Biodiesel Preparation without a Cosolvent in an Opposite-Side Micro-Fixed-Bed Reactor

Abstract

In the process of preparing biodiesel using microreactors, cosolvents and homogeneous catalysts are generally used, which adds to the separation cost. Hence, this study explored the use of an opposite-side feed micro-fixed-bed reactor with an inner diameter of 2 mm to produce biodiesel by the transesterification of soybean oil and methanol without any cosolvent, in which a fixed solid base acted as catalyst and obstacles. A biodiesel yield of 99.4% was achieved when the methanol-to-oil molar ratio was 12, the temperature was 70 °C, and the residence time was 8.05 min. A mixing study was conducted for this microreactor through computational fluid dynamics (CFD) simulations. The results showed that the best mixing index could reach 0.94 within 25 mm (length). Then, the transesterification kinetics in this reactor were found to be consistent with the proposed pseudo-homogeneous secondary reaction model. With the kinetics data and mixed-model, the biodiesel yield could be predicted. The simulated results were consistent with the experimental data.

1. Introduction

Biodiesel, which is based on sustainable biomass, is a typical non-fossil “green energy” [1]. The majority of biodiesel is produced by the transesterification of animal fats or vegetable oils with methanol or ethanol in the presence of homogeneous or heterogeneous catalysts in different reactors. Both intermittent and continuous processes can be used to produce biodiesel. The traditional method of producing biodiesel is an intermittent process, although its input is low and the production process is simple; long reaction times and the high energy consumption in the production process seriously affect its carbon reduction effect. A continuous process is more suitable because of its significantly lower energy consumption [2].

The continuous process would improve the environmental performance of producing biodiesel where the reaction equipment is an integral part of this improvement. Fixed-bed reactors [3], membrane reactors [4], tube reactors [5], and microreactors [6] have been investigated for application in biodiesel production. In recent years, microreactors have gradually become a research hotspot for the continuous preparation of biodiesel. Babak et al. [7] used graphene-oxide-doped magnesium oxide (GO@MgO) nanocatalysts in a 0.8 mm diameter microreactor for the transesterification of waste edible oils and fats and methanol: a maximum biodiesel yield of 99.2% was achieved. Gholami et al. [8] carried out the transesterification reaction of Norouzak oil and methanol in a microreactor using ionic liquid choline hydroxide as a catalyst: the biodiesel yield reached 96.8% at a methanol/oil volume ratio of 0.4, a catalyst concentration of 6.1 wt.%, and a reaction time of 12.5 min. The reaction time and energy consumption of this microreactor-based process were only 0.1 times and 0.03 times greater than those in the conventional mechanical stirrer process. Mohadesi et al. [9] studied the transesterification reaction of used cooking oil in a microreactor with calcined beef bone as the catalyst. Under optimum conditions, the yield of biodiesel could reach 99.2% when using a catalyst concentration of 8.5 wt.%, a methanol/oil volume ratio of 0.4, a temperature of 63.1 °C, and a reaction time of less than 1 min. Laboratory-scale microreactors can reduce operational requirements while obtaining high-quality biodiesel products [10]. The mobility in the microreactor was generally considered as laminar flow, where molecule diffusion is the main mass transfer mechanism [11]. However, at the scale of widely used microreactors for biodiesel production, the disadvantage of the diffusion between two phases would greatly affect the biodiesel yield. Thus, how to reduce the barrier of mass transfer between phases is key to directly improving the yield of biodiesel. Adding cosolvent is the most common method to solve the mass transfer resistance between liquid and liquid phases. Chueluecha et al. [12] used isopropanol as cosolvent of palm oil and methanol, the yield of biodiesel could reach 98.3% under the optimum conditions. Mohadesi et al. [13] used acetone as a cosolvent, the biodiesel yield reached 97.2%. Hassan et al. [14] added supercritical CO₂ as a co-solvent; the highest biodiesel yield (98.1%) was obtained after 20 min at 240 °C and 12 MPa. Cosolvent in the reaction system might result in a homogeneous reaction without interphase mass transfer, which can improve the reaction rate and obtain a higher biodiesel yield in a relatively shorter time. At the same time, cosolvent can also reduce the viscosity of the reaction system, which can lower the pressure drop during the flow process. However, a cosolvent will complicate the subsequent biodiesel separation and purification process, resulting in considerable energy consumption and reducing the green credentials of the biodiesel production process.

