graphical technique of relative deviation. Step two is the model accuracy comparison with two general composition-based prediction models and one linear relationship model of surface tension and kinematic viscosity. The first model is the Melo-Espinosa et al. (MLR) model which is presented a surface tension predictive model that is based on only the quantity of fatty acid compositions such as palmitic acid (C16:0), stearic acid (C18:0), oleic acid (C18:1), linoleic acid (C18:2), and linolenic acid (C18:3) at 293.15 K and use the multiple linear regression model to find the relationship of surface tension and fatty acid composition. The second model is the Parachor model which is the surface tension predictive model base on the combination of Sugden Parachor expression and Dalton-typemass-average equation. Esteban’s model is the linear regression model which predicts the surface tension from the relationship between the surface tension and kinematic viscosity [16].According to step one, the linear coefficientsa and b and the coefficients of determination (R2) in Table 4 are represented that the proposed model is agreed with the surface tension and temperature relationship in the Flingoh model through slightly the same a and b coefficients. Moreover, the Esteban model in Table 4 which is a similar linear form of kinematic viscosity and surface tension relationship is shown as the lowest coefficient of determination (R2). For this reason, the proposed model demonstrated the highly accurate trend in the group of linear form which is compared in Table 4.Table 4
Linear coefficient terms of the surface tension relationship.
Vegetable oilsProposed modelFlingoh model [6]Esteban model [8]abR2abR2abR2Rapeseed56.05−0.0763155.74−0.07541−77.10121.010.9993Sunflower55.82−0.0753156.15−0.07611−76.06119.810.9997Soybean55.49−0.0741155.34−0.07361−74.09117.390.9998Palm53.59−0.0709153.70−0.07111−64.90107.160.9994Corn55.54−0.0745155.94−0.07591−79.12123.040.9999Grapeseed55.73−0.0747155.34−0.07361−75.29118.710.9999The presented percent differences between the compared surface tension of the predicted model and the experimental values of vegetable oil from Figures2 to 4 show that the relative deviation of percent difference is approximately±1.0% in all predicted models. The proposed model’s deviation (Figure 2) is in agreement with the supported literature models such as the Flingo model (Figure 3) and Esteban model (Figure 4) which have a high deviation of a percent difference in the group of dominant C18:2 composition such as sunflower oil, soybean oil, corn oil, and grapeseed oil. The literature models demonstrate a good prediction than the proposed model in the case of corn oil due to the specific oil model generation of the Flingo model and its density-based model of the Esteban model. However, the specific oil model generation and density-based model led to improper handle using in all cases of vegetable oil which can be solved by the proposed model technique. In addition case of Sahasrabudhe et al. [17] measurement data, the proposed model can predict and compare with the literature model by using its measurement data which gives a similar AAD as the literature model of about 3%, but has less model mismatch of about 1%.Figure 2
Relative deviations (%Δ) between the surface tension of the proposed model by equation (23) and the experimental values reported by Esteban et al. [8].Figure 3
Relative deviations (%Δ) between the surface tension of the Flingo model by equation (1) and the experimental values reported by Esteban et al. [8].Figure 4
Relative deviations (%Δ) between the surface tension of the Esteban model by equation (3) and the experimental values reported by Esteban et al. [8].


### 3.1. The Melo-Espinosa et al. MLR Model [13]

Linear regression of the Melo-Espinosa et al. MLR model is used to make an estimated surface tension data from the quantity of fatty acid compositions in Table2 and the MLR model in Melo-Espinosa et al. The estimated surface tension is compared with the proposed model in Table 5 (in the case of six vegetable oils at 293.15 K). The % AAD results of each vegetable oil compared with the proposed model in Table 5 show that it is not a significantly different error but the proposed model is a lower %AAD than the Melo-Espinosa et al. MLR model. Moreover, the proposed model can be predicted in various temperatures while the Melo-Espinosa et al. MLR model cannot predict them. The advantage of the proposed model is the simplified linear model structure form which can be easily calculated without using the knowledge of chemical structure and composition. It can be explained by the estimation of surface tension to another oil as well.Table 5
The comparison of model deviation in the proposed model and previous study models (Eliezer model at 293.15 K/Parachor model at 313.15 K and Esteban model at 293.15 K).
Vegetable oilsz(ave)nd(ave)Temperature (K)293.15313.15293.15Proposed modelEliezer model [13]Proposed modelParachor’s model [9]Proposed modelEsteban’s model [16]Rapeseed17.91.333.732.5232.1729.2533.730.38(−0.30)(−3.79)(−0.22)(−8.88)(−0.30)(−1.95)Sunflower17.861.4233.7533.2432.2429.3133.7524.31(−0.74)(−2.25)(−0.19)(−9.26)(−0.74)(−1.99)Soybean17.771.5133.7633.2932.2829.3433.7630.54(−0.41)(−1.81)(−0.37)(−9.44)(−0.41)(−1.80)Palm17.060.6——31.428.72—32.29——(−0.32)(−8.83)—(−10.58)Corn17.771.4133.732.7132.2129.2833.730.36(−0.30)(−3.21)(−0.03)(−9.07)(−0.30)(−1.88)Grapeseed17.851.5833.8533.7832.3529.3933.8530.56(−0.15)(−0.35)(−0.15)(−9.01)(−0.15)(−1.86)Overall %AAD0.38%2.28%0.21%9.08%0.38%2.74%


