**1. Introduction**

The energy shortage and environmental threats caused by greenhouse gas emissions have promoted the demand for renewable energy. One of the most promising alternatives is biodiesel, which is known as a biomass-derived fuel produced from the transesterification of triglycerides [1,2]. Biodiesel is a biodegradable, renewable, and green fuel and has superior combustion efficiency [3,4]. Biodiesel use can result in lower hydrocarbon, CO, CO2, and particulate matter emissions compared with petroleum use [5,6]. Therefore, biodiesel has increasingly been used as a petroleum substitute.

Biodiesel is mainly derived from food oils [7,8], but the biodiesel produced is expensive due to the high cost of these feedstocks. To address this problem, waste and nonedible materials have been proposed as alternatives for producing biodiesel [9–11]. These oils can reduce biodiesel production cost, and they have no competition with the food supply [12,13]. Nevertheless, waste and nonedible oils contain high levels of free fatty acid, which must be reduced via esterification prior to transesterification [14,15]. Commonly, fatty acid is esterified using a liquid acid-catalyzed process [12,16]. Although this method effectively converts fatty acid into biodiesel, the downstream process for catalyst removal is difficult [17]. Notably, liquid acid catalysts cause environmental pollution and corrosive damage to equipment, raising concerns about their use [7]. To overcome the obstacles of liquid acid catalyst use, different solid acid catalysts have been proposed for the esterification such as Propyl-SO3H-SBA-15 [18], Amberlyst-36 [18], WO3-USY zeolite [19], HZSM-5 [20], Amberlyst-15 [21], biocatalysts [21], and ZrO2-TiO2 nanorods [22]; however, these solid catalysts exhibit low stability and catalytic activity, thus requiring high catalyst quantities and long reaction times, and they result in a low conversion yield [19,20]. In addition, these catalyst residues can cause a negative impact on the environment. Consequently, the applications of those solid catalysts in the esterification reaction are still limited.

With the growing awareness regarding environmentalism, the efforts have been undertaken to develop green methods for chemical production. For several decades, the enzymatic process has been developed as a substitute for chemically catalyzed esterification for biodiesel synthesis [23,24]. Enzyme-catalyzed esterification is superior to conventional processes in terms of the mild reaction conditions, reduced environmental impact, and selectivity of enzymes toward the substrate [24,25]. However, the high price of enzymes restricts its applications [24,26]. On the other hand, the noncatalytic esterification is considered another eco-friendly process for biodiesel synthesis. This process induced the esterification of fatty acids under thermal conditions without a catalyst [27], eliminating the problems associated with both chemical and enzymatic processes. The supercritical alcohol method has been applied in noncatalytic fatty acid esterification to prepare biodiesel [28,29]. Although this process produces a high yield of esters (up to 97%) within a short reaction time (10–30 min), it proceeds at a high temperature (280–320 ◦C) and high pressure (10–25 MPa) and consequently requires expensive reactors and extensive safety measures [28,29]. To overcome this, Cho et al. [30] proposed another form of noncatalytic esterification for less-than-supercritical conditions to produce biodiesel. Although their process proceeded at moderate pressure (0.85–1.2 MPa), it still required a high temperature (290 ◦C) for the reaction [30]. Therefore, another efficient esterification method is required for biodiesel production.

Microwave irradiation has been successfully applied in chemical reactions to enhance the reaction rate [31,32]. This method employs microwaves, which are electromagnetic waves, to transfer energy to the reactants [33]. Through relaxation or resonance generated by microwaves, reactant molecules are induced to vibrate at extremely high frequencies, resulting in fast molecular mobility of reaction species [34,35]. Consequently, microwave irradiation increases reaction rate and conversion yield whilst decreasing energy consumption and reaction time [33,36]. Furthermore, microwave-mediated processes are effective for large-scale operations [37]. Such advantages mean that the microwave approach has been extensively applied in catalytic transesterification and esterification for producing biodiesel [38–40]. Recently, our previous studies have demonstrated the potential use of microwave-mediated noncatalytic/autocatalytic synthesis of fatty acid [41], phytosterol esters [42], and ethyl levulinate [43]. However, its use in noncatalytic esterification to produce biodiesel remains limited.

