
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)12484-X
10.1016/j.heliyon.2024.e36453
e36453
Research Article
Using the group contribution method to predict the flash temperature of biodiesel and ethanol mixtures
Bahrani Esmaeil a
Shafeeyan Mohammad Saleh ms.shafeeyan@gmail.com
a⁎
Banihashemi Morteza b
a Department of Chemical Engineering, Faculty of Engineering, Golestan University, Aliabad Katoul, Iran
b Department of Chemical Engineering, Babol University of Technology, Babol, Iran
⁎ Corresponding author. ms.shafeeyan@gmail.com
16 8 2024
15 9 2024
16 8 2024
10 17 e3645325 1 2024
21 7 2024
15 8 2024
© 2024 The Author(s)
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
This article presents a novel approach to predict the flash temperature of biodiesel and ethanol mixtures using the Group Contribution Method (GCM). Expanding on the pioneering work by Liaw et al. (2003), our method employs GCM to calculate the activity coefficients of biodiesel and ethanol components in the mixture. Estimating these coefficients, crucial for accurate flash temperature prediction, involves a comprehensive analysis of composition, functional groups, and vapor-liquid equilibrium (VLE) data. For this purpose, the composition of the mixture components in biodiesel, the functional groups within each biodiesel component, the composition ratios of biodiesel and ethanol in the mixture, and the functional groups present in ethanol are considered. Given that the use of UNIQUAC and NRTL models requires estimating adjustable parameters, VLE data for ethanol and biodiesel mixtures are employed to calculate the activity coefficients. This approach not only aids in estimating these coefficients but also facilitates determining the values associated with each functional group. Flash temperature predictions for biodiesel and ethanol mixtures obtained through various models, including the ideal solution, UNIQUAC, NRTL, and our proposed GCM, are rigorously assessed. The results indicate that the GCM method outperforms the alternatives, exhibiting the lowest error with a deviation of just 1.72 K compared to deviations of 1.77 K, 1.75 K, and 1.73 K for the ideal solution, UNIQUAC, and NRTL models, respectively. This research offers a promising approach for flash point estimation in complex systems, such as biodiesel-ethanol blends, contributing to the ongoing exploration in this field.

Keywords

Flash point estimation
Biodiesel
Group contribution method
Vapor liquid equilibrium
==== Body
pmc1 Introduction

Nowadays, the widespread utilization of biodiesel as a fuel source is a notable trend across numerous nations. Biodiesel, a compelling alternative to conventional petroleum-based fuels, stands out for its eco-friendly attributes and renewable nature. It boasts characteristics such as energy renewability, biodegradability, non-toxicity, and environmental friendliness [[1], [2], [3], [4]]. Additionally, biodiesel offers advantages like enhanced lubricity, reduced carbon dioxide emissions, and environmental compatibility, rendering it an efficacious combustion fuel. Its elevated oxygen content and high flash point further contribute to its appeal [[5], [6], [7], [8], [9]]. The synthesis of biodiesel conventionally revolves around the transesterification of triglycerides obtained from renewable sources with short-chain alcohols. This process yields fatty acid monoalkyl esters with lengthy hydrocarbon chains [5,10,11]. Four primary methods are employed for biodiesel production: 1) catalytic cracking, 2) esterification, 3) microemulsion, and 4) direct utilization of compounds [12]. In the case of using plant oil-based biodiesels like soybean oil and canola oil, challenges can emerge in the combustion process within automobile engines due to their high flash temperatures. To ameliorate this issue, ethanol is often introduced as a flash temperature-reducing agent in biodiesel. Vapor pressure serves as a critical thermal property, offering insights into evaporation capability, stability, and fuel safety. Flashpoint and vapor pressure bear relevance to ignition, combustion, and fuel volatility. Flashpoint data, especially for flammable liquids, holds pivotal importance for ensuring operational safety and enhancing our understanding of volatility [13,14]. Even a minor ethanol admixture can significantly impact the flash temperature, underscoring the importance of accurate flash temperature prediction.