In addition to adding cosolvent, another popular method is to mix two liquids more evenly, which changes the flow patterns of oil and methanol. Saavedra et al. [15] used conventional alkaline and ultrasound-assisted transesterification to improve the mixing of jatropha and palm oil. The results showed that the jatropha oil optimized the fluid mixing of biodiesel. Additionally, ultrasound-assisted technology largely shortened the transesterification time for a gratifying biodiesel yield. However, this method was not ideal for large-scale production. Some researchers have designed reactors with extra mixing cells or different geometric structures to enhance the mixing of fluids [16]. Yu Hongpeng et al. [17] devised a micromixer for expansion–contraction channels and conducted CFD simulations to study the mixing behavior of fluids. They found that when the velocity ratio (ratio of vibration amplitude to initial velocity) was 2, the phase difference was 180°, and the St number was 0.2~0.4, the mixing index could reach 0.04 (depending on the mix indicator used; 0 meant complete mixing). Baydir et al. [18] studied the effect of T-type mixing cells with different inner diameters in continuous biodiesel production; 99.8% oil conversion was obtained when the inner diameter was 0.8 mm and the residence time was 120 s with KOH as the catalyst. The size of the microreactor has an obvious effect on the reaction efficiency. Xu et al. [19] found that when the inner diameter of the microreactor was increased from 0.25 mm to 0.53 mm, the yield decreased from 99% to 95% with the same reaction time. One reason could be that when the inner diameter was below a certain value, the diffusion path was small, leading to a significant improvement in the mass transfer effect. However, a narrower channel usually causes a greater pressure decrease as the viscosity of the fluid system is high. In order to improve the mixing behavior in larger channel microreactors, many researchers have added mixed elements [20] or designed microreactors with special structures such as Zigzag-type [21] or the so-called “Z” type [22], to improve the mixing degree of two phases and enhance the mass transfer between them. Santana et al. [23,24,25] studied the effects of circular obstacles and static baffle elements in micromixers or microchannels. Catalyzed by KOH or NaOH, a better biodiesel yield was obtained than from common microchannel reactors. Then, an “Elis”-shaped microreactor with circular obstacles inside was designed [6]. The structure achieved high efficiency on the micro scale, and the oil conversion could reach 91.3%. A stable slug and high specific surface area could be formed by internal components of the reactor [26]. Abdulla Yusuf [27] compared the performance of T-shaped, Y-shaped, and Tesla-shaped microreactors. Catalyzed by NaOH, the best biodiesel yield reached 97.9% with the Tesla-shaped microreactor when the methanol-to-oil molar ratio was 9 and the reaction time was only 4.9 s. Although internal obstacles in microchannels or special structures can significantly increase the biodiesel yield, they place high demands on the manufacturing technology.

In this study, an opposite-side feed micro-fixed-bed reactor (OS-MFBR), in which catalyst particles acted as internal obstacles, was applied to the transesterification of soybean oil and methanol without a cosolvent. The flow behavior inside the OS-MFBR was simulated using commercial ANSYS Fluent software with the mixing index as an evaluation criterion. In addition, the kinetics of the transesterification process were studied.

2. Experimental Section

Materials and Methods

Materials. The physicochemical properties of the raw materials are summarized in

Table 1. Physicochemical properties of raw materials (25 °C).