This study proposed noncatalytic esterification of fatty acid with ethanol using microwave irradiation as an environmentally friendly and energy-efficient method to produce ethyl biodiesel. The esterification was performed under selected microwave powers to enhance the reaction conversion. Ethanol (95%) was used as an acyl acceptor instead of methanol for biodiesel synthesis because ethyl biodiesel is superior to methyl biodiesel due to its oxidation stability, calorific value, cetane number, and cold flow properties [44,45]. Oleic acid was chosen as a model molecule for the esterification, because it is a predominant fatty acid presented in animal fats and vegetable oils [46]. The reaction parameter effect on the reaction conversion was investigated. Furthermore, a mathematical model was proposed for representing the kinetics of noncatalytic esterification reactions.

#### **2. Materials and Methods**

#### *2.1. Chemicals*

Ethanol (95%), oleic acid (98%), and other reagents were obtained from Tokyo Chemical Industry (Tokyo, Japan).

#### *2.2. E*ff*ects of Di*ff*erent Heating Processes*

The effect of two heating approaches (conventional heating and microwave irradiation) on the noncatalytic esterification of oleic acid with ethanol was examined. For microwave-assisted esterification, 10-mL glass reactor containing the reaction solution (5 mL, 2:1 molar ratio of ethanol to oleic acid) was sealed and placed in a CEM 908005 microwave oven (Matthews, NC, USA). The microwave was equipped with a gas cooling system to maintain the temperature at a desired level. The reaction was then carried out at a microwave power of 150 W, 433 K, and different reaction times (60–360 min) with stirring. The reaction using conventional heating was undertaken in a 10 mL-sealed stainless-steel reactor placed in an oil bath under the same reaction conditions: 2:1 molar ratio of ethanol to oleic acid, 433 K, and different reaction times (60–360 min). After the reaction was completed, the sample was withdrawn to determine the oleic acid conversion. Each experiment was independently performed in triplicate for each reaction time.

#### *2.3. Analysis*

The amount of ethyl oleate synthesized was quantified using a Gas Chromatograph system (GC-2014, Shimadzu, Japan) equipped with a flame ionization detector and Stabilwax column (60 m × 0.25 mm id, 0.25 μm film thickness; Restek, Bellefonte, PA, USA) [1]. Nitrogen was used as the carrier gas and set at 1.0 mL/min. The temperatures of detector and injector were maintained at 250 and 220 ◦C, respectively. The column temperature was held at 140 ◦C for 5 min, increased to 240 ◦C with a ramp rate of 4 ◦C/min, and maintained for 15 min. Ethyl oleate standard (Sigma-Aldrich, Louis, MO, USA) was used to identify and determine the amount the produced ethyl oleate. One mol of oleic acid could stoichiometrically produce 1 mol of ethyl oleate; the oleic acid conversion (*X*) was consequently calculated as follows:

$$\begin{array}{l} \text{Olecic acid conversion}, X & = \text{Ethyl oleate conversion} \\ & = \frac{\text{Amount of oleic acid reacted}}{\text{Initial amount of oleic acid}} \\ & = \frac{282.47 \times \text{amount of ethyl oleate produced}}{310.51 \times \text{initial amount of colic acid}} \end{array} \tag{1}$$

#### *2.4. Kinetics of Noncatalytic Esterification Using Microwave Irradiation*

Noncatalytic esterification of oleic acid (*A*) with ethanol (*B*) to produce ethyl oleate (*C*) and water (*D*) is demonstrated as follows:

$$aA + bB \overset{k\_1}{\underset{k\_2}{\rightleftharpoons}} cC + dD$$

The model established for depicting esterification is considered elementary and reversible. The rate law is therefore as follows:

$$-\frac{d\mathbf{C}\_A}{dt} = k\_1 \mathbf{C}\_A^a \mathbf{C}\_B^b - k\_2 \mathbf{C}\_C^c \mathbf{C}\_D^d \tag{2}$$

where *k*<sup>1</sup> and *k*<sup>2</sup> denote forward and reverse reaction rate constants (L mol−<sup>1</sup> min<sup>−</sup>1), respectively; *CA*, *CB*, *CC*, and *CD* are concentrations of oleic acid, ethanol, ethyl oleate, and water, respectively; *a*, *b*, *c*, and *d* are the reaction orders of involved species. The reaction is assumed to follow the second-order kinetics (*c* = *d* = *a* = *b* = 1), the rate law is shown in terms of oleic acid conversion (*X*) as follows Equation (3):