The flash temperature, denoting the lowest temperature at which a fuel emits adequate vapor to ignite when exposed to an ignition source [15], plays a pivotal role in assessing the potential for combustion and explosion, especially in contexts involving fuel production, storage, transportation, and in identifying highly hazardous fuels. Presently, ASTM standards encompass two test methodologies for flash point measurement: the Close Cup and Open Cup methods. Typically, the flashpoint recorded via the Close Cup method is a few degrees lower than that determined using the Open Cup method [16]. Various methods exist for predicting flash temperature, particularly for pure substances and binary mixtures. When it comes to predicting the flash temperature of binary and multicomponent mixtures, Le Chatelier's rule often serves as a foundational principle, despite the complexities associated with predicting flash points for multi-component mixtures. This approach presents both advantages and limitations. On the positive side, it accommodates mixtures containing non-flammable compounds [17]. On the flip side, it necessitates knowledge of the flash point for each component in isolation. In cases where the flash point of a pure substance within the mixture is unknown, estimation can be achieved experimentally using methods like the group contribution method [18,19], the QSPR approach [20], or other calculation-based techniques [21]. For mixtures consisting of a maximum of six flammable components, Catoire and colleagues introduced an equation known as the Catoire-Naudet (CN) model for flash point prediction [22]. Predicting flash point temperatures for multi-component mixtures through non-linear methodologies [[23], [24], [25], [26], [27], [28]] involves leveraging the Liaw reference model [29,30], along with a fusion of group contribution methods and intelligent algorithms [19]. Additionally, Guo and associates [31] proposed a dependable model for forecasting the flash point temperature of biodiesel-ethanol mixtures using the Liaw method [16] and standard mixture activity coefficient models.

In this article, we harness the combined prowess of the GCM and Liaw et al. [16] model to predict the activity coefficients of biodiesel-ethanol mixtures and calculate their flash temperatures. Our approach incorporates factors such as the type and number of functional groups within each biodiesel component and ethanol, as well as the mole fraction of each constituent, to estimate the activity coefficients of both biodiesel and ethanol. This comprehensive methodology promises insights into flash temperature behavior and the critical role of ethanol in altering the flash characteristics of biodiesel, offering valuable information for fuel engineering and safety considerations.

2 Materials and methods

2.1 Biodiesel synthesis

Biodiesel was synthesized in the laboratory using soybean oil, methanol, and potassium hydroxide (KOH) as the catalyst. The synthesis process involved mixing 100 g of soybean oil, 100 g of methanol, and 100 g of KOH in an isolated 500 mL Erlenmeyer flask. This mixture was stirred on a heater under full reflux conditions. The reaction proceeded as the fatty acids in soybean oil reacted with methanol at a specific temperature of 60 °C, with a reaction time of 2 h. The presence of KOH as the catalyst facilitated the conversion of these fatty acids into methyl esters, constituting biodiesel. After allowing the mixture to stand for 24 h in a decanter and separating the two formed phases, the synthesized biodiesel was obtained. High-purity chemicals were employed in this study, including methanol (99.9 % purity), potassium hydroxide (99.98 % purity), and ethanol (99.9 % purity), all sourced from Merck. The properties of the synthesized biodiesel are summarized in Table 1. The density of biodiesel was measured using a precision balance with an accuracy of 0.01 g. The volume was determined using a volumetric flask with a precision of 0.1 mL in a 100 mL volume, and the calculations were performed with an accuracy of ±0.1 %.Table 1 Properties of the synthesized biodiesel.

Table 1Properties	Value	
Density (at 20 °C) (kg/m3)	890 ± 1	
Flashpoint (°C)	179 ±0.5	
Molar weight (pure biodiesel)	310.83	
Purity	>99 %	

To determine the constituents of the synthesized biodiesel, gas chromatography (GC) analysis was carried out using Clarus 580 Gas Chromatogram (PerkinElmer, USA) equipped with a flame ionization detector (FID) and a BP20 wax capillary column (30 m length × 0.25 mm i.d., 0.25 μm thickness) with helium as a carrier gas. 1 μL of sample was injected by an autosampler. The oven temperature was equilibrated at 60 °C for 5 min, heated at 10 °C/min to 180 °C with 5 min hold, and finally raised at 5 °C/min to 240 °C with 7 min hold. The temperature of the detector and injector was set at 260 °C. The results of this analysis are presented in Table 2, providing information on the biodiesel components and their respective fractions as determined by GC analysis.Table 2 Biodiesel components and their fraction by GC analysis.