	Molecular Weight (g mol−1)	Density (kg m−3)	Dynamic Viscosity (kg (m·s)−1)	Interfacial Tension (mN m−1)
Soybean oil	750	909.15	0.06910	0.015
Methanol	32	781.80	0.00058

. The raw material for transesterification was soybean oil purchased from a local market. The saponification value was 219.6 mg KOH/g, and the acid value was 0.15 mg KOH/g. The fatty acid composition was as follows: methyl palmitate, 11.5 wt.%; methyl stearate, 5.8 wt.%; methyl oleate, 21.5 wt.%; methyl linoleate, 60.8 wt.%; methyl linolenate, 2.5 wt.%. Methanol was purchased from Sinopharm Chemical Reagent Co., Ltd., Shanghai, China (PRC). The catalyst was KF/Ca-Mg-Al LDHs, produced by the research group, and had high catalytic activity and good reuse effect.

Geometric design of the OS-MFBR. The OS-MFBR structure is shown in the diagram in Figure 1. The diameter of the two inlets (L_w) was 2 mm, the length of the inlet (L_m) was 5 mm, the inner diameter of the reactor channel (W_m) was 2 mm, and the length of the reactor section (L₁) was 300~550 mm. The diameter of the catalyst sphere (2D) was 0.63 mm. The void ratio was 0.76 and the spacing void, L₂, was 0.61 mm.

Experimental design. KF/Ca-Mg-Al LDHs (Ca-Mg-Al layered double hydrotalcites loaded with KF) were prepared following the methods in previous studies [28]. The obtained catalyst powder was shaped, crushed, and sieved. Particles with an average diameter of 0.63 mm (void ratio of 0.76) were chosen to fix into the tube to obtain the designed OS-MFBR. The entire reaction process also included two constant-current pumps to control the flow of reaction materials, and a heating oven to place the OS-MFBR. The transesterification process took place as shown in Figure 1. Soybean oil and methanol were initially pumped in at room temperature, through a heated pipeline section, and then entered the OS-MFBR from inlet 1 and inlet 2, respectively. The pipeline and the OS-MFBR were heated to a specific temperature before the experiment began. In this system, the reaction only took place in the part of the system filled with catalyst particles.

The operation parameters as methanol/soybean oil molar ratio (6~24), temperature (65~100 °C), reactor length (300~550 mm) and residence time (2~8 min) were studied.

Table 2. Volumetric flow rates and reactor lengths of soybean oil and methanol at different molar ratios (t = 8.04 min).

Methanol/Soybean Oil Molar Ratio	Soybean Oil Vol. Flow Rate (mL h−1)	Methanol Vol. Flow Rate (mL h−1)	Reactor Length (mm)
6:1	6	1.68	329
10:1	6	2.76	374
12:1	6	3.36	400
15:1	6	4.20	436
24:1	6	6.72	544

shows the flow rates and reactor lengths at different molar ratios. The residence time was controlled by adjustment of the flow rates of two pumps, while the molar ratio of methanol/soybean oil was determined by the flow ratio of the two pumps. After the reaction, the liquid was condensed and collected in the product tank. The supernatant after centrifugation was diluted and then analyzed by gas chromatography, and subsequently used to calculate the biodiesel yield.

Product Analytical Method. The FAME content of the sample was determined by FULI model 9790Ⅱgas chromatography GC (with FID and capillary column (DB-5 HT, 15 m × 0.2 mm × 0.1 μm)), following the method from previous studies [29]. Then, the biodiesel yield was calculated using the following formula.

$$\mathrm{Biodiesel} \mathrm{yield}=\frac{C_{FAME}\times M_{\mathsf{oil}}}{m_{oil}\times 3\times M_{FAME}}\approx \frac{C_{FAME}\times V}{m_{oil}}$$

(1)

where C_FAME is the FAME content tested by GC, V is the volume of the collected supernatant, m_oil is the oil added in the system, M_oil is the average molecular weight of oil, and M_FAME is the average molecular weight of FAME, which is approximately one-third of that of oil.