$$\begin{split} \frac{d\mathbf{X}}{dt} &= \frac{1}{\mathbb{C}\_{A0}} (k\_1 \mathbb{C}\_A \mathbb{C}\_B \quad -k\_2 \mathbb{C}\_C \mathbb{C}\_D) \\ &= \frac{1}{\mathbb{C}\_{A0}} \Big[ k\_1 \mathbb{C}\_{A0} (1 - X) \mathbb{C}\_{A0} (\theta\_B - X) - \frac{k\_1}{\mathbb{K}\_q} \mathbb{C}\_{A0} X \mathbb{C}\_{A0} (\theta\_D + X) \Big] \\ &= k\_1 \mathbb{C}\_{A0} \Big[ (1 - X)(\theta\_B - X) - \frac{X(\theta\_D + X)}{\mathbb{K}\_q} \Big] \end{split} \tag{3}$$

where *CA*<sup>0</sup> is the initial oleic acid concentration; θ*<sup>B</sup>* and θ*<sup>D</sup>* are the initial ethanol:oleic acid molar ratio and the initial molar ratio of water to oleic acid, respectively; and *Ke* is the equilibrium rate constant. The reaction rate (*dX*/*dt*) will be zero at equilibrium, and *Ke* is then calculated as follows:

$$K\_{\mathfrak{c}} = \frac{k\_1}{k\_2} = \frac{X\_{\mathfrak{c}}(\theta\_D + X\_{\mathfrak{c}})}{(1 - X\_{\mathfrak{c}})(\theta\_B - X\_{\mathfrak{c}})} \tag{4}$$

where *Xe* is the equilibrium conversion of oleic acid. After determination of *Ke*, Equation (3) can be integrated into Equation (5) using the derivation described by Su [47]. Equation (5) can subsequently be used for determining the rate constant *k*<sup>1</sup> by linearly plotting ln ⎡ ⎢⎢⎢⎢⎢⎣ <sup>−</sup>1−θ*B*−θ*<sup>D</sup> Ke* +α<sup>2</sup> *X*+2θ*<sup>B</sup>* <sup>−</sup>1−θ*B*−θ*<sup>D</sup> Ke* <sup>−</sup>α<sup>2</sup> *X*+2θ*<sup>B</sup>* ⎤ ⎥⎥⎥⎥⎥⎦ versus <sup>α</sup>2*CA*0*t*.

$$\ln\left[\frac{\left(-1-\Theta\_{\mathcal{B}}-\frac{\mathcal{O}\_{\mathcal{D}}}{K\_{\mathrm{r}}}+\alpha\_{2}\right)\mathcal{X}+2\mathcal{O}\_{\mathcal{B}}}{\left(-1-\Theta\_{\mathcal{B}}-\frac{\mathcal{O}\_{\mathcal{D}}}{K\_{\mathrm{r}}}-\alpha\_{2}\right)\mathcal{X}+2\mathcal{O}\_{\mathcal{B}}}\right]=\alpha\_{2}\mathcal{C}\_{A0}k\_{1}t\tag{5}$$

$$\text{where } \alpha\_2 = \left[ \left( 1 + \theta\_B + \frac{\theta\_D}{K\_t} \right)^2 - 4\alpha\_1 \theta\_B \right]^{\frac{1}{2}} \tag{6}$$

$$\text{and } \alpha\_1 = 1 - \frac{1}{K\_{\mathfrak{e}}} \tag{7}$$

The relationship between temperature and the rate constants are represented by the Arrhenius equation:

$$k\_1 = A\_1 \exp\left(-\frac{E\_a}{RT}\right) \tag{8}$$

$$K\_{\varepsilon} = A\_{\varepsilon} \exp\left(-\frac{\Delta h}{RT}\right) \tag{9}$$

where *Ae* and *A*<sup>1</sup> (L mol−<sup>1</sup> min<sup>−</sup>1), respectively, denote pre-exponential factors for the equilibrium and forward rate constants; Δ*h* (J mol<sup>−</sup>1) and *Ea* (J mol<sup>−</sup>1), respectively, represent the molar reaction heat and activation energy of forward reaction; *T* (K) and *R* (J mol−<sup>1</sup> K<sup>−</sup>1) are the reaction temperature and the ideal gas constant, respectively. These parameters (*A*1, *Ae*, *Ea*, and Δ*h*) can be obtained from the Arrhenius–Van't Hoff plot [Equations (10) and (11)]:

$$
\ln k\_1 = \ln A\_1 - \frac{E\_a}{RT} \tag{10}
$$

$$
\ln K\_t = \ln A\_t - \frac{\Delta h}{RT} \tag{11}
$$