Table 2Component Name	Weight percent	
C16:0	10.59	
C18:0	4.57	
C18:1	24.22	
C18:2	51.44	
C19:0	3.22	
C18:3	4.67	
C18:3 cis6	0.58	
Unknown components	0.72	

In addition to the composition analysis, Table 3 presents the names, molecular weights, and structural information of the biodiesel compounds.Table 3 Names, molecular weights and structures of biodiesel compounds.

Table 3Formula	Component name	Molecular weight	Structure	
C16:0	Methyl hexadecanoic methyl ester	270	Image 1	
C18:0	Methyl octadecanoic methyl ester	299	Image 2	
C18:1	Elaidic Acid methyl ester	282.5	Image 3	
C18:2	Linolenic acid methyl ester	280.4	Image 4	
C19:0	Nonadecanoic acid methyl ester	313	Image 5	
C18:3	linolenic acid methyl ester	292.5	Image 6	
C18:3 cis6	Linolenic acid methyl ester	292.5	Image 6	

2.2 Flash point measurement and vapor-liquid equilibrium data for biodiesel-ethanol mixtures

Flash point measurements were conducted using an Open Cup configuration apparatus, model B130 manufactured by Controls Italy, complying with the ASTM D1310 standard. The apparatus allowed precise control of the heating rate, with options ranging from 1 °C/min to 10 °C/min. For our experiments, a heating rate of 1 °C/min was selected for flashpoint determinations.

To record VLE data for biodiesel, ethanol, and their mixtures, a specialized apparatus detailed in Fig. 1 was employed. This equipment featured temperature and pressure sensors, a 500 mL stainless steel cylindrical chamber, and a data logger for capturing temperature and pressure data, stored in Excel files. The temperature sensor used was an RTD Pt1000 type, operating based on the Wheatstone bridge principle, a configuration with four resistors commonly used for measuring resistance values and calibrating measurement devices. For pressure measurements, the Hogller 10 bar pressure sensor was employed. It accurately measures pressure up to 10 bar for various liquids and gases compatible with stainless steel 304. The chamber's design incorporated a groove with an inner diameter of 110 mm and a thickness of 5 mm on top to accommodate an O-ring, preventing vapor leakage. Calibration was performed using ethanol, with our results aligning with graphs presented in previous articles [[32], [33], [34], [35], [36], [37]] for these three substances. Before each test, the chamber underwent vapor evacuation using a vacuum pump. The volume of liquid ethanol in the chamber was precisely calibrated, with the same procedure applied to biodiesel and biodiesel-ethanol mixtures. In the measurement of equilibrium pressure and temperature, a metallic chamber filled with oil experienced gradual temperature increase at a rate of 0.1 °C/min, allowing sufficient time for equilibrium to be reached. Equilibrium data were recorded every 10 s.Fig. 1 Schematic of the experimental system used for the VLE measurements.

Fig. 1

In each test, 200 mL of the mixture was introduced into the chamber, and it was securely sealed to prevent vapor or air exchange. Subsequently, the vapor within the chamber was evacuated. The chamber was then immersed in a larger oil bath, and the heating process was initiated with a controlled, slow rate to ensure gradual heating and accurate pressure and temperature measurements at equilibrium. The duration of each test varied depending on the specific mixture composition and the percentage of biodiesel-ethanol present. Higher biodiesel content required extended heating times to reach 1 bar gauge pressure.