3. Results and Discussion

Transesterification in the OS-MFBR

The tube-based reactor could easily achieve pressure higher than atmospheric values. In order to maintain a relative rapid reaction, temperatures of 65 °C [29,30] or higher (70~140 °C) [31,32,33] were studied in the continuous transesterification process in the tube. A sufficient residence time (reaction time) ensures a high conversion. In traditional tubular reactors of membrane reactors, transesterification usually needs 1~6 h to achieve a satisfactory conversion [29,31]. Microreactors are effective in reducing reaction times, which ranged from 1 to 15 min in previous studies [7,9,34]. The effect of residence time and temperature on biodiesel yield is shown in Figure 2; the biodiesel yield increases as the residence time increases. Transesterification usually takes place with a high methanol/soybean oil molar ratio; a longer reaction time leads to a higher oil conversion and biodiesel yield. This was consistent with some prior research results [30].

From Figure 2, it can also be seen that when the temperature rises from 65 °C to 70 °C, the yield increases and reaches a maximum of 99.4%. The reason for this is that an increase in temperature raises the miscibility of the two phases, which enhances their contact area and mass transfer rate. Although some studies have also shown that there is a positive correlation between temperature and reaction efficiency [31], in this study, the biodiesel yield with a longer residence time (8.05 min) was almost unchanged, and even slightly dropped when the temperature was above 70 °C. This is because a high temperature caused methanol in the reactor to vaporize, forming a gas–liquid mixture. The whole system became a gas–liquid–liquid triple phase, and the mass transfer was affected. The high temperature also accelerated side reactions such as saponification, which ultimately led to lower biodiesel yields. Although this effect was not obvious in the tubular reactor, too high a temperature may still lead to the rapid vaporization of methanol, resulting in increased reaction pressure, which might increase the mass transfer resistance and shorten the catalyst life.

The theoretical molar ratio of methanol/soybean oil for transesterification is 3:1. However, as this is a reversible reaction and the viscosity of the system is high, excess methanol is usually added to promote the reaction in the positive direction and to reduce the viscosity of the system. Although a high methanol/soybean oil molar ratio will lead to a high biodiesel yield, it also might increase the cost. For this reason, an appropriate molar ratio is vital in the production of biodiesel. The effects of the molar ratio were investigated; the results are shown in Figure 3. A higher biodiesel yield (99.4%) could be obtained when the molar ratio was 12. When the methanol/soybean oil molar ratio was above 12, the excess methanol promoted the reaction to move in the positive direction, which improved the biodiesel yield. When the molar ratio exceeded 12, the yield decreased; the concentration of soybean oil may have been diluted by the excess methanol. This reduced the chance of oil reaction molecules colliding with the catalyst sites. At the same time, excess methanol increased the dissolution of glycerol and weakened the separation of the glycerol phase [24], this prevented the equilibrium from shifting in a positive direction and caused the biodiesel yield to decrease. This was confirmed by the fact that glycerol was still detectable in the isolated samples. Therefore, a methanol/soybean oil molar ratio of 12 in the OS-MFBR is the best choice for biodiesel yield enhancement, which is clearly lower than previous research [32,33].

4. Mixing Study of Soybean Oil and Methanol in the OS-MFBR

4.1. Foundation of the Simulation Model

The mixing behaviors of soybean oil/methanol in the OS-MFBR and OS-MR (opposite-side feed microreactor) were analyzed in the case of L₁ = 25 mm. The transesterification of methanol and oil in the fixed bed was a liquid–liquid biphasic reaction in which the blending of two phases had a significant effect on biodiesel yield. The momentum transfer mechanism controlled the interfacial force. In the liquid–liquid system, the drag force was dominant and the Ishii–Zuber model was added [34].

Fluent Meshing software was used for grid division; there were 1.39 million grids for the OS-MFBR and 1.29 million grids for the OS-MR. The feasibility of the model was verified through the pressure drop experiment, as shown in Figure 4. The results show that the numerical simulation agreed with the experiment and the error was small, proving that the model was reasonable.