### 3.2. Parachor Model

According to the Sugden expression of pure component surface tension, this model used the Parachor, density, and molecular weight of each vegetable oil component in Table1 in order to predict the surface tension following the theory of mixture in Dalton-typemass-average Equation. The surface tension results from these equations are compared with the experimental data in Table 5 (in the case of six vegetable oils at 313.15 K). The %AAD of each vegetable oil is almost the same at 9% while the proposed model %AAD is around 0.21% at this temperature. This model is based on the linear relationship between the surface tension of the mixture and the pure component. Thus, the nonlinear rate of change in surface tension of unsaturated compositions is not accurate in this model.


### 3.3. Esteban’s Model

The linear model from the relationship between surface tension and kinematic viscosity is compared with the experimental data in Table5 (in the case of six vegetable oils at 293.15 K). The % AAD of vegetable oils is indicated in the range between 1.80 and 10.58 while the proposed model % AAD is around 0.38% at this temperature. The highest %AAD as shown in Table 5 may cause by the kinematic viscosity human error test. According to different factors in the Esteban model from the kinematic viscosity factor invert variation (1/η) of temperature (T) effect to % AAD is high. Thus, the proposed model is the best performance when compared with this model.


### 3.4. Other Predictive Model for Surface Tension of Vegetable Oils

Although (21) is a precise and simple model for the prediction of vegetable oils at different temperatures, the determination of the z(ave) and average nd(ave) requires the knowledge of fatty acid composition data. The analysis must be done with a GC or an HPLC. Saponification number (SN) and iodine value (IV) have long been used in the characterization of oil and do not require special instrumentation. However, both SN and IV can also be calculated from the fatty acid composition. Thus, it will be convenient if the surface tension of vegetable oils can be predicted from these two values. The nd and z can be converted to IV and SN according to the following equations:(27)nd=IV×M25400,(28)z=M+2nd−4614,(29)M=56000SN.Equation (30) is obtained by combining equations (21), (27) to (29) and it can be used for the prediction of surface tension of vegetable oils at different temperatures from the SN and IV values.(30)γ=80.111−0.1125T−0.4327M+9.98IVSN.The surface tension of vegetable oils predicted by (30) (at different temperatures) are very close to those calculated using the zave and nd(ave) in equations (22) and (23)


## 3.1. The Melo-Espinosa et al. MLR Model [13]

Linear regression of the Melo-Espinosa et al. MLR model is used to make an estimated surface tension data from the quantity of fatty acid compositions in Table2 and the MLR model in Melo-Espinosa et al. The estimated surface tension is compared with the proposed model in Table 5 (in the case of six vegetable oils at 293.15 K). The % AAD results of each vegetable oil compared with the proposed model in Table 5 show that it is not a significantly different error but the proposed model is a lower %AAD than the Melo-Espinosa et al. MLR model. Moreover, the proposed model can be predicted in various temperatures while the Melo-Espinosa et al. MLR model cannot predict them. The advantage of the proposed model is the simplified linear model structure form which can be easily calculated without using the knowledge of chemical structure and composition. It can be explained by the estimation of surface tension to another oil as well.Table 5
The comparison of model deviation in the proposed model and previous study models (Eliezer model at 293.15 K/Parachor model at 313.15 K and Esteban model at 293.15 K).
Vegetable oilsz(ave)nd(ave)Temperature (K)293.15313.15293.15Proposed modelEliezer model [13]Proposed modelParachor’s model [9]Proposed modelEsteban’s model [16]Rapeseed17.91.333.732.5232.1729.2533.730.38(−0.30)(−3.79)(−0.22)(−8.88)(−0.30)(−1.95)Sunflower17.861.4233.7533.2432.2429.3133.7524.31(−0.74)(−2.25)(−0.19)(−9.26)(−0.74)(−1.99)Soybean17.771.5133.7633.2932.2829.3433.7630.54(−0.41)(−1.81)(−0.37)(−9.44)(−0.41)(−1.80)Palm17.060.6——31.428.72—32.29——(−0.32)(−8.83)—(−10.58)Corn17.771.4133.732.7132.2129.2833.730.36(−0.30)(−3.21)(−0.03)(−9.07)(−0.30)(−1.88)Grapeseed17.851.5833.8533.7832.3529.3933.8530.56(−0.15)(−0.35)(−0.15)(−9.01)(−0.15)(−1.86)Overall %AAD0.38%2.28%0.21%9.08%0.38%2.74%