In the initial phase, biodiesel-ethanol mixtures with varying mass fractions of ethanol were prepared. For each mixture, 70 mL was dispensed into a cup, and the flash temperature was measured using a flash point tester. When dealing with mixtures containing a high mass fraction of ethanol, the mixture and the cup were placed in an ice bath due to the lower temperature of these mixtures. The heater was consistently set at 1 °C/min for all mixtures during flash point determination. As the percentage of biodiesel increased, the heater temperature was adjusted accordingly for flash point measurements. The mass fractions of ethanol ranged from 0 to 100, with each test repeated three times. Estimating the flash point using the Liaw et al. [16] method necessitates equilibrium temperature and pressure data. VLE measurements for ethanol, biodiesel, and biodiesel-ethanol mixtures were conducted in the subsequent step. The equilibrium data for ethanol and biodiesel were plotted separately in Fig. 2, Fig. 3. The choice to use open-cup flash point measurements was influenced by the availability of equipment in our laboratory. Although open-cup and closed-cup flash point measurements differ in their operational conditions, both methods rely on vapor-liquid equilibrium (VLE) data for accurate flash point determination. Previous studies have demonstrated the modeling of open-cup flash point data based on equilibrium data, providing a basis for our approach [18,38,39]. While equilibrium data for biodiesel were not readily available, we provided experimental data for biodiesel and compared ethanol VLE data with overall data from existing literature for validation purposes. Furthermore, previous works have successfully utilized both closed-cup and open-cup flash point data in conjunction with Group Contribution Method (GCM) calculations for flash temperature prediction, demonstrating high accuracy in both cases [18]. Therefore, the utilization of open-cup flash point measurements, combined with equilibrium data-based modeling, aligns with established practices in the field and ensures the robustness of our approach.Fig. 2 VLE data of pure ethanol.

Fig. 2

Fig. 3 VLE data of biodiesel.

Fig. 3

3 Results and discussion

3.1 Flash temperature modeling for biodiesel-ethanol mixtures

A model was constructed using the least squares method, specifically tailored to fit the equilibrium data for pure ethanol and biodiesel. This model was based on the well-established Antoine equation, a widely recognized formula for predicting vapor pressures as a function of temperature. The Antoine equation coefficients, along with their respective error values, are meticulously documented in Table 4. Moreover, equations (1), (2) were applied to calculate the molecular weight of biodiesel precisely. Each component's molecular weight was determined meticulously through rigorous GC analysis, resulting in an average molecular weight of 310.83.(1) xi=∑i=1nwiMwi

(2) Mwav=∑i=1nxiMwi

Table 4 Antoine equations, coefficients and error values for biodiesel and ethanol.

Table 4Component	Equation	Antoine coefficient	R2	
A	B	C	
Biodiesel	Ln(P)=A−BT−C	36.2	29810	−1043	0.9914	
Ethanol	Ln(P)=A−BT−C	18.17	1284	−116.7	0.9996	

In the above table, T is in degrees Celsius and P is in Pa.

3.2 Flash point estimation using the Liaw equation

In the case of binary mixtures, the determination of the flash point, as proposed by Liaw et al. [16], is described by the following equation:(3) x1γ1p1satp1,fpsat+x2γ2p2satp2,fpsat=1

in ideal mixtures, where the activity coefficients (γ1 and γ2) are assumed to be equal to 1, the equation simplifies to:(4) x1p1satp1,fpsat+x2p2satp2,fpsat=1

Here, in this binary equation, component 1 represents biodiesel, and component 2 represents ethanol. The vapor pressure values are derived from the Antoine equation. In these equations, pisat is the vapor pressure of component i at the flash temperature of the mixture, which needs to be predicted. On the other hand, pi,fpsat is the vapor pressure of component i at its pure flash temperature, which is obtained experimentally and is a known value. xi represents the mole fraction of component i in the mixture, and γi is the activity coefficient of component i. It is important to note that, unlike simple binary mixtures, biodiesel is considered as component 1, which is itself a combination of several methyl esters. Therefore, in this study biodiesel can be regarded as a pseudo-component, and its properties need to be obtained either through experiments (such as biodiesel flash point or VLE data) or by interpolating the properties of mixtures to this pseudo-component. Certain parameters, like activity coefficients must be calculated using methods such as UNIQUAC [40], NRTL [41] and GCM. In UNIQUAC, the values of R and Q are calculated by UNIFAC. For biodiesel, R and Q are calculated using equations (5), (6)). Biodiesel is a mixture of methylesters, so R and Q for biodiesel calculated by mole average of methylester's r and q calculated by UNIFAC.(5) R=∑i=1nxiri