ANSYS Fluent 2019 R1 was used to simulate the mixing situation in the two models. The calculation involved the Navier–Stokes equation (Equation (1)), the mass balances of species based on the finite volume element method (Equation (2)), and continuity (Equation (3)), as follows:

$$\rho \left(U\cdot \nabla U\right)=-\nabla p+\mu {\nabla }^{2}U+\rho g$$

(2)

$$\rho \left(U\cdot \nabla Y_{\mathrm{i}}\right)=\rho D_{\mathrm{i}}{\nabla }^2{Y}_{\mathrm{i}}+S_{\mathrm{i}}$$

(3)

$$\nabla \cdot U=0$$

(4)

where ρ is the specific mass, g is the acceleration of gravity, μ is the dynamic viscosity, U is the velocity vector, p is the pressure, Y is the mass fraction chemical species, S is the mass source given by (4), and D_i is the molecular diffusion coefficient.

$$S_{\mathrm{i}}=\left({\displaystyle \sum _{r_{\mathrm{r}}}^n{v_{\mathrm{r}}}^{″}{r}_{\mathrm{r}}-{\displaystyle \sum _{r_{\mathrm{r}}}^n{v_{\mathrm{r}}}^{\prime }{r}_{\mathrm{r}}}}\right)M_{\mathrm{Wi}}$$

(5)

${v_r}^{\prime }$ and ${v_r}^{″}$ are the stoichiometric coefficients of the chemical species as reactants or as products; $r_r$ is the rate of reaction r.

Most flows in the OS-MR were laminar flow; a laminar model and a two-phase mixture model were selected, in which oil was defined as the first phase and methanol was the second phase. The boundary conditions were as follows:

Inlet 1: Oil phase (A) inlet and the velocity inlet. The velocity inlet and mass fraction were:

u = U_A; Y_A = 1; Y_G = 0

Inlet 2: Methanol phase (B) inlet and the velocity inlet. The velocity inlet and mass fraction were:

u = U_B; Y_B = 1; Y_G = 0

Outlet: Pressure outlet; the average pressure at the outlet was 0.

Wall surface: No-slip.

The under-relaxation factors were adjusted in order to achieve better convergence: pressure, 0.2; density, 0.7; body forces, 0.5; momentum, 0.4; slip velocity, 0.1; volume fraction, 0.3.

The dependence of the mixing effect on the Reynolds number was divided into three different stages [35]. The mixing quality was also improved to varying degrees through the development of different stages. The instability of the flow would be greatly increased if the Reynolds number was greater than a certain value, which would lead to an improvement in the performance of the mix.

The calculation of Reynolds number is as follows:

$$\mathrm{Re}=\frac{du\rho }{\mu }$$

(6)

where d is the pipe diameter, u is the fluid flow rate, ρ is the density of the fluid, and μ is the dynamic viscosity.

The apparent viscosity of the two-phase mixture was obtained using the following formula:

$$\mu ={\mu }_{\mathrm{c}}\left[1+2.5{\mathsf{\Phi }}_{\mathrm{d}}(\frac{{\mu }_{\mathrm{d}}+\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.{\mu }_{\mathrm{c}}}{{\mu }_{\mathrm{c}}+{\mu }_{\mathrm{d}}})\right]$$

(7)

The apparent fluid density of two phases was obtained using the following formula [36]:

$$\rho ={\mathsf{\Phi }}_{\mathrm{d}}{\rho }_{\mathrm{d}}+(1-{\mathsf{\Phi }}_{\mathrm{d}}){\rho }_{\mathrm{c}}$$

(8)

where ${\Phi }_d$ is the volume fraction of the dispersed phase, ρ_d and ρ_c are the density of the dispersed phase and continuous phase, respectively, µ_c and µ_d are the dynamic viscosity of the dispersed phase and continuous phase, respectively.

The statistical method of standard deviation was used to calculate the component mass fraction at each section. After data processing, the standard deviation value of each section node sample was obtained with the following standard deviation expression:

$$\sigma =\sqrt{\frac{{\displaystyle \sum {\left(Y_{\mathrm{n}}-\stackrel{\_}{Y}\right)}^2}}{N}}$$

(9)

The mixing index, M, is defined as [37]:

$$M=1-\sqrt{\frac{{\sigma }^2}{{\sigma }_{\max }{}^2}}.$$

(10)

where $Y_n$ is the mass fraction at the sampling point, $\stackrel{-}{Y}$ is the average mass fraction of the sampling cross-section, N is the number of sampling points, σ is the standard deviation, σ² is the standard deviation of the statistical sample, and ${\mathsf{\sigma }}_{\max }^2$ is the maximum standard deviation (the best mixing is the maximum value; the worst mixing is the minimum value). Therefore, the closer the M value is to 1, the better distributed the two phases are.