## 3.2. Parachor Model

According to the Sugden expression of pure component surface tension, this model used the Parachor, density, and molecular weight of each vegetable oil component in Table1 in order to predict the surface tension following the theory of mixture in Dalton-typemass-average Equation. The surface tension results from these equations are compared with the experimental data in Table 5 (in the case of six vegetable oils at 313.15 K). The %AAD of each vegetable oil is almost the same at 9% while the proposed model %AAD is around 0.21% at this temperature. This model is based on the linear relationship between the surface tension of the mixture and the pure component. Thus, the nonlinear rate of change in surface tension of unsaturated compositions is not accurate in this model.


## 3.3. Esteban’s Model

The linear model from the relationship between surface tension and kinematic viscosity is compared with the experimental data in Table5 (in the case of six vegetable oils at 293.15 K). The % AAD of vegetable oils is indicated in the range between 1.80 and 10.58 while the proposed model % AAD is around 0.38% at this temperature. The highest %AAD as shown in Table 5 may cause by the kinematic viscosity human error test. According to different factors in the Esteban model from the kinematic viscosity factor invert variation (1/η) of temperature (T) effect to % AAD is high. Thus, the proposed model is the best performance when compared with this model.


## 3.4. Other Predictive Model for Surface Tension of Vegetable Oils

Although (21) is a precise and simple model for the prediction of vegetable oils at different temperatures, the determination of the z(ave) and average nd(ave) requires the knowledge of fatty acid composition data. The analysis must be done with a GC or an HPLC. Saponification number (SN) and iodine value (IV) have long been used in the characterization of oil and do not require special instrumentation. However, both SN and IV can also be calculated from the fatty acid composition. Thus, it will be convenient if the surface tension of vegetable oils can be predicted from these two values. The nd and z can be converted to IV and SN according to the following equations:(27)nd=IV×M25400,(28)z=M+2nd−4614,(29)M=56000SN.Equation (30) is obtained by combining equations (21), (27) to (29) and it can be used for the prediction of surface tension of vegetable oils at different temperatures from the SN and IV values.(30)γ=80.111−0.1125T−0.4327M+9.98IVSN.The surface tension of vegetable oils predicted by (30) (at different temperatures) are very close to those calculated using the zave and nd(ave) in equations (22) and (23)


## 4. Conclusions

In this paper, the three commonly predictive models of vegetable oil surface tension are reviewed and compared in model accuracy with the developed model based on the total Gibbs energy (ΔG) theorem of this research through the six economic vegetable oil samples. Moreover, the proposed model is used as the aspect model in order to compare the agreement trend with the two literature predictive models such as the Flingoh model and the Esteban model. The model accuracy is dependent on the value of %AAD which is calculated and analyzed by the prediction value and experimental surface tension value of six vegetable oil samples. With the comparison results given by the proposed model, it can be concluded that the estimated surface tension by the average number of the double bond and atom carbon of the proposed model provides the suitable accuracy of %AAD under the surface tension of six vegetable oils. In the case of 293.15 K, the % AAD of the proposed model is the lowest % AAD at 0.38% while the % AAD of other models is also 2.28% for the Eliezer model and 2.74% for the Esteban model. In another case at 313.15 K, the %AAD of the proposed model is also the lowest at 0.21% while the %AAD of the Parachor model is 9.08%. However, the proposed model still shows a less accurate prediction model in the group of dominant C18:2 compositions such as sunflower oil, soybean oil, corn oil, and grapeseed oil. This problem was caused by the highly unsaturated content of dominant C18:2 which affected the nonlinear mismatch for the linearized-based model such as the proposed model. In the literature agreement trend, the comparison results show very good agreement between the proposed model and the Flingoh model with linear relation constant terms (a and b) and the coefficient of determination (R2). And then, the graphical technique of the relative deviation also shows the acceptable percent difference at approximately±1.0% in a proposed model. From the overall results, it can be inferred that the proposed model is suitable for predicting vegetable oil surface tension compared with other previous comparison models, and the model performance was statistically confirmed to explain the development predicted in other vegetable oil surface tension.



---

*Source: 4477071-2023-01-18.xml*