(6) Q=∑i=1nxiqi

In the above equations, xi represents the fractional composition of each methyl ester in biodiesel, and ri and qi are obtained via UNIFAC for each methylester. By using the UNIFAC equation and obtaining the UNIFAC coefficients for each methyl ester, the values of R and Q for biodiesel can be calculated. This calculation process is also applied to pure ethanol using the UNIFAC method. By calculating the UNIFAC coefficients for the CH3, CH2, and OH groups, the values of R and Q for ethanol can be determined. Therefore, by obtaining the adjustable parameters in the UNIQUAC and NRTL, the flash temperature is derived by solving equation (3). Additionally, in the case of GCM, the values of each functional group must be determined to calculate the activity coefficient of each component. Then, by using equation (3), the flash temperatures of biodiesel-ethanol mixtures are computed. After obtaining all the flash temperatures using NRTL, UNIQUAC or GCM, the flash temperature error in each method is calculated using equation (7):(7) err=∑i=1n|TflashReal−TflashPredicted|n

3.3 Finding adjustable parameters and functional group values using VLE data/parameter and functional group optimization using VLE data

In the calculation of activity coefficients, whether through the group contribution method or activity equations, there exist adjustable parameters that can be optimized to minimize the error in equation (8). To achieve this optimization, a genetic algorithm is employed to fine-tune the adjustable parameters in the NRTL and UNIQUAC models or functional group values in GCM. It's worth noting that, to enhance efficiency, rather than optimizing the general flash equation for all components (Eq. (7)), an alternative equation (Eq. (8)) can be optimized, and its optimal coefficients can be calculated. These coefficients are then incorporated into the Liaw equation. For this purpose, VLE data in terms of temperature and pressure are obtained for the ethanol-biodiesel mixture (with ethanol mass fraction = 0.1, biodiesel mass faction = 0.9). Subsequently, using equation (9), the equilibrium pressure of the mixture is computed. Through genetic algorithm optimization, by minimizing the error in equation (8), the adjustable parameters and functional group values were obtained. The optimization process involves 100 generations, with a chromosome creation probability of 0.8 for each parent. In shaping the next generation, the uniform mutation method with a rate of 0.1 is applied to produce mutant chromosomes. The optimization process halts when the number of generations reaches 100. Here are the equations involved in this process:(8) err=∑i=1n|PiReal−PiCalculated|n

(9) PCalculated=x1γ1P1sat+x2γ2P2sat

Subsequently, the optimal coefficients obtained through this equation are integrated into the Liaw equation. This allows for solving the nonlinear Liaw equation for each mixture. Consequently, by substituting the obtained coefficients into the Liaw equation for each mixture and solving the nonlinear Liaw equation, the flash temperature of the mixture is determined. For reference, experimental VLE data (equilibrium temperature and pressure) for ethanol-biodiesel mixture (%wtethanol=10), biodiesel and ethanol are presented in Table 5.Table 5 Experimental VLE data (equilibrium temperature and pressure) for ethanol-biodiesel mixture (%wtethanol=10).