4.2. Mixing Analysis with Two Reactors

The mixing index was obtained using Equations (6) and (7) above, as shown in Figure 5. Figure 5 shows that the OS-MFBR greatly improved the mixing situation, among which the mixing index could reach 0.94 with Re = 171. Although the increase in the flow rate can promote mixing, the effect is very small if there are no obstacles. Therefore, the mixing index in the OS-MR was kept at a relatively low level. This is not advantageous for transesterification.

Figure 6 shows the oil volume fraction distribution in the two mixers when Re = 1. From Figure 6a, the interface between methanol and oil is clear when there are no obstacles. Additionally, with obstacles, at the outlet of the mixer, the mixing situation between oil and methanol was much better (Figure 6b).

The oil volume fraction distributions for the OS-MFBR at different Re values are shown in Figure 7. There are two main ways to improve mixing: fluid splitting and recombination, and then vortex formation. The existence of obstacles breaks the parallel flow state of the initial fluid at Re = 53 (Figure 7a), but not sufficiently to promote fluid mixing. With the increase in the Reynolds number, the vortex formation is more developed and favors mixing (Figure 7b,c). The velocity and direction of the fluid changed sharply when it collides at the front of the particle. The compact structure increased the complexity of the fluid flow path, with large local variations (Figure 8a–c). Then, an effective flow could be formed in the original dead flow area (Figure 8d), increasing the surface contact area between the oil and methanol phase. As mentioned above, the original parallel fluids were cut and recombined to improve the diffusion process. Alam et al. [37] proved that the increase in the number of obstacles could effectively promote the mixing degree of ethanol and water, which was also applicable to the mixing of oil and methanol in this study.

5. Kinetics Model of Transesterification Process in the OS-MFBR

From Section 4, it can be seen that at 25 mm from the feed inlet, oil and methanol almost uniformly mixed. Therefore, a pseudo-homogeneous second-order reaction model was used to describe the transesterification process in the OS-MFBR.

The transesterification of triglycerides was a serial reaction involving three steps, as follows:

Triglyceride (TG) + Methanol = FAME + Diglyceride (DG)

(11)

Diglyceride (DG) + Methanol = FAME + Monoglyceride (MG)

(12)

Monoglyceride (MG) + Methanol = FAME + Glycerol (G)

(13)

The proportion of DG and MG in the system was relatively low when excess methanol was added. Thus, the reaction can be simplified as (14):

Triglyceride (TG) + 3 Methanol = 3 FAME + Glycerol (G)

(14)

Defining TG as A, methanol as B, FAME as C, and glycerol as D, the kinetic equation can be derived as:

$$r_A=K_{\mathsf{\theta }}(k_1{C}_{\mathrm{A}}{C}_{\mathrm{B}}-k_2{C}_{\mathrm{C}}{C}_{\mathrm{D}})$$

(15)

where k₁ is a positive reaction rate constant, k₂ is an inverse reaction rate constant, ${\mathit{K}}_{\mathsf{\theta }}$ is the effect of catalyst dosage on the rate of an intrinsic reaction, and C_A, C_B, C_C, and C_D are the concentrations of TG, methanol, FAME, and glycerol, respectively.