Table 5Pressure (Kpa)	Ethanol-biodiesel mixture (%wethanol = 10)
Temperature (°C)	Biodiesel Temperature (°C)	Ethanol Temperature (°C)	Pressure (Kpa)	Ethanol-biodiesel mixture (%wethanol = 10)
Temperature (°C)	Biodiesel Temperature (°C)	Ethanol Temperature (°C)	
4	22.91	30.52	12.77	53	68.05	132.58	59.38	
5	23.82	32.25	18.41	54	68.73	133.08	59.831	
6	25.53	35.78	19.95	55	69.38	133.58	60.28	
7	27.10	39.40	21.40	56	70.05	134.05	60.73	
8	28.59	43.18	22.90	57	70.71	134.51	61.18	
9	30.08	47.14	24.45	58	71.33	135.01	61.61	
10	31.53	50.82	25.96	59	71.87	135.46	62.00	
11	32.86	54.08	27.5	60	72.46	135.94	62.40	
12	34.07	57.34	28.92	61	73.04	136.57	62.80	
13	35.30	61.01	30.27	62	73.64	137.21	63.23	
14	36.36	64.63	31.63	63	74.24	137.80	63.67	
15	37.36	68.30	32.81	64	74.83	138.45	64.08	
16	38.46	71.69	33.99	65	75.40	139.07	64.4	
17	39.46	75.15	35.18	66	75.95	139.81	64.81	
18	40.39	78.72	36.37	67	76.48	140.55	65.24	
19	41.37	82.24	37.43	68	77.02	141.36	65.65	
20	42.31	85.88	38.46	69	77.55	142.1	66.02	
21	43.27	89.27	39.47	70	78.04	142.86	66.40	
22	44.19	92.90	40.45	71	78.55	143.74	66.77	
23	45.09	96.75	41.37	72	79.06	144.60	67.13	
24	46	100.11	42.24	73	79.56	145.45	67.47	
25	46.95	103.15	43.11	74	80.01	146.30	67.8	
26	47.92	105.67	44.01	75	80.52	147.03	68.18	
27	48.77	107.59	44.88	76	81.03	147.90	68.55	
28	49.69	109.33	45.67	77	81.45	148.6	68.89	
29	50.52	111.08	46.32	78	81.84	149.39	69.18	
30	51.21	112.72	46.94	79	82.31	150.14	69.54	
31	51.93	114.39	47.61	80	82.78	150.82	69.94	
32	52.66	115.92	48.18	81	83.25	151.44	70.30	
33	53.40	117.38	48.78	82	83.74	152.18	70.62	
34	54.07	118.73	49.44	83	84.22	152.93	70.97	
35	54.70	120.12	50.03	84	84.73	153.66	71.33	
36	55.39	121.42	50.58	85	85.2	154.27	71.65	
37	56.13	122.58	51.12	86	85.62	155.03	72.03	
38	56.86	123.49	51.63	87	86.07	155.68	72.41	
39	57.56	124.37	52.21	88	86.50	156.40	72.72	
40	58.29	125.19	52.81	89	86.90	156.97	73.01	
41	58.98	126.13	53.34	90	87.32	157.54	73.32	
42	59.75	126.98	53.88	91	87.72	158.18	73.64	
43	60.52	127.61	54.44	92	88.11	158.85	73.92	
44	61.32	128.22	55.02	93	88.43	159.52	74.15	
45	62.07	128.80	55.56	94	88.80	160.25	74.41	
46	62.79	129.33	56.02	95	89.20	160.95	74.74	
47	63.56	129.89	56.50	96	89.56	161.75	75.1	
48	64.37	130.40	57.06	97	89.93	162.46	75.37	
49	65.17	130.91	57.61	98	90.34	163.19	75.62	
50	65.94	131.36	58.11	99	90.71	163.82	75.85	
51	66.64	131.74	58.54	100	91.03	164.42	76.12	
52	67.34	132.13	58.95	101	91.21	165.02	76.39	

3.4 Activity coefficient estimation using UNIQUAC and NRTL models

The UNIQUAC equation, a widely accepted method for estimating activity coefficients in non-ideal solutions, plays a pivotal role in our flash temperature modeling. The UNIQUAC equation is structured as a series of interrelated mathematical expressions (Eqs. (10), (11), (12), (13), (14), (15), (16), (17), (18))), each contributing to the accurate estimation of activity coefficients. The key parameters necessary for this calculation include binary interaction coefficients and the computation of R and Q.(10) lnγi=lnγiR+lnγiC

(11) li=Z2(ri−qi)−(ri−1)

(12) ∅i=xiri∑jxjrj

(13) θi=xiqi∑jxjqj

The remaining activity coefficient for each pure substance is calculated using the following equations:(14) lnγiR=∑kvk(i)(lnΓk−lnΓk(i))

(15) lnΓk=Qk[1−ln(∑mΘmΨmk)−∑mΘmΨkm∑nΘnm]

(16) Θm=QmXm∑nQnXn

(17) Xm=∑ivm(i)xi∑i∑nvn(i)xi

(18) Ψmn=exp(−AmnT)

To calculate the values of R and Q for biodiesel components, equations (5), (6)) were utilized, which consider the number of functional groups within each component. For instance, for C16:0, which comprises one methyl group, 14 CH2 groups, and one COOCH3 group, the value was calculated as follows: (rc16:0 = 0.8480 + 14*0.54 + 1.7380). The interaction coefficient (Amn) represents the net energy of interaction between different components within a binary mixture. Specifically, A11 and A22 denote the interactions between like molecules (biodiesel-biodiesel and ethanol-ethanol), while A12 and A21 represent the interactions between unlike molecules (biodiesel-ethanol and ethanol-biodiesel). When molecules m and n are identical (i.e., biodiesel-biodiesel or ethanol-ethanol interactions), their interaction is considered negligible, and thus, the corresponding Amn values are set to zero. In the determination of interaction coefficients within the UNIQUAC equation, the optimal values of A12 and A21 were derived through an optimization process using a genetic algorithm. The resulting values for A12 and A21 were found to be −36.83 K and 100.53 K, respectively.

The following correlations were used to estimate the activity coefficients from NRTL equation:(19) Lnγ1=x22[τ21(G21x1+x2G21)2+τ12G12(x2+x1G12)2]

(20) Lnγ2=x12[τ12(G12x2+x1G12)2+τ21G21(x1+x2G21)2]

(21) G12=exp(−α12τ12)

(22) G21=exp(−α12τ21)

(23) τ12=U12−U22RT

(24) τ21=U21−U11RT

For the NRTL equation, another essential component of our flash temperature modeling, we derived optimal values for parameters such as α, U12−U22 and U21−U11. These values, critical to the NRTL equation's accuracy, were determined as follows: α = 0.4, U12−U22 = 5.439 cal/mol, and U21−U11 = 8.685 cal/mol.

3.5 Flash point estimation using GCM

The group contribution method has proven to be a highly effective approach for predicting thermodynamic properties of pure substances. While its primary application lies in the prediction of pure substance properties, it has found occasional use in estimating the properties of mixtures. GCM operates by assigning specific properties to individual structural groups within a substance. These groups, such as hydroxyl, methyl, ethyl, and others, are characterized by values that contribute to a general equation for each substance, enabling the determination of its properties. In this study, GCM was harnessed to predict the activity coefficients of individual components. Equation (8) was employed to acquire the optimal values for coefficients associated with each structural group. Subsequently, these optimized coefficients were leveraged to predict the flash point of each mixture, employing the Liaw equation. This involved obtaining the equilibrium pressure for a mixture containing 10 % ethanol in biodiesel at various temperatures. As a result, there are numerous equations in which the values of the group contributions are unknown, but the objective function, which is the computational error, should be minimized by optimizing these coefficients. Consequently, with the use of a genetic algorithm, these values were optimized. In this work, the activity coefficient function is expressed as follows, based on the group contributions:(25) γ1=x1+x2(∑(niGi)2∑(niGi)1+α)

(26) γ2=x1(∑(niGi)1∑(niGi)2+β)+x2

in these equations, n represents the number of each functional groups xi denotes mole fraction of each component (biodiesel and ethanol), Gi is the value associated with each functional group. α and β are constants. To determine ∑(niGi)1 for biodiesel, we employed the following expression:(27) ∑(niGi)1=∑xi′(∑ni′Gi′)

in this context, xi′ represents mole fraction of each methyl ester within biodiesel, ni′ is the number of functional groups within each methyl ester and Gi′ denotes the functional groups of each methyl ester.

3.6 Integrating functional group analysis and GCM for flash point estimation of ethanol-biodiesel mixtures

In this section, we present the results of our flash point estimation model for ethanol-biodiesel mixtures, emphasizing the functional groups of methyl esters and ethanol, along with the optimum values associated with these groups (Table 6).Table 6 Functional groups of methyl esters and ethanol optimum values.

Table 6Functional group	Value	
CH3	−36.684	
CH2	−3.402	
CH2=CH2	−10.812	
COOCH3	21.613	
OH	−10.328	
α	61.368	
β	−19.241	

These values are critical in our flash point estimation model. Banihashemi and Movagharnejad [18] employed equation (3) to determine the optimal values of activity coefficients based on these functional group values, rather than relying on conventional methods like UNIQUAC. Table 7 compares the real versus predicted flash points of ethanol-biodiesel mixtures using four models: GCM, Ideal Solution, UNIQUAC, and NRTL.Table 7 Real vs. predicted flash point of ethanol-biodiesel mixture.