Assuming that the molar concentration of oil is C_A0 (mol/L) at the initial reaction, and the methanol to oil molar ratio is n, when the conversion rate reached x after t minutes, Equation (15) can be simplified as:

$$r_{\mathrm{A}}=K_{\mathsf{\theta }}{C}_{\mathrm{A}0}^2\left[k_1\left(1-x\right)\left(n-3x\right)-3k_2{x}^2\right]$$

(16)

There was an excess of catalyst in our system; therefore, ${\mathit{K}}_{\mathsf{\theta }}$ was considered as 1. In the transesterification process in a micro-fixed-bed reactor, the volume of the system does not vary, and the reaction rate can be defined as the change in oil concentration in unit residence time (t = $\frac{\mathrm{W}}{\rho \mathrm{F}}$, min, where W is the loading quality of the catalyst, g; ρ is the catalyst density, 0.029 g cm⁻³; F is the feed rate, mL min⁻¹; and x is the conversion rate at moment t). According to the simplified reaction, there were no side reactions at different temperature levels in this system; the biodiesel yield was equal to the oil conversion. Additionally, because the data obtained in the experiment were yields of biodiesel, in Equation (17), x refers to biodiesel yields.

$$r_{\mathrm{A}}=C_{\mathrm{A}0}\frac{dx}{dt}=C_{\mathrm{A}0}\frac{dx}{d\frac{W}{\rho F}}$$

(17)

Substituted (17) into (16), and ${\mathrm{K}}_{\mathsf{\theta }}$ = 1, the following can be derived:

$$r_{\mathrm{A}}=C_{\mathrm{A}0}\frac{dx}{d\frac{W}{\rho F}}=C_{\mathrm{A}0}^2\left[k_1\left(1-x\right)\left(n-3x\right)-3k_2{x}^2\right]$$

(18)

Namely,

$$\frac{dx}{d\frac{W}{\rho F}}=C_{\mathrm{A}0}\left[k_1\left(1-x\right)\left(n-3x\right)-3k_2{x}^2\right]$$

(19)

According to the model shown in Equation (19) and the data in Figure 2, ${\mathrm{k}}_1$ = 0.2396 and ${\mathrm{k}}_2$ = 0.0365 were estimated using the integral method and nonlinear regression.

Comparing the value of model and experiment, as shown in Figure 9, the difference between the two results was small; thus, it can be considered that this model is applicable and the kinetic model of the reaction at 70 °C can be obtained as follows:

$$r_{\mathrm{A}}=C_{\mathrm{A}0}^2\left[0.2396(1-x)\left(n-3x\right)-0.1095x^2\right]$$

(20)

Figure 9 indicates that the kinetic model was appropriate. Then, the kinetic data were introduced into the mixed-model presented in Section 4.1, and the yields of biodiesel in this reactor could have been predicted. The yields of biodiesel predicted by the simulation and obtained by experiment with different methanol/oil molar ratios are shown in Figure 10.

Figure 10 shows that the simulation results agreed with the experimental results; the absolute error was less than 5%, and the relative average error was 4.8%. Additionally, the trends in biodiesel yields were the same, and both obtained the maximum yield when the methanol/oil molar ratio was 12:1. However, the experimental results were lower than the simulation results. This is likely because the simulation data ignored the solubility of biodiesel in glycerol, which reduced the recovery of biodiesel compared with that which was generated by the transesterification.

6. Conclusions

In this study, an opposite-side feed micro-fixed-bed reactor with an inner diameter of 2 mm was used to produce biodiesel through the transesterification of soybean oil and methanol without any cosolvent, in which the fixed solid base was used as catalyst and obstacles. Without a cosolvent, when the molar ratio of methanol to oil was 12, temperature was 70 °C, residence time was 8.05 min, and in the presence of a solid base catalyst, the biodiesel yield was 99.4%. The commercial ANSYS Fluent was applied to analyze the fluid mixing behavior. The results showed that the mixing index could reach 0.94 (a value of 1 would indicate complete mixing) within 25 mm in length due to the twisted flow trajectories and increased collision points which were caused by the blocking of the catalyst. According to the results of the mixing investigation, a pseudo-homogeneous secondary reaction model was used to describe the transesterification kinetics in the OS-MFBR. Additionally, the experimental data fitted well with the model results. With the kinetic data and mixed model, the biodiesel yields in the OS-MFBR could be predicted; the simulation results were consistent with the experimental results. A continuous process is more energy-efficient than a batch process and produces lower carbon emissions; therefore, the OS-MFBR represents a simple way to produce biodiesel in a microreactor. Its applications could also be expanded to heterogeneous reactions at higher viscosity.