Table 7Weight percent of ethanol	Real FPa (°C)	FP by GCM (°C)	FP by Ideal Solution (°C)	FP by UNIQUAC (°C)	FP by NRTL (°C)	
0	179	179	179	179	179	
2	59.9	52.3	54	53.8	51.5	
4	39.8	40.5	42	41.1	39.9	
6	36.5	34.6	36	35.2	34.2	
8	27.5	31.5	32.5	31.8	30.9	
10	25.5	29	30	29.5	28.8	
20	25	23.3	24	23.6	23.4	
30	20.9	21	21.5	21	21	
40	19	19.7	20	19.3	20	
60	18.4	18.4	18.5	17.2	18.5	
80	17.4	17.8	17.9	16.4	17.8	
100	17.3	17.3	17.3	17.3	17.3	
Absolute error values	1.72	1.77	1.75	1.73	
a Experimental flash point data.

The results, as depicted in Table 7 and Fig. 4, highlight the superior accuracy of the GCM in predicting flash points across various ethanol percentages compared to the Ideal Solution, UNIQUAC, and NRTL models. As the ethanol concentration increases, the flash point temperature decreases due to ethanol's increased volatility. For mixtures with higher ethanol percentages, all four methods provide reasonably accurate predictions. However, across the full range of ethanol content (0 %–100 %), the GCM method consistently emerges as the most reliable, showing the lowest absolute error.Fig. 4 Flash estimation by UNIQUAC and group contribution method (GCM).

Fig. 4

Notably, the prediction errors for the equilibrium data of a mixture with 10 wt% ethanol in biodiesel (presented in Table 5) using the GCM, Ideal Solution, UNIQUAC, and NRTL models were calculated to be 1.17, 6.85, 1.77, and 3.1 kPa, respectively. This further underscores the superior performance of the GCM and UNIQUAC models, as only these two models exhibit acceptable accuracy in predicting equilibrium data. Given the low prediction errors for both flash point and equilibrium data, the GCM and UNIQUAC models are validated as the most accurate and reliable methods. This dual reliability is critical for practical applications, where both flash point temperatures and equilibrium data need to be accurately predicted to ensure safety and efficiency in processes involving ethanol-biodiesel mixtures. The findings of this study highlight the robustness of the GCM model in particular, which not only outperforms in flash point predictions but also maintains high accuracy in equilibrium data estimation. Therefore, the GCM model provides a comprehensive solution for modeling such non-ideal mixtures, making it a valuable tool for researchers and industry professionals alike.

4 Conclusion

This study employed a highly accurate method to predict the flash point of a biodiesel-ethanol mixture. Given the inherent complexity of biodiesel, comprising a blend of diverse methyl esters, it represents a multifaceted, multi-component system. Consequently, treating biodiesel as a pseudo-component was pivotal in estimating the flash point of this intricate mixture. The activity coefficients were determined via various methodologies, including UNIQUAC, NRTL, and the group contribution method. Adjustable coefficients within the NRTL and UNIQUAC equations, alongside the optimized values of group contributions for each constituent of biodiesel and ethanol, were meticulously ascertained through genetic algorithms. These coefficients were fine-tuned using vapor-liquid equilibrium temperature and pressure data for a mixture consisting of ethanol and biodiesel, featuring a 10 % weight fraction of ethanol. The overall accuracy of each method in flash point prediction was gauged by optimizing coefficients and applying the Liaw equation to predict flash points. The results unequivocally demonstrate that, among the four methods evaluated - Ideal Solution, UNIQUAC, NRTL, and GCM - the GCM outperformed the rest, exhibiting significantly higher accuracy. The absolute error for flash point estimation across these four methods (Ideal Solution, UNIQUAC, NRTL, and GCM) stood at 1.77 K, 1.75 K, 1.73 K, and 1.72 K, respectively. In light of these findings, the GCM emerges as a compelling choice for accurately predicting the flash point of multi-component, miscible mixtures. This research serves as a valuable contribution to the field, offering a robust approach for flash point estimation in complex systems, such as biodiesel-ethanol blends.

Data availability statement

Data will be made available on request.

CRediT authorship contribution statement

Esmaeil Bahrani: Writing – review & editing, Writing – original draft, Validation, Software, Resources, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. Mohammad Saleh Shafeeyan: Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. Morteza Banihashemi: Writing – review & editing, Writing – original draft, Validation, Resources, Project administration, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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